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Contents
1 Principles of Thermodynamics
1.1 Introduction . . . . . . . . . . . . . . .
1.2 State variables and differential forms
1.3 Equation of state . . . . . . . . . . . .
1.4 Zeroth law . . . . . . . . . . . . . . . .
1.5 Internal energy . . . . . . . . . . . . .
1.6 Work . . . . . . . . . . . . . . . . . . .
1.7 First law . . . . . . . . . . . . . . . . .
1.8 Second law . . . . . . . . . . . . . . . .
1.9 Carnot’s cycle . . . . . . . . . . . . . .
1.10 Third law . . . . . . . . . . . . . . . . .
1.11 Problems . . . . . . . . . . . . . . . . .
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1
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1
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2
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4
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6
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6
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6
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8
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9
. 10
. 12
. 12
2 Thermodynamic potentials
2.1 Fundamental equation . . . . . . . . . . . . . . .
2.2 Internal energy U and Maxwell relations . . . .
2.3 Enthalpy H . . . . . . . . . . . . . . . . . . . . . .
2.4 Free energy F . . . . . . . . . . . . . . . . . . . .
2.5 Gibbs function G . . . . . . . . . . . . . . . . . . .
2.6 Grand potential Ω . . . . . . . . . . . . . . . . . .
2.7 Thermodynamic responses . . . . . . . . . . . . .
2.8 Thermodynamic stability conditions . . . . . . .
2.9 Stability conditions of matter . . . . . . . . . . .
2.10 Thermodynamic potentials in electromagnetism
2.11 Problems . . . . . . . . . . . . . . . . . . . . . . .
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15
15
16
17
19
19
20
21
24
24
25
27
3 Applications of thermodynamics
3.1 Classic ideal gas . . . . . . . . .
3.2 Free expansion of gas . . . . . .
3.3 Mixing entropy . . . . . . . . .
3.4 Dilute solution, osmosis . . . .
3.5 Chemical reaction . . . . . . . .
3.6 Phase equilibrium . . . . . . . .
3.7 Phase transitions and diagrams
3.8 Coexistence . . . . . . . . . . . .
3.9 Van der Waals equation of state
3.10 Problems . . . . . . . . . . . . .
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29
29
31
32
33
36
39
40
42
44
47
v
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CONTENTS
vi
4 Classical phase space
4.1 Phase space and probability density .
4.2 Flow in phase space . . . . . . . . . . .
4.3 Microcanonical ensemble and entropy
4.4 Problems . . . . . . . . . . . . . . . . .
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49
49
52
54
57
5 Quantum-mechanical ensembles
5.1 Density operator and entropy . .
5.2 Density of states . . . . . . . . . .
5.3 Energy, entropy and temperature
5.4 Problems . . . . . . . . . . . . . .
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59
59
63
66
67
6 Equilibrium distributions
6.1 Canonical ensemble . . . . . . . . .
6.2 Grand canonical ensemble . . . . .
6.3 Connection with thermodynamics .
6.4 Thermodynamic fluctuation theory
6.5 Reversible minimal work. . . . . . .
6.6 Problems . . . . . . . . . . . . . . .
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69
69
75
77
78
82
83
7 Ideal equilibrium systems
7.1 Free spin system . . . . . . . . . .
7.2 Classical ideal gas . . . . . . . . .
7.3 Diatomic ideal gas . . . . . . . . .
7.4 Statistics of bosons and fermions
7.5 Problems . . . . . . . . . . . . . .
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86
86
91
97
101
107
8 Bosonic systems
8.1 Bose gas and Bose condensation .
8.2 Black body radiation . . . . . . .
8.3 Lattice vibrations . . . . . . . . .
8.4 Problems . . . . . . . . . . . . . .
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109
109
115
121
127
9 Fermionic systems
128
9.1 Conduction electrons in metals . . . . . . . . . . . . . . . . . . 128
9.2 Magnetism of degenerate electron gas . . . . . . . . . . . . . . 135
9.3 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
10 Phase transitions
10.1 Description of phase transitions
10.2 Landau theory . . . . . . . . . .
10.3 Ginzburg–Landau theory . . . .
10.4 Fluctuations in Landau theory
10.5 Problems . . . . . . . . . . . . .
Index
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144
144
147
151
157
160
162
1. Principles of Thermodynamics
1.1 Introduction
Thermodynamics gives a phenomenological and very general description
of matter – largely independent of models of microscopic structure (which
where practically nonexistent at the time of foundation of thermodynamics in the 19th century). It is based on very few basic laws plus rules of
calculus. Properties of matter or concrete systems are taken from outside
(experiment, statistical mechanics).
System. Macrophysical entity under consideration, may interact with
its environment. It is often homogeneous or consists of homogeneous
phases. The usual classification according to the possibility of exchange
of energy and matter between the two goes as follows: Open system: both
energy and matter may be exchanged. Closed system: particle number(s)
fixed, energy may be exchanged. Isolated system: no exchange of matter or
energy.
Thermodynamic equilibrium. State of matter without any macroscopic changes or flows. A genuine equilibrium state is unambiguously
determined by externally imposed state variables like pressure, volume,
electric and magnetic fields. There are no memory effects like hysteresis.
Traditionally, in the description of thermodynamic equilibrium there are
three different equilibria:
• mechanical equilibrium: no changes of form or other processes accompanied by production of macroscopic mechanical (or electromagnetic)
work;
• chemical equilibrium: no changes in the macroscopic chemical composition of the system;
• thermal equilibrium: no macroscopic energy flows in a system in mechanical and chemical equilibrium. In plain words: no heat flows.
In local thermodynamic equilibrium macroscopic subsets of the system
are in equilibrium, but in neighbouring subsystems the equilibria are different, so that the system is not in equilibrium as a whole. Currents, heat
flow etc. may occur; this is the realm of hydrodynamics. In most practically
important cases local equilibrium is reached in macroscopically short time.
State variables. State variables are parameters needed for characterization of the equilibrium state. Usually there is only a handful of them,
1
2
1. PRINCIPLES OF THERMODYNAMICS
in many cases like the prototypical one-component gas two is enough to
determine the equilibrium state, in which the rest are then functions of
these parameters, state functions. State variables are either extensive or
intensive, the former being proportional to the number of particles (the volume VR , particle number N , internal energy U , entropy S, magnetic moment
m = d3 r M (r) etc.), whereas the latter (the temperature T , the pressure
p, the chemical potential µ, magnetic field strength H) are independent of
the number or particles. In thermodynamic differential forms these variables appear as conjugate pairs of extensive and intensive variables.
For quantities like energy and entropy the extensiveness requires weakness of interaction energy (or correlations) between macroscopic subsystems of the original system in comparison with the "bulk" quantities prescribable to the subsystems themselves. Gravity might cause problems in
this respect at very large scales. Electromagnetic interaction is usually
screened in matter and thus of short range.
Process. A change of state is called a process in thermodynamics. In
a reversible process the direction may be inverted in the ”whole universe”
(system plus environment). These processes are always quasistatic, i.e. so
slow that the state of the system is infinitesimally close to thermodynamic
equilibrium. Not all quasistatic processes are reversible, however. An irreversible process is often a sudden or spontaneous change (e.g. mixing of
gases, explosion), during which the system may be far from equilibrium
and the description by the state variables is no sufficient. An irreversible
process may occur quasistatically, though. A cyclic process (cycle) consists
of repeating periods during which the system always returns to the initial
state.
1.2 State variables and differential forms
State variables are macroscopic quantities related to the equilibrium. Not
all of them are independent in equilibrium, though. Once independent
variables are chosen, the rest are unambiguous functions of them, say
p = p(T, V, N ), U = U (T, V, N ), S = S(T, V, N ) etc.
y
y
1
2
x
1=2
x
Figure 1–1: Examples of processes leading from state 1 to state 2.
When the change is infinitesimal, the rules of calculus yield the following relation between the differential of a function and the differentials of
1.2. STATE VARIABLES AND DIFFERENTIAL FORMS
3
indepent variables:
∂p
∂p
∂p
dp =
dT +
dV +
dN.
∂T V,N
∂V T,N
∂N V,T
This implies that in a cyclic process the net change vanishes:
I
I
dp =
dU = · · · = 0.
1→1
1→1
Differential and differential form. Consider the differential form
−
dF
≡ F1 (x, y) dx + F2 (x, y) dy,
(1.1)
where F1 and F2 are given functions. An example familiar from mechanics
is the differential form of work exerted by a force F on a body:
−
dW
= F · dr = Fx dx + Fy dy + Fz dz .
−
in (1.1) means that the differential form is not necessarily
The notation dF
R2 −
may depend on the integration path. If the
a differential, therefore 1 dF
−
= dF (x, y) is a differential (ofcondition ∂F1 /∂y = ∂F2 /∂x holds, then dF
ten referred to as the exact differential in this context). Then the integral
R2 −
dF = F (2) − F (1) is independent of the path and F1 (x, y) = ∂F (x, y)/∂x
1
ja F2 (x, y) = ∂F (x, y)/∂y are coordinates of the gradient of some function F .
−
= F1 dx + F2 dy is not a differential,
Integrating factor. If the form dF
in case of two variables a function integrating factor λ(x, y) may be found
such that, in a vicinity of the point (x, y) the condition
−
≡ λ F1 dx + λ F2 dy = df
λ dF
holds, which implies ∂(λF1 )/∂y = ∂(λF2 )/∂x. Both the integrating factor λ
and the function f are then state variables.
In case of three or more variables the integrating factor may not exist, in
general. In thermodynamics, however, the integrating factor of the differential form of heat always exists, this is partially the content of the second
law.
Legendre transform. Legendre transform generates changes of variables between conjugate variable pairs. Consider, for simplicity, the function f (x) and define the variable conjugate to x as
y≡
df (x)
.
dx
(1.2)
The Legendre transform of f is the following function of y:
g(y) ≡ f (x) − yx ,
(1.3)
4
1. PRINCIPLES OF THERMODYNAMICS
where on the right-hand side x is expressed as a function of y from relation
(1.2). Direct calculation yields
dg(y)
= −x ,
dy
(1.4)
so that df = ydx and dg = −xdy.
Mathematical identities. In thermodynamics, fixed variables are usually indicated explicitly when calculating partial derivatives. This is because several sets of independent variables are in wide use and infer different physical meaning for partial derivatives in different sets. Examples of
useful relations for various changes of variables are listed below.
Jacobi determinants. The use of Jacobi determinants
∂u ∂u ∂(u, v) ∂x ∂y =
∂(x, y) ∂v ∂v ∂x ∂y
is often convenient when carrying out changes of variables in differential
relations. This is due to the properties
∂(u, y)
∂(u, v) ∂(s, t)
∂u
∂(u, v)
=
=
,
,
∂(x, y)
∂x y
∂(x, y)
∂(s, t) ∂(x, y)
valid for Jacobians of arbitrary order and easily checkable by direct calculation for 2 × 2 determinants.
Example 1.1. Consider the function of two variables F (x, y). If by some
reason we want to use the pair (x, z) as independent variables, we may
writw
F (x, y) = F x, y(x, z) .
The chain rule then yields
∂F
∂x
=
z
∂F
∂z
∂F
∂x
+
y
=
x
∂F
∂y
∂F
∂y
x
x
∂y
∂z
∂y
∂x
,
z
.
x
1.3 Equation of state
Equation of state expresses the relation of state variables of the system in
equilibrium. It is usually written in a form involving ”mechanical” variables and the temperature. Equation of state does not usually include internal energy or other extensive variables of dimensions of energy, and in
this sense the equation of state does not give a complete thermodynamic
description of the system. A few widely used equations of state are listed
below.
1.3. EQUATION OF STATE
5
Classical ideal gas. The equation of state of classical ideal gas is
(1.5)
pV = N T .
Here, p = pressure, V = volume, N = number of molecules and T = absolute
temperature.
Mixture
P pV = N T ,
Pof ideal gases: Equation of state remains the same
with N = i Ni . The total pressure may be expressed as p = i pi , where
pi = Ni T /V = partial pressure of the ith component.
Virial expansion of real gas. The equation of state of the ideal gas
may be amended so that the intermolecular interaction is taken into account. Denote the (particle) number density by n ≡ N/V . In the limit of
small density the pressure may be expanded in powers of the density (the
virial expansion
p=T
n + n2 B2 (T ) + n3 B3 (T ) + · · · ,
(1.6)
where the virial coefficients Bn depend on the temperature only.
Curie’s law. Magnetic field strength H, magnetic induction B and magnetization M are related as.
B = µ0 (H + M ) .
Further the magnetic moment of the system shall often be denoted by m;
in case of homogeneous field then m = V M .
The magnetic equation of state expresses the dependence of magnetization M on the field strength H. Many paramagnetic materials (no spontaneous magnetic ordering) obey Curie’s law
M=
C
H,
T
(1.7)
where C is a material constant proportional to the number density of paramagnetic atoms.
Responses. Thermodynamic responses describe the reaction of state
variables to the change of other state variables. They usually are easily
measurable quantities. The equation of state determines ”mechanical” responses like thermal expansion coefficient
1 ∂V
,
(1.8)
αp =
V ∂T p,N
isothermal compressibility
κT = −
1
V
∂V
∂p
=
T,N
1
n
∂n
∂p
T
,
(1.9)
6
1. PRINCIPLES OF THERMODYNAMICS
or isothermal susceptibility
χT =
∂M
∂H
(1.10)
T
of magnetic material.
Under an adiabatic (thermally isolated) change the responses are adiabatic compressibility
1 ∂n
1 ∂V
=
(1.11)
κS = −
V
∂p S,N
n ∂p S
and adiabatic susceptibility
χS =
∂M
∂H
.
(1.12)
S
1.4 Zeroth law
Zeroth law of thermodynamics is the observation that there is a quantity
called temperature characterizing the thermal equilibrium and a thermometer to measure and compare temperatures. This comparison is transitive:
if two bodies are separately in equilibrium with a third one they are in equilibrium with each other.
1.5 Internal energy
In thermodynamics internal energy is the total energy of the system at rest.
Usually the potential energy of the system in an external field is excluded.
Then internal energy consists of the kinetic energy of relative motion of
particles, energy of their interaction and structural energy of the particles.
Due to interaction between the system and its environment care has to
taken in dividing the world in the system and the environment, especially
when long-ranged interactions occur.
At the present state of our knowledge of the structure of matter it is
quite obvious that the internal energy of the system is a state variable and
thus an unambiguous function of the state. It is also clear that internal
energy may determined even in systems which are not in a state of thermodynamic equilibrium.
1.6 Work
Work is energy exchange between the system and environment which may
be described in terms of work of macroscopic mechanics and electromagnetic theory.
1.6. WORK
7
There are different sign conventions. Here, the elementary work (dif−
is the work exerted to the environment by the
ferential form of work) dW
system. In this case positive work means loss of energy by the system. In
the paradigmatic SVN - system1 the work related to the change of the volume is
−
dW
= p dV.
(1.13)
The work related to the surface energy of a liquid may be written as
(1.14)
−
dW
= −σ dA ,
where σ = surface tension and A = free surface area. With positive surface
energy σ > 0 and the surface tension tends to decrease the area. An elastic
deformation gives rise to the work form:
(1.15)
−
dW
= −F dL ,
where F = the force stretching the rod and L = the rod length. The tension
is σ = F/A = force/cross-section area. According to Hooke’s law σ = E(L −
L0 )/L0 , where E is Young’s modulus and L0 the rest length of the rod.
The general expression of the differential form of work is
−
dW
=
X
i
(1.16)
fi dXi = f · dX ,
where fi are the coordinates of the generalized force and Xi the coordinates
of the generalized displacement.
Work in electromagnetism. Treatment of energetic quantities in a
system in electromagnetic field requires considerable care in the definition
of the system, because usually the introduction of polarizable or magnetizable body in an electromagnetic field changes the field everywhere, not
only in the body itself. Unambiguous definition may be obtained, if the
whole electromagnetic field is considered a part of the system. In this case
the elementary work required for a change of fields in the form familiar
from electrodynamics (with the sign corresponding to our convention)
Z
−
dW = − d3 r (E · dD + H · dB)
(1.17)
may be interpreted as the work carried out by the system.
With the aid of a special experimental setup in some cases it is possible
to arrive at a situation in which the polarized body does not affect the fields
E and H and the field energy outside the body may be dropped from the
energy balance of the system so that
−
dW
= −V0 (E · dD + H · dB) ,
(I)
(1.18)
1
One-component isotropic homogeneous material with the state variables S, V, N
(natural variables of the internal energy), often referred to as the pVT system as
well.
8
1. PRINCIPLES OF THERMODYNAMICS
when the volume V0 is small enough so that the fields may be regarded as
uniform.
If this is not possible, the energy corresponding to fields with the same
sources (free charges and conducting currents) but without the polarizable
body is nevertheless often subtracted from the energy related to the system including this body. The point here is that thermodynamics is brought
about in the problem by the presence of polarizable material. Without it,
the problem would be that of "pure" electrodynamics.
For simplicity, consider still the case in which the fields E and H are the
same both with and without the polarizable body. In uniform fields then the
total energy might be written as
1
1
(1.19)
Etot = U + V0
ε0 E 2 + µ0 H 2
2
2
thus excluding the energy of the "empty space" from the internal energy of
the system considered. In this case the differential form of electromagnetic
work related to the change of the internal energy defined as (1.19) assumes,
according to (1.18) the form
−
dW
= −V0 (E · dP + µ0 H · dM ) .
(II)
(1.20)
This convention is often used in condensed matter and solid state physics.
1.7 First law
The first law of thermodynamics is the law of conservation of energy.
−
−
.
− dW
dE = dQ
(1.21)
Here, dE is the differential of the energy of the system. With the usual
convention of thermodynamics, it may be identified by the differential of
the internal energy: dE = dU , provided the momentum, angular momentum
and the potential energy in external field of the system remain unaltered.
If the particle number may change, the chemical potential µ is introduced by the definition
−
−
+ µdN .
− dW
dU = dQ
In a general form for several particle species the first law is
X
−
µi dNi .
− f · dX +
dU = dQ
(1.22)
i
Heat capacity. The ability of a body to receive heat is described by the
heat capacity
∆Q ,
CA =
(1.23)
∆T A
1.8. SECOND LAW
9
where the subscript A refers to fixed variables, e.g.: CV , Cp . Specific heat
is the heat capacity per unit mass. Heat capacities are usually easy to
measure contrary to the internal energy.
Cyclic process. Cyclic processes (cycles) are especially important in the
theory of heat engines. In a cycle the system return to its initial state again
and again after certain periodic stages. In a simple SVN system, in which
−
− p dV , the area enclosed by the curve describing the process in
dU = dQ
the (V, p) plane is
I
(1.24)
p dV = W .
H
Since dU = 0, the work during a cycle is equal to the difference of the
amounts of heat received and delivered by the system. The thermal efficiency of a cycle is η = ∆W/∆Q+ , where ∆Q+ is the amount of heat received
by the system during a cycle.
1.8 Second law
From the formal point of view the second law states two things:
• for the differential form of heat there is an integrating factor (the temperature) giving rise to the extensive state variable entropy S; in a
reversible process:
dS =
−
dQ
.
T
(1.25)
dS >
−
dQ
.
T
(1.26)
• In an irreversible process
There are several traditional equivalent formulations of the second law:
(1.8a) Heat cannot be transferred from a colder heat reservoir to a
warmer heat reservoir without any other changes. (Clausius)
(1.8b) There is no cyclic process with the sole result of transferring the
heat received to work. (Kelvin)
(1.8c) Of all heat engines working between the temperatures T1 and T2
the Carnot engine has the highest efficiency. (Carnot)
All these statements are equivalent in the sense that each of them yields
the others. Here, we shall not dwell on demonstration of this equivalence,
however.
The first law in a reversible process may now be cast in the form
dU = T dS − f · dX +
X
i
µi dNi .
(1.27)
10
1. PRINCIPLES OF THERMODYNAMICS
1.9 Carnot’s cycle
The notion of entropy may be approached by analyzing Carnot’s cycle consisting of four reversible stages(Fig. 1–2):
a) isothermal
b) adiabatic
c) isothermal
d) adiabatic
T2
T2 → T1
T1
T1 → T2
∆Q2 > 0
∆Q = 0
∆Q1 > 0
∆Q = 0
T2
p
a
∆Q 2
∆Q 2
d
∆W
b
∆Q1
c
∆Q1
T1
V
Figure 1–2: Carnot’s cycle.
The thermal efficiency of the process is
η=
∆Q1
∆W
=1−
.
∆Q2
∆Q2
(1.28)
Since the cycle is reversible, it may be also used as a heat pump. The efficiency of Carnot’s cycle depends only on the temperatures T1 and T2 of the
heat reservoirs but not on the details of realization.
T3
∆Q3
∆W 23
T2
∆Q 2
∆W 12
T1
∆Q 1
Figure 1–3: Determination of absolute temperature scale.
1.9. CARNOT’S CYCLE
11
Absolute temperature. An absolute temperature scale may be determined with the aid of a serial connection of Carnot’s cycles as in Fig. 1–3.
The efficiency depends only on the reservoir temperatures, therefore
1−η =
∆Qout
= f (Tmax , Tmin ) .
∆Qin
(1.29)
From relations f (T3 , T2 ) = ∆Q2 /∆Q3 , f (T2 , T1 ) = ∆Q1 /∆Q2 , f (T3 , T1 ) =
∆Q1 /∆Q3 the functional identity follows
f (T3 , T2 )f (T2 , T1 ) = f (T3 , T1 ),
which has to hold for all Ti . The simplest choice is
f (T2 , T1 ) =
T1
T2
(1.30)
which defines the thermodynamic (absolute) temperature scale up to the
choice of the unit. For the efficiency of Carnot’s cycle this yields
η =1−
Tmin
.
Tmax
(1.31)
Consider now a cyclic (quasistatic) process divided to a large number of
subprocesses with temperatures Ti and the amount of heat received ∆Qi .
Imagine that these portions of heat are transferred by Carnot engines working between the system a huge heat reservoir at the temperature T0 > Ti ,
so that the ith engine receives the heat ∆Q0i from the reservoir. Calculate
now the work done in one cycle by the system and all the Carnot engines.
In one cycle the work done by the system equals the heat received:
X
∆Qi .
(1.32)
Wsystem =
i
The work of ith Carnot engine is WCi = ∆Q0i −∆Qi , so that the total work is
equal to the heat received by our combined system from the heat reservoir:
X
X
X
∆Q0i = Q0 ≤ 0 ,
(1.33)
(∆Q0i − ∆Qi )
∆Qi +
Wtotal =
i
i
i
which cannot be positive according to Kelvins statement of the second law,
since the combined system consisting of the original cycle and the auxiliary
Carnot machines did not give any heat to a heat reservoir at a temperature
lower than T0 . Now ∆Q0i /T0 = ∆Qi /Ti . Therefore, replacing the sum over
subprocesses by a contour integral in the state variable space, we arrive at
the Clausius inequality
I
−
dQ
≤ 0.
T
(1.34)
12
1. PRINCIPLES OF THERMODYNAMICS
For any reversible process this is an equality, which means that the integrand is a differential of some state variable. This state variable is the
entropy S and
−
dQ
= dS .
T
If a finite portion of our process is reversible, say from state 1 to state 2, the
corresponding part of the contour integral in Clausius’s inequality (1.34)
yields the difference between the values of entropy in these states:
S2 − S1 =
Z2
−
dQ
,
T
(1.35)
1
and Clausius’s inequality takes the form
S2 − S1 ≥
Z2
−
dQ
.
T
(1.36)
1
In particular, in a thermally isolated system the entropy cannot decrease.
The second law seems to be in contradiction with the time-reversal invariance of the basic microscopic laws of physics, since it establishes a preferred direction of processes. The origin of this time-reversal symmetry
breaking in macroscopic physics remains unclear.
1.10 Third law
The third law thermodynamics, Nernst’s law, states that the entropy of an
equilibrium system vanishes, when the temperature approaches the absolute zero:
lim S = 0 .
(1.37)
T →0
In classical thermodynamics the conjecture is that this limit exists, the particular value 0 is explained in quantum statistical physics.
1.11 Problems
Problem 1.1. Show that
∂x
∂y
z
∂y
∂z
x
∂z
∂x
y
= −1
(1.38)
1.11. PROBLEMS
13
and that for any function F
∂F
∂y z
∂x
.
= ∂y z
∂F
∂x z
(1.39)
Problem 1.2. Which of the following differential forms are differentials? Find the integrating factor for those differential forms which are
not differentials.
(a) d− u =
x4
y
dx + y 2 dy.
(b) d− u = (10x + 6y)dx + 6x dy ,
(c) d− u = 12y 2 dx + 18xy dy .
Problem 1.3. Define the Legendre g(y) transform of the function f (x)
as
df
g(y) = f (x) − xy ,
y=
,
dx
where on the right-hand side x is assumed to be expressed as a function
of y from the condition y = f ′ .
(a) Show that
d2 f
dx2
d2 g
dy 2
= −1 .
(b) Construct the Legendre transform of the function f = 21 x2 .
(c) Construct the Legendre transform of the function f = −ax ln x − b,
where a and b are positive constants.
Problem 1.4. Thermal expansivity α and isothermal compressibility κ
of matter are defined as
α=
1
V
∂V
∂T
;
κ=−
p
1
V
∂V
∂p
.
T
Show that
∂α
∂p
T
=−
∂κ
∂T
;
p
α
=
κ
∂p
∂T
.
V
Problem 1.5. Calculate the virial coefficients B2 , B3 and B4 of Clausius’ matter. Clausius’ equation of state is
p+
aN 2
(V − bN ) = N T,
T (V + cN )2
where a, b and c are positive experimental constants and N the total
number of particles. Can you determine all the virial coefficients for
this matter?
14
1. PRINCIPLES OF THERMODYNAMICS
Problem 1.6. Consider a spherical capacitor with external radius b and
internal radius a charged to an initial charge Q. The capacitor is halffilled by a dielectric substance of permittivity ε in such a way that the
dielectric fills the space between the plates to one side of a cross-section
plane dividing the spheres in two halves, while to the other side of the
plane the capacitor is empty. Express the differential form of work in
terms of electric induction D and electric field E. Calculate the work
exerted on the capacitor, when the charge is increased by an infinitesimal amount δQ. Proceed by subtracting the differential form of work
required to increase the charge by δQ from Q in an empty capacitor.
Express the result in terms of the polarization vector P .
Problem 1.7. Consider the same capacitor but now with an initial potential difference ∆φ between the plates. Express the differential form
of work in terms of electric induction D and electric field E. Calculate the work exerted on the capacitor, when the potential difference
between the plates is increased by an infinitesimal amount δφ. Proceed
by subtracting the differential form of work required to increase the potential difference by δφ from ∆φ in an empty capacitor. Express the
result in terms of the polarization vector P .
Problem 1.8. Experimentally it has been found that a rubber band
obeys:
∂F
∂L
T
T
=a
L0
"
1+2
L0
L
3 #
,
∂F
∂T
L
L
=a
L0
"
1−
L0
L
3 #
,
where F is the tension and the constant a and the rest length of the
band L0 are parameters.
(a) Calculate (∂L/∂T )F and give a physical interpretation.
(b) Show that dF = ∂L F dL + ∂T F dT is a differential.
(c) Determine the equation of state F = F (L, T ) of the band.
Problem 1.9. In a perfect gas the internal energy obeys the relation
dU = CV dT . Find the equation of state for such a gas in a process, in
which the heat capacity C is a constant (polytropic process).
Problem 1.10. Stirling’s cycle consists of two isotherms at T1 and T2
and two isochores (processes with constant volume) at V1 and V2 . Calculate the coefficient of thermal efficiency of Stirling’s cycle working on
the ideal gas. Compare with the thermal efficiency of Carnot’s cycle.
2. Thermodynamic potentials
2.1 Fundamental equation
Thermodynamic potentials are extensive state variables of dimensions of
energy. Their purpose is to allow for simple treatment of equilibrium for
systems interacting with the environment.
In thermodynamics all variables are either extensive or intensive.
Mathematically this may expressed in homogeneity relations with respect to the system size. Thus, extensive variables (e.g. N, V, U, S, . . .)
are first-order homogeneous functions, whereas intensive variables (like
p, T, µ, . . .)are independent of the size of the system.
Natural variables. are those whose differentials appear in the differential form of the first law: dU = T dS − p dV + µ dN so that S, V ja N
are natural variables of internal energy. With all intensive variables fixed,
extensivity of all these variables means
(2.1)
U (λS, λV, λN ) = λU (S, V, N ).
Differentiating both sides with respect to the auxiliary parameter λ and
putting λ = 1 thereafter we arive at the identity (Euler equation for homogeneous functions):
U =S
∂U
∂S
+V
V,N
∂U
∂V
+N
S,N
∂U
∂N
.
S,V
From the first law it follows that
∂U
∂S
V,N
=T,
∂U
∂V
S,N
= −p ,
∂U
∂N
= µ.
S,V
Thus, we arrive at the fundamental equation
U = T S − pV + µN .
15
(2.2)
16
2. THERMODYNAMIC POTENTIALS
2.2 Internal energy U and Maxwell relations
The first law dU = T dS − p dV + µ dN yields
∂U
,
T =
∂S V,N
∂U
,
p = −
∂V S,N
∂U
µ =
.
∂N S,V
From the definition of heat capacity it follows that
!
−
dQ
∂U
.
=
CV =
dT
∂T V,N
(2.3a)
(2.3b)
(2.3c)
(2.4)
V,N
Since U may be assumed to be single-valued smooth state variable,
result of iterative differentiation does not depend on order ∂T /∂N =
∂(∂U/∂S)/∂N = ∂(∂U/∂N )/∂S = ∂µ/∂S. This procedure gives rise to
Maxwell relations:
∂T
∂p
= −
,
(2.5a)
∂V S,N
∂S V,N
∂T
∂µ
=
,
(2.5b)
∂N S,V
∂S V,N
∂p
∂µ
= −
.
(2.5c)
∂N S,V
∂V S,N
These and similar relations for other thermodynamic potentials are often
useful in expressing differential relations in terms of response functions
and state variables.
In an irreversible process T δS > δQ = δU + δW − µδN , therefore
δU < T δS − p δV + µ δN.
(2.6)
In an irreversible process with fixed S, V and N the internal energy decreases. Thus, in equilibrium U assumes the mimimal value with S, V and
N fixed (implying, of course, that something else may change).
If some other work may be done in a reversible process, then
∆U = R = −∆Wfree ,
where the free work ∆Wfree = −R is the work the system may carry out in
given circumstances.
If the process is irreversible, then
∆Wfree ≤ −∆U
(2.7)
even if (S, V, N ) are kept fixed. Thus, the minimal work needed to bring
about the change of internal energy ∆U is R = ∆U .
2.3. ENTHALPY H
17
2.3 Enthalpy H
Other thermodynamic potentials are Legendre transformsof the internal
energy U (S, V, N ) with respect to natural variables S, V or N . Enthalpy (or
the heat function) is obtained by using p instead of V :
H ≡ U + pV .
(2.8)
The differential follows from the definition and the first law:
dH = T dS + V dp + µ dN .
(2.9)
Natural variables are (S, p, N ). From the definition of heat capacity it follows that
!
−
dQ
∂H
Cp =
.
(2.10)
=
dT
∂T p,N
p,N
From the expression for dH three more Maxwell relations follow:
∂V
∂T
=
,
∂p S,N
∂S p,N
∂µ
∂T
=
,
∂N S,p
∂S p,N
∂V
∂µ
=
.
∂N S,p
∂p S,N
(2.11a)
(2.11b)
(2.11c)
Ia an irreversible change δQ = δU + δW − µ dN < T δS. Substitution of
δU = δ(H − pV ) yields δH − δ(pV ) + δW − µ dN < T δS, i.e
δH < T δS + V δp + µ δN .
(2.12)
If in the process S, p and N remain constant, spontaneous changes drive
the system to the state with mimimum enthalpy.
Many practically important processes (phase transitions, chemical reactions etc) take place at constant (ambient) pressure. If the conditions
include also thermal isolation, the enthalpy is the natural energy quantity
to use.
In hydrodynamics adiabatic flow is a popular approximation. Then the
specific (per unit mass) internal energy u appears in the energy equation
only in the combination u + p/ρ = h, which is the specific enthalpy and thus
the natural energy variable.
When (S, p, N ) are fixed, the portion of the energy of the system freely
exchangeable for work obeys the condition
∆Wfree ≤ −∆H .
The mimimum work required to bring about ∆H is thus R = ∆H.
(2.13)
18
2. THERMODYNAMIC POTENTIALS
Joule–Thomson process. Consider thermally isolated forced flow of
gas through a throttle valve or a porous wall. Movement of pistons is devised to keep pressures p1 > p2 fixed. Although the flow is far from equilibrium and not reversible, a hypothetical reversible process between the
same states is useful, because state variables are process-independent.
For the transfer of an infinitesimal quantity of matter the work by
−
= p2 dV2 + p1 dV1 .
the system is dW
Initially V1 = Vi and V2 = 0. Finally
p1
p2
V1 = 0 and V2 = Vf . For constant
pressures the work is
Z
−
Figure 2–1: Flow through porous
dW
= p2 V f − p1 V i .
W =
wall.
Thermal isolation means ∆Q = 0,
therefore ∆U = −W . From this it follows Uf + p2 Vf = Ui + p1 Vi . Thus,
the quantity U + pV , i.e. the enthalpy H remains constant: the process is
isenthalpic,
∆H = Hf − Hi = 0 .
(2.14)
Imagine now a reversible isenthalpic process of decreasing the pressure by
infinitesimal steps. The response of the temperature to this is given by the
Joule–Thomson coefficient
∂T
.
(2.15)
∂p H
To express this coefficient in terms of already introduced quantities, use the
Jacobi determinant method. It is good policy to introduce variables which
are the natural variables of the thermodynamic potential appearing in this
definition, because then its first derivatives are state variables. Thus,
∂T
∂p
∂(p, S) ∂(T, H)
1 ∂(T, H)
∂(T, H)
=
=
=
∂(p,
H)
∂(p,
H)
∂(p,
S)
T
∂(p, S)
H
"
# ∂T
∂T
V
∂H
∂H
∂T
1
−
−
=
. (2.16)
=
T
∂p S ∂S p
∂S p ∂p S
∂p S Cp
Using the Maxwell relation (2.11a) rewrite
∂T
∂S
T
∂V
T
∂V
∂V
=
=
=
.
∂p S
∂S p
Cp ∂S p ∂T p
Cp ∂T p
Thus, we arrive at the expression
#
"
V
T
∂V
V
∂T
=
(T αp − 1) .
=
−
∂p H
Cp
∂T p T
Cp
(2.17)
The latter form follows form the definition of the thermal expansion coefficient : αp = V −1 (∂V /∂T )p .
2.4. FREE ENERGY F
19
In the process the pressure decreases, so that the gas is cooled, if
T αp > 1, or heated, if T αp < 1. For the ideal gas the Joule–Thomson coefficient vanishes, so that the temperature of an ideal gas remains the same.
For real gases the coefficient is positive below a certain pressure-dependent
inversion temperature, so that the gas is cooled. Thus, the Joule-Thomson
process may be and is used for cooling and eventually liquifying gases.
2.4 Free energy F
The Legendre transform of the internal energy with respect to S yields the
free energy (Helmholtz free energy): F = U − S(∂U/∂S)V,N i.e.
F ≡ U − TS .
(2.18)
The corresponding differential is
dF = −S dT − p dV + µ dN .
The natural variables are T , V and N . The Maxwell relations are
∂S
∂p
=
,
∂V T,N
∂T V,N
∂S
∂µ
= −
,
∂N T,V
∂T V,N
∂p
∂µ
= −
.
∂N T,V
∂V T,N
(2.19)
(2.20a)
(2.20b)
(2.20c)
As before, for an irreversible process
δF < −S dT − p δV + µ δN .
(2.21)
Thus, with fixed T , V and N , the system evolves towards the minimum of
the free energy. For the free work at fixed T, V, N it follows
∆Wfree ≤ −∆F .
(2.22)
Free energy is an extremely important tool in statistical mechanics: in
many cases it is the natural macroscopic quantity to calculate for a given
microscopic model.
2.5 Gibbs function G
The Legendre transform of U with respect to both S and V leads to the
Gibbs function (Gibbs free energy)
G ≡ U − T S + pV ,
(2.23)
20
2. THERMODYNAMIC POTENTIALS
with the differential
(2.24)
dG = −S dT + V dp + µ dN .
The natural variables are T , p and N
∂S
=
∂p T,N
∂S
=
∂N T,p
∂V
=
∂N T,p
and the Maxwell relations
∂V
−
,
∂T p,N
∂µ
−
,
∂T p,N
∂µ
.
∂p T,N
(2.25a)
(2.25b)
(2.25c)
With fixed T , p and N , a non-equilibrium system evolves towards the minimum of the Gibbs function:
δG < −S δT + V δp + µ δN ,
(2.26)
∆Wfree ≤ −∆G .
(2.27)
G = µN ,
(2.28)
and the free work is
The Gibbs function is a suitable thermodynamic potential for systems
which change at fixed pressure and temperature (no thermal isolation).
Since these parameters are perhaps most easily of all adjustable, the Gibbs
potential has a wide scope of applications both in physics and chemistry.
From the fundamental equation it follows that
i.e. the chemical potential is the Gibbs function per particle in a singlespecies system. Since from (2.35) it follows that dG = µdN + N dµ and,
taking into account the alternative form (2.24), we arrive at the Gibbs–
Duhem equation
V
S
dµ = − dT + dp ,
(2.29)
N
N
showing that the natural variables of the chemical potential are T, p.
2.6 Grand potential Ω
The grand potential is also an important quantity for calculations in statistical mechanics when the number of particles cannot be fixed. The definition is
Ω ≡ U − T S − µN
(2.30)
leading to the differential
dΩ = −S dT − p dV − N dµ ,
(2.31)
2.7. THERMODYNAMIC RESPONSES
21
showing that the natural variables are T , V and µ. The Maxwell relations
are
∂S
∂p
=
,
(2.32a)
∂V T,µ
∂T V,µ
∂S
∂N
=
,
(2.32b)
∂µ T,V
∂T V,µ
∂p
∂N
=
.
(2.32c)
∂µ T,V
∂V T,µ
In an irreversible change the inequality
δΩ < −S δT − p δV − N δµ ,
(2.33)
holds revealing that in a process with fixed T , V and µ the system tends to
state with the minimum of Ω. The free work under these conditions is
∆Wvapaa ≤ −∆Ω .
(2.34)
From the fundamental equation it follows that
(2.35)
Ω = −pV
revealing that knowledge of Ω is tantamount to knowing the equation of
state (although in non-standard variables).
2.7 Thermodynamic responses
Thermodynamic responses have the form of partial derivatives
(∂K/∂A)B,C,... and reveal the effect of an infinitesimal change of a state
variable (A) to some quantity (K) describing the system at equilibrium.
Usually these are (the most) directly measurable quantities.
Coefficient of (volumninal) thermal expansion. Definition
1 ∂V
.
αp =
V ∂T p,N
(2.36)
In terms of number density n = N/V :
αp = −
1
n
∂n
∂T
.
(2.37)
p
In isotropic substance the coefficient of linear thermal expansion is one
third of this, since a small change of volume is three times the change of
length.
22
2. THERMODYNAMIC POTENTIALS
Isothermal compressibility. Reaction to pressure at constant temperature
1 ∂V
1 ∂n
κT = −
=
.
(2.38)
V
∂p T,N
n ∂p T
Adiabatic compressibility. Pressure acting in thermal isolation
1 ∂n
1 ∂V
=
.
(2.39)
κS = −
V
∂p S,N
n ∂p S,N
This quantity determines the speed of sound:
cs = √
1
.
mnκS
(2.40)
Here, m is the particle mass and mn = mN/V is the mass density.
Isochoric heat capacity. Definition by reversible process, thus ∆Q =
T ∆S, and the heat capacity in general
∆Q ∆S C≡
=
T
.
∆T condition
∆T condition
Due to the chosen scale of temperature, heat capacities are dimensionless.
Specifically at constant volume we obtain
∂S
.
(2.41)
CV = T
∂T V,N
−
=
Since under these conditions dU = T dS −p dV +µ dN reduces to dU = dQ
T dS and, on the other hand, S = −(∂F/∂T )V,N , we arrive at relations
CV =
∂U
∂T
V,N
= −T
∂2F
∂T 2
.
(2.42)
V,N
Isobaric heat capacity. Analogously
Cp = T
∂S
∂T
(2.43)
,
p,N
−
and S = −(∂G/∂T )p,N yield
(dH)p,N = T dS = dQ,
Cp =
∂H
∂T
p,N
= −T
∂2G
∂T 2
.
p,N
(2.44)
2.7. THERMODYNAMIC RESPONSES
23
Connections. Relations between heat capacities under different conditions are due to differences in work. For Cp , change variables
∂S
∂S(V (p, T ), T )
∂S
∂S
∂V
=
=
+
.
∂T p
∂T
∂T V
∂V T ∂T p
p
Free energy Maxwell (∂S/∂V )T = (∂p/∂T )V (2.20a) yields
∂V
∂p
.
Cp = CV + T
∂T V ∂T p
(2.45)
The change of variables in (∂p/∂T )V gives rise to the result
Cp = CV + V T
αp2
.
κT
(2.46)
Compressibility is positive in stable matter, therefore Cp > CV .
Construction of potentials. An equation of state like p = p(T, V ) and
a thermal response, say CV , are required to this end. Consider, for instance,
van der Waals matter:
N2
p + a 2 (V − N b) = N T .
V
The heat capacity CV is directly a partial derivative of the internal energy:
∂U
.
CV =
∂T V
To calculate the other one, change variables
∂U
∂(U, T ) ∂(V, S)
∂(U, T )
=
=
∂V T
∂(V, T )
∂(V, S) ∂(V, T )
∂S
∂U
∂T
∂T
∂U
=
.
−
∂T V
∂V S ∂S V
∂S V ∂V S
(2.47)
Here, derivatives of U are −p and T , this is the point of introducing the
natural variables of U . Thus
∂U
∂S
∂p
= −p + T
=T
− p,
∂V T
∂V T
∂T V
where the free-energy Maxwell (2.20a) has been used once more. This relation shows, in particular, that for the van der Waals equation of state
∂2U
∂CV
= 0,
=
∂V T
∂V ∂T
i.e. CV is a function of temperature only! Then the integration is simple:
Z
Z
∂p
N2
U (T, V ) = CV dT + T
.
− p dV = CV (T )dT − a
∂T V
V
24
2. THERMODYNAMIC POTENTIALS
2.8 Thermodynamic stability conditions
Let the near-equilibrium system be
divided to (semi)macroscopic subsystems
(labeled by index α) each in a local equilibrium, but with different pressure, temperature etc. in neighbouring subsystems. Extensive
quantities
remain
PaddiP
P
tive: S = α Sα , V = α Vα , U = α Uα .
Let Njα be the particle number ofP
species
j in the subsystem α. Then Nj = α Njα
for the jth species.
Due to local equilibrium
For a small change of Sα
∆Sα =
α
pα T α V α
Figure 2–2: System near equilibrium
Sα = Sα (Uα , Vα , {Njα }) .
X µjα
pα
1
∆Uα +
∆Vα −
∆Njα .
Tα
Tα
Tα
j
Assume a system isolated as a whole, then U , V and Nj remain constant. To
simplify notation, consider two subsystems: α = A, B. Conservation laws
yield ∆UB = −∆UA , ∆VB = −∆VA and ∆NjB = −∆NjA . Thus,
X
∆S =
∆Sα
α
=
1
1
−
TA
TB
∆UA +
pA
pB
−
TA
TB
∆VA −
X µjA
j
TA
−
µjB
TB
∆NjA .
At equilibrium ∆S = 0 identically. Since the fluctuations ∆UA , ∆VA and
∆NjA are arbitrary, the equilibrium conditions:

TA = TB 
pA = pB
(2.48)

µjA = µjB
follow. Thus, in equilibrium the temperature is the same everywhere, as
well as the pressure (provided no external fields impose inhomogeneity) and
the chemical potential for each particle species. The conditions hold also in
the case system consists of different phases (constant pressure requires flat
interfaces, however).
2.9 Stability conditions of matter
In stable equilibrium the entropy must be at maximum. To analyze this,
the second variation of the entropy with respect to {∆Uα , ∆Vα and ∆Njα }
may be used.
2.10. THERMODYNAMIC POTENTIALS IN ELECTROMAGNETISM 25
Let T , p and {µj } be the common equilibrium values. For simplicity,
assume one species. In Taylor expansion at the equilibrium point S = S0 +
dS + 21 d(dS) the linear term vanishes. Since d2 X = 0 for any independent
variable X, we obtain
1
S = S0 + d(dS) =
2 1
1X 1
1X
d
(dpα dVα − dµα dNα ) .
(dUα + pα dVα − µα dNα ) +
2 α
Tα
2 α Tα
Here, dUα + pα dVα − µα dNα = Tα dSα , so that (denote dX → ∆X)
1 X
∆Stot ≡ S − S0 = −
(∆Tα ∆Sα − ∆pα ∆Vα + ∆µα ∆Nα ) .
2T α
(2.49)
The condition of a stable equilibrium is that this expression is negative
definite.
Since any subsystem α is at local equilibrium, only three of fluctuations
of the quantities Tα , Sα , pα , Vα , µα , Nα are independent, the rest must be
expressed as functions of the chosen three.
Let ∆Nα = 0. Then only two independent variables remain. Choose
∆Tα and ∆Vα and express ∆Sα as ∆pα functions thereof. Maxwell relations
allow for simplification and the result is
1
1 X CV,α
(2.50)
(∆Tα )2 +
(∆Vα )2 .
∆Stot = −
2T α
T
κT V α
Another possibility ∆Vα = 0 with ∆Tα and ∆Nα as independent variables
leads to
(
)
1 X CV,α
∂µ
∆Stot = −
(∆Tα )2 +
(2.51)
(∆Nα )2 .
2T α
T
∂Nα T,Vα
From these expressions it is readily seen that the total entropy is at maximum, when the following stability conditions hold: :

CV
> 0 



κT
> 0 
.
(2.52)

∂µ


> 0 

∂N
T,V
Otherwise the equilibrium is unstable and small spontaneous disturbances
give rise to growing changes which lead to another state.
2.10 Thermodynamic potentials in electromagnetism
The starting point here is the basic differential form of work (1.17)
Z
−
dW
= − d3 r (E · dD + H · dB) ,
26
2. THERMODYNAMIC POTENTIALS
whose addition to previously introduced differentials gives rise to differentials of thermodynamic potentials in electromagnetism, say (relevant parameters only explicit)
Z
dU = T dS + d3 r (E · dD + H · dB) .
Material parameters contained in vectors E and B like the permittivity ε
and permeability µ should be expressed here as functions of the entropy
S. This is inconvenient, therefore a preferable choice is the free energy, for
which
Z
dF = −SdT + d3 r (E · dD + H · dB) .
(2.53)
Here, ε and µ are functions of the temperature.
Thermodynamics potentials assume minimum values at equilibrium,
when their natural variables are fixed. Since free charges are sources of
the electric induction D and the vector potential A the source of the magnetic induction B, the free energy (2.53) is the choice for problems with
fixed charges of conductors and fixed vector potentials (the latter might be
difficult to control in real world, though).
For other cases, new potentials should be formed
R by suitable Legendre
transforms. For instance, the potential FeE = F − d3 r E · D gives rise to
the differential
Z
dFeE = −SdT − d3 r D · dE ,
(2.54)
which reveals that the natural variables are T and E. Thus, this potential
minimizes at equilibrium when the field E (or the electric potential) is kept
constant.
R
Similarly, the potential FeH = F − d3 r H · B with the differential
Z
dFeH = −SdT − d3 r B · dH ,
(2.55)
and natural variables T and H is suitable for cases with fixed currents.
Combinations of these transform may appear useful as well. Unfortunately,
there seems to be no standard nomenclature of the different potentials in
the electromagnetic case (cf. the Helmholtz free energy and the Gibbs function of an S, V , N system).
Example 2.1. Consider a vertical parallel-plate capacitor in contact
with a liquid reservoir. Let us calculate, how high the liquid with the
dielectric constant εr rises between the vertical plates, when the capacitor is charged and disconnected from any voltage source.
The potential energy of the liquid in the gravitational field is
Wg =
1
gρwdy 2 ,
2
where g is the acceleration of gravity, ρ the density of the liquid, y the
height of the liquid slab between the plates, d the separation of the
2.11. PROBLEMS
27
plates and w the width of the plates. The energy of the electric field
between the plates is
WQ =
Q2
Q2 d
=
,
2C
2wε0 [h − y + εr y]
where Q is the charge of the capacitor and h the height of the plates.
Minimization of the free energy F with respect to y leads to the thirdorder equation
2
Q2
h
y y+
=
,
ε0 − 1
2ρw2 gε0
which only has one real solution most conveniently obtained by some
symbolic calculation programme like Maple or Mathematica.
2.11 Problems
Problem 2.1. Calculate the value of the expression
∂T
∂p
V
∂S
∂V
−
p
∂T
∂V
p
∂S
∂p
.
V
Problem 2.2.
(a) During a thermally isolated free expansion of a gas no work is
carried out and no heat exchanged, thus the internal energy of
the gas remains constant (the expansion is not necessarily a quasistatic process, however). Show that the Joule coefficient for a
free expansion of a gas is
∂T
∂V
1
CV
=
U,N
p−
T αp
κT
.
(b) Is the van der Waals gas heated or cooled in the free expansion?
Hint: it is more convenient not to calculate αp and κT separately.
The van der Waals equation of state is
p+a
N2
V2
(V − N b) = N T.
Problem 2.3. Find out in which systems the heat capacity CV does not
depend on the volume of the system.
Problem 2.4. The free energy of a crystal in which the ions have just
two quantum states is
F = −N T ln 1 + e−ǫ/T
,
where ǫ is a constant.
(a) Find the entropy S of the system as a function of the internal energy U and the number of particles N , i.e. the fundamental relation S = S(U, N ).
28
2. THERMODYNAMIC POTENTIALS
(b) Calculate the heat capacity Cǫ as a function of temperature T .
Plot it and find the position of the peak, known as the Schottky
anomaly. Such a peak is characteristic of a system in which atoms
have a few low-lying closely spaced energy levels, and at low temperatures may dominate all other contributions to the heat capacity of the solid.
Problem 2.5. Show (N is kept fixed) that the internal energy of the
Clausius gas is
U (T, V ) = N u1 (T ) −
2aN 2
,
T (V + N c)
where u1 (T ) is a function of temperature only, whose explicit form cannot be determined thermodynamically. ZHowever, there is another relation (show this as well) U (T, V ) =
CV (T, V ) , dT , establishing a
connection between u1 and the isochoric heat capacity of the gas. The
equation of state of Clausius’s gas is
p+
aN 2
(V − bN ) = N T .
T (V + cN )2
Problem 2.6. For a unit volume of dielectric at constant density find
the difference cE − cD between the heat capacities of a homogeneous
isotropic dielectric at constant electric field strength E and electric induction D.
Problem 2.7. Show – without resorting to the connection between Cp
and CV – that in a stable thermodynamic equilibrium Cp > 0 and κS >
0
Problem 2.8. By minimizing a suitable thermodynamic potential, find
how high dielectric liquid rises between the vertical plates of a parallelplate capacitor connected to a voltage source with the constant electromotive force E. Express your answer in terms of the electric field in the
capacitor rather than the emf.
3. Applications of thermodynamics
3.1 Classic ideal gas
For a complete thermodynamic description of a system, knowledge of the
equation of state and some thermodynamic potential (energy function) is
required. From the equation of state mechanical responses may be inferred
and vice versa: mechanical responses suffice to reconstruct the equation of
state. To determine internal energy or some other thermodynamic potential
a thermal response is needed, however.
The equation of state of the perfect gas pV = N T immediately yields
coefficient of thermal expansion
1 ∂V
1
N
αp =
=
(3.1)
=
V ∂T p,N
Vp
T
and isothermal compressibility
1
NT
1 ∂V
= .
=
κT = −
V
∂p T,N
V p2
p
(3.2)
It is en empirical observation that in conditions in which the equation of
state of a real gas coincides with that of the ideal gas (low pressure, high
temperature) the heat capacity is constant. Denote
CV =
1
fN .
2
(3.3)
The quantity 21 f is the specific heat capacity, (specific heat), i.e. heat capacity per molecule. In chemistry heat capacity per mole and in hydrodynamics
per mass unit are preferred. The factor f is the number of the effective degrees of freedom, whose classic value depends on the number of modes of
translational, rotational and vibrational motion of the molecule:
monatomic molecule
diatomic molecule
polyatomic molecule
f =3
f =5
f =6
3 translations
3 transl. + 2 rotations
3 transl. + 3 rot.
Each active vibrational mode adds two effective degrees of freedom (for both
kinetic and potential energy of the corresponding mode of harmonic oscillation). In what follows f is assumed constant.
29
30
3. APPLICATIONS OF THERMODYNAMICS
The differential of the entropy
∂S
∂S
dS =
dT +
dV
∂T V
∂V T
∂p
1
CV dT +
=
dV
T
∂T V
(3.4)
is readily integrated in the (T, V ) plane (N is fixed) with the use of the heat
capacity and mechanical responses (see Fig. 3–1) Assuming constant CV
and using the equation of state we obtain
Z V
Z T
T
V
N
CV
+
= S0 + CV ln
+ N ln
.
dV
S = S0 +
dT
T
V
T
V
0
0
V0
T0
The integration constants S0 and V0 – as extensive quantities – may be
written as S0 = N s0 , V0 = N v0 , where s0 and v0 are the the specific entropy
and volume at the reference point. The entropy of the ideal gas is thus
" #
f /2
T
V
S = N s0 + N ln
.
(3.5)
T0
N v0
The specific entropy at the reference point s0 shall be defined later with the
aid of statistical mechanics.
The expression (3.5) for entropy does not
vanish in the limit T → 0 but even diverges
V
in contradiction with the III law. For real
gases, however, lowering the temperature
leads either to phase transitions to liquid or
V0
solid or the appearance of quantum correcT0
T
tions.
All thermodynamic information may be
Integration
calculated starting from the known entropy. Figure 3–1:
path
for
entropy.
The internal energy may also be calculated
in the same fashion as before for the van der
Waals gas by integration in the (T, V ) plane:
dU S(T, V ), V
= T dS(T, V ) − p dV
∂S
∂S
dT + T
dV − p dV
= T
∂T V
∂V T
∂S
= CV dT + T
− p dV .
∂V T
From the Maxwell (2.20a) and the equation of state it follows that (∂S/∂V )T
= (∂p/∂T )V = N/V = p/T , so that the coefficient of dV vanishes. The equation of state renders the heat capacity CV independent of volume as well.
Thus, the internal energy of the ideal gas is independent of the volume and
may be written as
1
U = U0 + f (T − T0 )N .
2
3.2. FREE EXPANSION OF GAS
31
or, relabeling the normalization term, as
1
U = N µ0 + f T .
2
(3.6)
According to 2.7 we obtain Cp = CV +V T αp2 /κT = CV +V T p/T 2 = N ( 21 f +1).
The usual notation is
Cp = γCV ,
where γ is the heat capacity ratio (adiabatic constant)
γ=
f +2
Cp
=
.
CV
f
(3.7)
3.2 Free expansion of gas
Free expansion (Joule process) takes place, when a valve is opened or a
wall removed from between two chambers with different pressures. A sudden leveling of pressures is a typical irreversible process during which the
system is not in equilibrium. The initial and final states, however, are equilibrium states.
Let the volume grow from V1 to V2 during the expansion. Assume thermal isolation: ∆Q = 0. Since opening the valve
ideally does not involve work ∆W = 0 as
well. Therefore, internal energy does not
change ∆U = 0; the process is isergic..
Changes in state variables may again be
calculated along a hypothetic reversible
path 1 → 2.
Figure 3–2: Free expansion of
Ideal gas. Since in the ideal gas U = gas to vacuum.
the temperature must remain the
same T1 = T2 . The change in entropy may be calculated with the aid of
relation (3.5).
1
2fTN,
∆S = N ln
V2
.
V1
(3.8)
It should be noted that due to thermal isolation no entropy changes in the
environment occur: the entropy production is completely of internal origin.
Other equations of state. In analogy with the Joule-Thomson process
the following Joule coefficient may be defined and expressed in terms of
mechanical responses:
∂T
αp
1
p−T
.
(3.9)
=
∂V U,N
CV
κT
The result yields the temperature change in an infinitesimally small expansion. For finite changes integration is needed.
32
3. APPLICATIONS OF THERMODYNAMICS
3.3 Mixing entropy
Consider two gases (A and B) separated by a wall. When the wall is
removed, the gases mix. Let the
A
B
temperature and the pressure in
p
the final state be the same as before
mixing, so that the process may be
thought of as isothermal and isobaric. Obviously, disorder increases Figure 3–3: Isobaric and isothermal
and increase of the entropy is ex- mixing of gases.
pected.
Consider a mixture of ideal gases, so that partial pressures obey pj Dalton’s law pj V = Nj T , with Nj being the particle
Pnumber of species j. Its
concentration is xj = Nj /N = pj /p, where p = j pj is the total pressure.
In the following the increase of the entropy shall be calculated in two ways,
of which the latter is easier to generalize to non-ideal systems.
Way 1. Both gases are imagined to freely expand in turns to the total
volume and the entropy changes related to these stages are added. This is
possible, because interaction between the compounds is negligible. Since
pA = pB and TA = TB , the initial volumes are Vj = V xj , and the entropy
change according to (3.8) is
∆S =
X
Nj ln
j
V
.
Vj
In terms of concentrations this is
∆Ssek = −N
X
xj ln xj .
(3.10)
j
This is always ≥ 0, because 0 ≤ xj ≤ 1.
Way 2. Since the process takes place at constant pressure and temperature, the Gibbs function is useful. For a single-species ideal gas the Gibbs
function is
G(p, T, N ) = N T [φ(T ) + ln p] = N µ(p, T ) ,
(3.11)
where the most general form of the function φ is φ(T ) = µ0 /T − ζ − ( 12 f +
1) ln T . In the case of non-interacting gases the Gibbs function of the mixture is the sum of Gibbs functions of the compounds.
Prior to mixing the pressure of all compounds is p, and the Gibbs function is the sum of those of compounds:
X
Gi =
Nj T [φj (T ) + ln p] .
(3.12)
j
3.4. DILUTE SOLUTION, OSMOSIS
33
After mixing the partial pressures are pj and thus the Gibbs function
X
Nj T [φj (T ) + ln pj ] .
(3.13)
Gf =
j
Since pj = pxj , the difference is
∆Gmixing ≡ Gf − Gi =
X
Nj T ln xj .
j
The entropy is calculated as the partial derivative S = −(∂G/∂T )p,{Nj } .
Thus, the mixing entropy is
X
Nj ln xj .
(3.14)
∆Smixing ≡ Sf − Si = −
j
In isobaric mixing of real gases the volume of the system is not preserved
due to interactions and the Gibbs function of the mixture is not the simple
sum (3.13).
Gibbs paradox. What happens, if the gases are the same, A = B? If the
expressions obtained are used as such, increase of entropy follows ∆S > 0,
although there is no macroscopic physical change. This is the Gibbs paradox that cannot be satisfactorily explained by classical physics. In quantum
statistics the solution is that microstates differing only by permutation of
numbers of identical particles are considered one physical state.
3.4 Dilute solution, osmosis
Consider a system consisting of a solvent and a small amount of added
solute. Molecules of the solute are far away from each other and do not
interact and the whole system looks like a dilute ideal gas of the solute
mixed with the solvent. Thus, the mixing entropy has to be taken into
account.
Denote the concentration of the solute by x = N1 /N = N1 /(N0 + N1 ),
where N0 is the number of molecules of the solvent and N1 that of the solute.
Differentiation yields

∂x
x 

= −
∂N0
N 
.
(3.15)
1−x 
∂x


=
∂N1
N
In the Gibbs function of the solution
G(p, T, N0 , N1 ) = N0 µ0 (p, T, x) + N1 µ1 (p, T, x)
(3.16)
the chemical potentials µ0 and µ1 may depend on the particle numbers
through the concentration x only, so that from the Maxwell relation
∂µ0
∂µ1
∂2G
=
=
∂N1
∂N0
∂N0 ∂N1
34
3. APPLICATIONS OF THERMODYNAMICS
the consistency condition follows
∂µ0 ∂x
∂µ1 ∂x
=
,
∂x ∂N1
∂x ∂N0
or, in view of (3.15),
(1 − x)
∂µ0
∂µ1
+x
= 0.
∂x
∂x
(3.17)
This must hold for all 0 ≤ x ≤ 1.
Obviously, for the chemical potential of the solvent µ0 the limit x →
0 must be well-behaved and the existence of a power expansion in x is a
natural assumption:
µ0 (p, T, x) = µ0 (p, T, 0) + xν(p, T ) + O(x2 ) .
(3.18)
In µ1 the logarithmic mixing entropy term must be present, therefore we
write
µ1 (p, T, x) = T ln x + ψ(p, T ) + O(x) .
(3.19)
The consistency condition (3.17) immediately yields
ν(p, T ) = −T .
(3.20)
Taking into account that xN0 ≈ N1 we arrive at the expansion of the
Gibbs function
h
i
x
G(p, T, N0 , N1 ) = N0 µ0 (p, T, 0) + N1 T ln + ψ(p, T ) + · · · .
(3.21)
e
P
Since dG = −S dT + V dp + i µi dNi , the volume is obtained as
∂G
V =
∂p T,{Nj }
∂ψ(p, T )
∂µ0 (p, T, 0)
+ N1
∂p
∂p
= N0 v 0 + N 1 v 1 .
= N0
(3.22)
Here, v0 = ∂µ0 /∂p is the specific volume of the pure solvent and v1 = ∂ψ/∂p
the additional volume required by one molecule of the solute of small concentration. For the entropy we obtain
∂G
S = −
∂T p,{Nj }
x
∂ψ(p, T )
∂µ0 (p, T, 0)
− N1 ln − N1
∂T
e
∂T
= S(0) + Ssek + S(1) ,
= −N0
(3.23)
where S(0) = −N0 ∂µ0 /∂T is the entropy of the pure solvent; Ssek =
x
−N1 ln = −N0 ln(1 − x) − N1 ln x is the mixing entropy, when both come
pounds are considered ideal gases; finally S(1) = −N1 ∂ψ/∂T contains the
3.4. DILUTE SOLUTION, OSMOSIS
35
entropy of the solute and additional entropy due to, say, interactions between the molecules of the solute and the solvent.
Osmosis. If the free transport of the solute is blocked by a semipermeable membrane, a pressure difference, osmotic pressure is brought about
between both sides of the membrane. The osmotic pressure may easily be
measured by hydrostatic means (see Fig. 3–4). If the density of the solution
is nearly the same as that of the pure solvent, then ∆p = ρm gh, where ρm is
the density of the solvent and h is the height of the surface of the solution
above the surface of the solvent.
According to Chapter 2.8 at equilibrium the
chemical potential of the solvent must be the
same everywhere:
B
A
h
µ0 (pA , T, 0) = µ0 (pB , T, x) ,
pB
pA
where x is the concentration of the solute.
For small concentrations relations (3.18) and
(3.20) yield
Figure 3–4: Osmosis.
µ0 (pA , T, 0) = µ0 (pB , T, 0) − xT .
(3.24)
Due to the Maxwell relation (2.25c)
∂V
V
∂µ
=
=
= v,
∂p T,N
∂N T,p
N
where v is the specific volume. Therefore, we may approximate
µ0 (pB , T, 0) ≈ µ0 (pA , T, 0) + v0 ∆p ,
where v0 is the specific volume of the solvent. Combining with relation
(3.24) we arrive at the Van’t Hoff equation
∆p V = N1 T .
(3.25)
Here, ∆p = pB − pA is the osmotic pressure, and N1 = xN is the number of
solute molecules in the volume V .
The result is of the same form as the equation of state of the ideal gas.
Thus, osmosis causes in a dilute solution additional pressure following exactly the ideal gas law regardless of the nature of the solute, the solvent
and their interactions.
Osmotic pressures calculated by (3.25) may be significant, even orders
of magnitude larger (of order of a few MPa in biological applications) than
the ambient pressure (in natural conditions about 0.1 Mpa). This might
raise some suspicion about the applicability of relation (3.25), because its
derivation was based, among other assumptions, on the expansion of the
total pressure in the solution pB in powers of the osmotic pressure ∆p with
36
3. APPLICATIONS OF THERMODYNAMICS
only two leading terms retained. A closer inspection reveals, however, that
for liquid solutions the Van’t Hoff equation remains applicable also for osmotic pressures much larger than the ambient pressure. The point is that
the comparison of the remainder of the Taylor expansion
µ0 (pB , T, 0) = µ0 (pA , T, 0) + v0 ∆p +
1 ∂v0 (p∗ , T, 0)
∆p2 ,
2
∂p
where pA ≤ p∗ ≤ pB , with the second term v0 ∆p contains the factor
∂v (p∗ , T, 0)
, i.e. the isothermal compressibility, which for practiκT = − 0
v0 ∂p
cal liquid solvents is very small (for water, e.g., about 0.5 · 10−9 Pa−1 ). Thus,
the numerical smallness of the second derivative of the chemical potential
overweighs the relatively large values of the osmotic pressure ensuring the
applicability of Van’t Hoff equation (3.25) for fairly large osmotic pressures.
3.5 Chemical reaction
In the equation of a chemical reaction
0=
X
νj M j
(3.26)
j
integer-valued stoichiometric coefficients νj express the proportions in
which the amount of different species of molecules Mj change in the reaction. For instance, to the burning reaction
2H2 S + 3O2 −→
←− 2H2 O + 2SO2
(3.27)
corresponds the notation
−νA A − νB B −→
←− νC C + νD D ,
where A = H2 S, B = O2 , C = H2 O and D = SO2 with the stoichiometric
coefficients νA = −2, νB = −3, νC = 2 and νD = 2.
The progress of reaction is described by the degree of reaction (degree of
advancement ξ, whose differential is defined by 1
dNj = νj dξ .
(3.28)
Thus, when ξ is increased (decreased) by one exactly one reaction takes
place to the right (left) in the equation of reaction. The usual convention
is that ξ = 0, when the reaction is at its leftmost state, i.e. when one of
the compounds on the right-hand side is used up completely. With this
normalization ξ ≥ 0 always.
1
In chemistry the definition goes by number of moles instead of the number of
molecules used here.
3.5. CHEMICAL REACTION
37
Assume constant p and T . Consider the Gibbs function
X
µj Nj
G=
j
and its differential with the account of definition (3.28):
X
X
νj µj .
µj dNj = dξ
dG =
(3.29)
Define the unit change of the Gibbs function in the reaction as
X
∂G
∆r G ≡
νj µj = −A .
=
∂ξ p,T
j
(3.30)
j
j
With the opposite sign this is the affinity A. At constant (p, T ) the thermodynamic equilibrium corresponds to the minimum of G. Since the degree of
reaction ξ is the only independent dynamic variable [apart from (p, T )], the
equilibrium condition is
X
∆r Geq =
νj µeq
(3.31)
j = 0.
j
In a non-equilibrium state dG/dt < 0. If ∆r G > 0, then dξ/dt < 0 and the
reaction proceeds to the left.
In rarified gas or dilute solution in a passive solvent the ideal gas description is reasonable for the reacting compounds. Then
µj = T [φj (T ) + ln p + ln xj ] ,
(3.32)
where the most general form of the function φj is φj (T ) = µ0j /T − ζj − (1 +
1
2 fj ) ln T . Calculation yields
P Y
X
∆r G = T
νj φj (T ) + T ln p νj
xj νj .
(3.33)
j
The equilibrium condition (∆r G = 0) may now be cast in the form of the
law of mass action:
P
Y
(3.34)
xj νj = p− j νj K(T ) .
j
Here, the equilibrium constant
K(T ) = e−
P
j
νj φj (T )
,
(3.35)
of the reaction has been introduced. For reaction (3.27) the equilibrium
condition is
xC 2 xD 2
= pK(T ) .
xA 2 xB 3
Heat of reaction. This is the heat ∆r Q acquired by the reacting system
in one step to the right. Two classes of reactions are distinguished:
38
3. APPLICATIONS OF THERMODYNAMICS
∆r Q > 0: endothermic reaction,
∆r Q < 0: exothermic reaction.
In an isobaric process the amount of heat is equal to the change of enthalpy,
since ∆Q = ∆U + ∆W = ∆U + p ∆V = ∆(U + pV ) = ∆H. This can be
straightforwardly calculated with the use of the unit change of the Gibbs
function. According to relations (3.33) and (3.35)
X
νj ln(pxj ) .
(3.36)
∆r G = −T ln K(T ) + T
j
If the total amount of matter is unchanged, then in a reversible process
dG = −S dT + V dp. From this it follows that
1
V
G
G
S
H
V
G
= dG − 2 dT = −
dT + dp = − 2 dT + dp ,
+
d
2
T
T
T
T
T
T
T
T
since G = H − T S. In particular,
H = −T
2
∂
∂T
G
T
.
(3.37)
p,N
From expression (3.36) of ∆r G we obtain
∆r G
d
∂
=−
ln K(T ) ;
∂T
T
dT
where comparison goes between the temperature dependence of the Gibbs
function before and after one reaction step. The change of enthalpy, i.e. the
heat of reaction, is
d
ln K(T ) .
∆r H = T 2
(3.38)
dT
The heat of reaction is often put together with the reaction formula as
CH4 + 2O2 −→ CO2 + 2H2 O,
∆r H = −890.35 kJ/mol.
(3.39)
It is usually quoted per stoichiometric molar changes at the reference temperature 25◦ C = 298 K.
Example 3.1. One mole of H2 S and 2 moles of H2 O is gaseous state are
mixed at the pressure p and the temperature T , which results in the
chemical reaction
H2 S + 2H2 O ⇐⇒ 3H2 + SO2 .
Calculate concentrations xi of all the substances as well as the standard
free energy for the reaction ∆r G as functions of the degree of reaction
ξ. Find also the equation for ξ and the volume of the system at equilibrium. Use the ideal gas approximation.
Here, the species labels and stoichiometric coefficients are A = H2 S, B
= H2 O, C = H2 , D = SO2 , νA = −1, νB = −2, νC = 3 and νD = 1.
3.6. PHASE EQUILIBRIUM
39
Since dNi = νi dξ and ξ = 0, when only the left-hand compounds are
present, we may write Ni = νi + Ni0 , where Ni0 are the initial numbers of molecules. Taking into account the initial condition we arrive at
relations (N0 is the Avogadro number here):
NA = N0 − ξ ,
NB = 2N0 − 2ξ ,
NC = 3ξ ,
ND = ξ .
Since the total number of particles is N = NA +NB +NC +ND = 3N0 +ξ,
the sought concentrations are
xA =
N0 − ξ
,
3N0 + ξ
xB =
2N0 − 2ξ
,
3N0 + ξ
xC =
3ξ
,
3N0 + ξ
xD =
ξ
.
3N0 + ξ
In the ideal gas approximation µi = T [φi (T ) + ln p + ln xi ]. Thus
∆r G = T
X
P
νi φi (T ) + T ln p
i
+ T ln
N0 − ξ
3N0 + ξ
−1 i νi
+ T ln
Y
i
2N0 − 2ξ
3N0 + ξ
−2 xνi i = −T ln K(T ) + T ln p
3ξ
3N0 + ξ
3 ξ
3N0 + ξ
.
This yields
∆r G = T ln
27pξ 4
4K(T )(3N0 + ξ)(N0 − ξ)3
.
In equilibrium ∆r G = 0 and the equilibrium value ξ0 is found from the
relation
4K(T )
ξ04
=
.
27p
(3N0 + ξ0 )(N0 − ξ0 )3
Thus, in equilibrium pV = N T = (3N0 + ξ0 )T and
V = (3N0 + ξ0 )
T
.
p
3.6 Phase equilibrium
Phases are macroscopically different homogeneous equilibrium states of
matter. They may coexist. If the contact surface allows free transfer of
molecules in both directions, the usual equilibrium conditions (2.48) hold
(provided all interfaces are flat, curvature leads to pressure differences), so
that the pressure p and the temperature T are the same in all phases. Further, the chemical potential of each species is the same in all phases: for
each pair of phases (A, B)
µjA = µjB ,
(j = 1, . . . , H)
(3.40)
With H species taking into account the common values of the pressure and
the temperature this yields

µ1A (p, T, x1A , . . . , xhA ) = µ1B (p, T, x1B , . . . , xhB ) 

..
,
(3.41)
.


µHA (p, T, x1A , . . . , xhA ) = µHB (p, T, x1B , . . . , xhB )
40
3. APPLICATIONS OF THERMODYNAMICS
where h = H − 1. With F coexisting phases F − 1 independent constraints
for each species in different phases follow. Altogether H(F − 1) constraints
are imposed.
Each chemical potential in (3.40) depends, apart from (p, T ), on concentrations of the particles xjP . There are H − 1 independent concentrations
in each phase. Therefore we arrive at the result
variables M = (H − 1)F + 2 pieces,
constraints Y = H(F − 1) pieces.
To have solutions, we must have M ≥ Y . For the number of coexisting
phases F this yields the Gibbs phase rule,
(3.42)
F ≤ H + 2,
according to which, e.g., in a pure single-species substance at most three
phase may coexist at the triple point (no more freedom is left by the two
equilibrium conditions imposed on the two variables p and T ). The total
number of phases is not restricted by the Gibbs rule, however. For instance,
ice has several different phases under high pressure, all coexisting according to the Gibbs rule (see Fig. 3–6).
3.7 Phase transitions and diagrams
In Figs. 3–5 typical phase diagrams of the SVN system have been plotted
in (T, p) and (V, p) planes. In the (V, p) plane coexistence regions of solid,
liquid and gaseous phases occupy finite area, whereas in the (T, p) plane
the coexistence region is a curve.
p
p
kiinteä
C
C
neste
kiinteä
neste
kylläinen höyry
K
isotermejä
kaasu
koeksistenssi
kaasu
K
T
V
Figure 3–5: Phase diagrams of a liquid in (T, p) and (V, p) planes.
Typical features include the triple point K, where solid, liquid and
gaseous phases meet. For instance, for water (H2 O) on pK = 610 Pa, TK
= 0.01◦ C. The coexistence curve of liquid and gas ends at the critical point
3.7. PHASE TRANSITIONS AND DIAGRAMS
41
C; for water pc = 22 MPa, Tc = 374.15◦ C. Gas in the coexistence limit is
called saturated vapour.
Typically, the chemical potential µ = ∂G/∂N is continuous at crossing of the coexistence curve, but the rest of the partial derivatives of G,
S = −∂G/∂T and V = ∂G/∂p, do not necessarily share this property. If
some of these derivatives are discontinuous, the phase transition is of first
order. If all the first derivatives are continuous, but discontinuities (or
worse singularities) appear in the second order derivatives, then we are
dealing with a second order or continuous phase transition.
The jumps in the entropy and volume in a first-order transition are
∆S = −
∆V =
∂G
∂T
∂G
∂p
(2)
+
p
(2)
T
−
∂G
∂T
∂G
∂p
(1)
,
p
(1)
.
T
It should be borne in mind that the Gibbs functions describing different
phases are separately perfectly smooth functions of state variables in a
discontinuous transition, and the apparent discontinuity stems from the
change in the description from the Gibbs function of one phase to that of
the other phase depending on which is less at constant pressure and temperature.
When the coexistence curve is crossed T and p remain constant. In such
a process the amount of heat is equal to the change of the enthalpy (heat
function), since ∆Q = T ∆S = ∆U + p ∆V = ∆(U + pV ) = ∆H. This is the
latent heat of the phase transition (heat of fusion, vapourization etc.)
p (kbar)
30
1608 atm
17.78 K
VII
VIII
kiinteä hcp
20
kiinteät faasit
10
neste
III
V
-50
I
0
135.4 atm
3.14 K
kiinteä bcc
33.5 atm
VI
II
kiinteä fcc
p
50 T (C)
neste
28.92 atm
0.32 K
C
kaasu
1.15 atm
3.32 K
T
Figure 3–6: Examples of phase diagrams of real substances: (a) water
(H2 O), (b) 3 He.
42
3. APPLICATIONS OF THERMODYNAMICS
3.8 Coexistence
On the coexistence curve in the (T ,p)
plane µ1 (p, T ) = µ2 (p, T ). According to
the Gibbs-Duhem equation
dµ1
dµ2
dp
dT
coex
=
s2 − s1
1 ∆h
=
.,
v2 − v1
T ∆v
1
2
dp
S1
V1
dT +
dp ,
N1
N1
V2
S2
dp .
= − dT +
N2
N2
= −
dT
Along the coexistence curve this yields
the Clausius–Clapeyron equation
p
T
Figure 3–7: Coexistence curve
of two phases.
(3.43)
where s is the specific entropy, v the specific volume and h the specific enthalpy, so that ∆h is the specific latent heat of the phase transition.
If the phase 2 corresponds to higher temperature, then ∆h > 0, because
the discontinuous transition takes place at the crossing of the chemical potentials of different phases µ2 (T ) = µ1 (T ). If the phase 2 is stable at higher
temperature, then
∂µ2
∂µ1
<
,
∂T
∂T
so that s2 > s1 and ∆h > 0.
For liquid-vapour transition the Clausius-Clapeyron equation may readily be integrated. In this case obviously v2 ≫ v1 . Neglecting the specific volume of the liquid and using the ideal gas equation of state for the saturated
vapour we obtain
∆h
p∆h
dp
=
= 2 .
dT
T v2
T
For constant latent heat of vapourization this yields the equation of the
coexistence curve in the form
1
1
∆h
−
T0
T ,
p = p0 e
which yields the temperature dependence of the pressure of saturated
vapour. This is a reasonable approximation if the temperature (and pressure) differences are small. If not, then it might be better to assume that
the specific entropy in the gaseous phase is much larger than that in the
liquid phase: s2 ≫ s1 , and neglect s1 . With the subsequent substitution of
the perfect-gas entropy an integrable equation is obtained in this case as
well.
Example 3.2. Show that the boiling temperature of a liquid at constant
pressure becomes higher upon dissolution in it a small amount of any
3.8. COEXISTENCE
43
non-volatile solute. Derive for the change of the boiling temperature in
the limit of small concentration the result
δT = x
T2
,
∆h
where x is the molar fraction of the solute and ∆h the specific (per
molecule) heat of evaporation of the solvent.
The chemical potential of the liquid solvent in the limit of small concentration of the solute is
µl (p, T, x) ≈ µl (p, T, 0) − xT .
In equilibrium this must be equal to the chemical potential of the solvent in the gaseous phase µg (p, T ),
µl (p, T, 0) − xT = µg (p, T ) ,
(3.44)
which determines the boiling temperature as a function of then pressure T = T (p, x). The response of the boiling temperature to a small
addition of the solute is
δT =
∂T
∂x
x.
p
From condition (3.44) at constant pressure it follows
−sg dT = −sl dT − xdT − T dx
so that
∂T
∂x
=
p
T
T2
T
≈
,
=
sg − sl − x
sg − sl
∆h
which yields the desired relation.
Example 3.3. A pot of soup boils at 103◦ C at the bottom of a hill of
height 300 m and boils at 98◦ C at the top. What is the latent heat of
vapourization of the soup?
Small difference in boiling (coexistence) temperature, therefore the latent heat may be considered constant and the Clausius-Clapeyron equation used in the form (this is independent of the molar fraction of the
soup vapour in the air and probably worth checking):
p∆h
dp
=
.
dT
T2
In the present problem we need the dependence of the boiling temperature T on the height rather than on the pressure. The latter are related
from the mechanical equilibrium condition of a gas column as
dp
p
dz ,
= −ρg = −µg
dz
Tair
where µ is the average mass of a molecule in the air. Combining the two
equations yields the relation
−µg dz = ∆h
Tair dT
.
T2
44
3. APPLICATIONS OF THERMODYNAMICS
Thus,
M g∆zTtop Tbottom
= 8.1 kJ/mol ,
Tair (Tbottom − Ttop )
where NA is the Avogadro number, M the molar mass of air and the
temperature of the ambient air assumed to be Tair = 20◦ C.
∆H = NA ∆h =
3.9 Van der Waals equation of state
In terms of the specific volume v = V /N the van der Waals equation of state
contains only intensive quantities
a
p + 2 (v − b) = T .
(3.45)
v
For the pressure p we obtain
p=
a
T
− 2.
v−b v
(3.46)
In Fig. 3–8 the phase diagram brought about by the van der Waals
equation of state is sketched in the (v, p) plane (i.e. the curves depicted are
isotherms). The critical point corresponds to the infliction point:
∂p
= 0,
∂v
∂2p
= 0.
∂v 2
The former of these equations implies, in particular, an infinite compressibility at the critical point. This is an example of singular behaviour typical
of response functions at the critical point.
The coordinates of the critical point are readily found:

vc = 3b



8a 
Tc =
.
(3.47)
27b 

a 

pc =
27b2
In terms of dimensionless variables p = p/pc , T = T /Tc and v = v/vc the
van der Waals equation assumes the form of the law of corresponding states
3
p + 2 (3v − 1) = 8T .
(3.48)
v
without any explicit dependence on the material parameters a and b. This
is an illustration of universality, which is a generic feature of (especially
continuous) phase transitions: important properties may be expressed in
mathematical form fairly insensitive to physical details and nature of the
phase transition.
Coexistence region. Isotherms in the phase diagram of the (v, p) plane
are not monotonically decreasing at T < Tc . In Fig. 3–8 between the points
3.9. VAN DER WAALS EQUATION OF STATE
p
45
T>Tc
C
T=Tc
C
D
A
T< Tc
B
Nb
V
Figure 3–8: Phase diagram of the van der Waals matter.
B and C there is a rising portion with κT < 0 on. This means unstable
states and in the corresponding regions the isotherms are excluded from
the phase diagram.
The unstable region is surrounded by regions in which the matter is
in metastable states , which are stable against small fluctuations. Large
enough disturbances cause phase separation to liquid and gaseous phases
The region AB corresponds to superheated liquid, whereas the region CD
corresponds supercooled vapour.
Global thermodynamic equilibrium corresponds to states beyond these
regions on the isotherm bounded by the points A and D. In coexistence pA =
pD . To find this common pressure of the coexistence state the remaining
equilibrium condition
µA = µD .
may be used. According to the Gibbs–
Duhem equation dµ = −s dT + v dp ,
on an isotherm dµ = v dp. Integration
along the isotherm yields
µD − µA =
Z
D
v dp.
p
A
II
I
D
(3.49)
V
A
From the geometric construction of Figure 3–9:
Fig. 3–9 we infer
tion.
Z D
v dp = Area I - Area II
Maxwell construc(3.50)
A
– the difference of areas. The points A and D must thus be chosen such
that the areas I and II are the same. This is the Maxwell construction.
Example 3.4. For the van der Waals matter in the vicinity of the critical point:
46
3. APPLICATIONS OF THERMODYNAMICS
(a) Calculate the jump ∆n = n+ (p, T ) − n− (p, T ) in the number density n = N/V , where n± are the number densities of the liquid
and gaseous phases on the coexistence curve. In the notation
∆n ∝ (Tc − T )β , calculate the value of the critical exponent β.
Such non-analytic dependence on state variables is also a typical
feature near the critical point and the corresponding non-integer
powers (critical exponents) are important quantities in the description of critical phenomena.
(b) Construct the phase diagram in the (T, p) plane.
(c) Calculate the latent heat ∆H.
(a) Expand first the law of corresponding states in small deviations
from the critical values: p = 1 + π, T = 1 + τ and n = 1/v = 1 + ν
to obtain
3
3
(3.51)
π ≈ 4τ 1 + ν + ν 3 .
2
2
Since τ and π are equal in both phases, the equation of state (3.51)
yields the connection
4τ (ν1 − ν2 ) = ν23 − ν13 = (ν2 − ν1 ) ν22 + ν2 ν1 + ν12
whose obvious solution ν1 = ν2 is physically uninteresting, so that
we are left with
ν22 + ν2 ν1 + ν12 + 4τ = 0 .
(3.52)
Another relation is obtained by the Maxwell construction, whose
geometric content is easier to realize by integration over ν than
over π. In these terms the condition (3.50) means
Zν2
π dν ≈
0=
ν1
, ν2
4τ
ν1
ν+
3 2
ν
4
+
3 4
ν .
8
This condition is simplified by the use of the equation of state
yielding
3 4
4τ ν22 − ν12 =
ν1 − ν24 .
2
Excluding the uninteresting solution ν1 = ν2 we arrive at
3
4τ (ν2 + ν1 ) = − (ν2 + ν1 ) ν12 + ν22 .
2
Now the solution ν12 + ν22 = −8τ /3 with the account of equation
(3.52) yields the uninteresting solution ν1 = ν2 again and we are
left with ν1 = −ν2 . This finally yields the desired temperature
dependence (let ν1 > 0)
√
n1 − n2 = 2ν1 = 4 −τ = 4
r
Tc − T
,
Tc
withe the value of the critical exponent β = 12 .
3.10. PROBLEMS
47
(b) The result ν1 = −ν2 allows to write the explicit equation of the
coexistence curve near the critical point as
π = 4τ .
(c) The latent heat is found from the Clapeyron-Clausius equation
dp
∆h
=
dT
T (v2 − v1 )
as
∆h = T (v2 − v1 )
dp
Tc
≈
dT
2b
n1 − n2
n2c
r
= −6Tc
Tc − T
.
Tc
3.10 Problems
Problem 3.1. Assume that Earth’s atmosphere consists of an ideal gas
with a molecule’s mass m and that the acceleration of gravity g is a
constant.
(a) Show that as a function of altitude the pressure changes as
mg
dp
=−
dz .
p
T
(b) For an isothermal atmosphere, derive the barometric formula:
mgz p(z) = p0 exp −
T
.
(c) Assume then that temperature changes due to adiabatic expansion and show that in such an adiabatic atmosphere
p(z) = p0 1 −
mg
T0
1−
1
γ
z
γ
γ−1
.
Problem 3.2. Given the following expression for the chemical potential
of the ideal gas
µ(T, p) = µ0 + T −ζ −
f
+ 1 ln T + ln p ,
2
where µ0 and ζ are constants, and the equation of state pV = N T , calculate the thermodynamic potentials U (S, V, N ), H(S, p, N ), F (T, V, N ),
G(T, p, N ) and Ω(T, V, µ) as functions of their natural variables.
Problem 3.3. In two isolated tanks (of volumes V1 and V2 ) there is
the same amount (N particles in each tank) of the same ideal gas at
the same temperature, but at different pressures p1 and p2 . Find the
entropy change, when the tanks are joined (the wall separating them is
removed).
48
3. APPLICATIONS OF THERMODYNAMICS
Problem 3.4. Show that in a mixture of perfect gases the heat of reaction is
X
X
1
∆r H =
νj µ0j + T
νj 1 + fj .
2
j
j
Here, the term with µ0j accounts for the changes in the chemical energies (electronic binding energies) of the molecules.
Problem 3.5. Consider production of atomic hydrogen in the reaction
e + H+ ↔ H. Show that for the equilibrium concentrations the Saha
equation
I
ne nH+
≈ nQ exp −
,
nH
T
holds, where I is the ionization energy of the hydrogen and nQ a function with a weaker (than exponential) dependence on the temperature.
What is this function?
Problem 3.6. The humidity of the air is defined as the ratio of its
steam pressure and the pressure of the saturated steam (i.e. steam in
equilibrium coexistence with water). Let the humidity of the air at 20
◦
C be 50 %. Estimate its humidity in a sauna at 80 ◦ C.
Hints: The heat of vapourization of water at the pressure of 1 atm is
2.26 MJ/kg. Construct the pressure of the saturated steam p = p(T )
assuming the perfect gas laws for the steam and that both the specific
volume and specific entropy are much larger than that of the coexistent
water.
Problem 3.7. Consider superconducting matter. In the normal phase
the magnetization M is negligible, whereas in the superconducting
phase the magnetic induction B = 0 in the bulk. The phase transition takes place at a constant temperature T < Tc , when the magnetic
field strength becomes less than the critical value
"
Hc (T ) = H0 1 −
T
Tc
2 #
.
Draw the phase diagram in the (T,H) plane, calculate the change in the
Gibbs function, the latent heat and the change in the specific heat CH
in the phase transition. What is the order of the transition, if H = 0?
Hint: For a homogeneous sample assumed here, the differential of the
magnetic Gibbs function dG = −SdT − V BdH + µdN .
Problem 3.8. Calculate the entropy of the van der Waals gas as a
function of the temperature and volume.
Problem 3.9. Analyse the behaviour of the mechanical responses αp
and κT of a van der Waals gas near the critical point. Show, in particular, that on the isobar pb = p/pc = 1
αp ∼
8
+ ···
9Tc (vb − 1)2
κT ∼
2
+ ···
9pc (vb − 1)2
where vb = v/vc . Find also αp and κT as functions of the pressure on the
isochore vb = 1 in the gaseous phase, when pb > 1.
4. Classical phase space
4.1 Phase space and probability density
Phase space. The mechanical state of a classical many-particle system
(point-like particles, for simplicity) is completely described by the position
vectors and velocities of the particles {r i (t), v i (t)}, i = 1, . . . , N . In Hamiltonian mechanics generalized coordinates and momenta are used instead:
q = (q1 , . . . , qN d )
.
(4.1)
p = (p1 , . . . , pN d )
N particles in d dimensional space gives rise to 2N d variables. The phase
space is a 2N d dimensional space or manifold whose each point P = (q, p)
corresponds to a possible mechanical (microscopic) state of the system. During the evolution of the system the image point P (t) moves in the phase
space along a trajectory in a manner determined by the canonical Hamilton equations of motion:
dqi
dt
dpi
dt
∂H
∂pi
∂H
= −
∂qi
=
(4.2)
Here H ≡ H(q1 , . . . , qN d ; p1 , . . . , pN d ; t) =
H(P, t) is the Hamilton function. For timeindependent H, the trajectories are stationary. Except for some special points, the trajectories cannot intersect, branch or merge.
The temporal evolution of an arbitrary
function F (q, p, t) related to properties of
the system may be expressed using Poisson
brackets
X ∂F ∂G ∂G ∂F −
.
{F, G} ≡
∂qi ∂pi
∂qi ∂pi
i
(4.3)
P(t)
qi
pi
Figure 4–1: Trajectories in
a phase space.
When the argument point P (t) of F is the image point of the system moving
on a trajectory the total time derivative of F is
dF
∂F
=
+ {F, H} .
dt
∂t
49
(4.4)
50
4. CLASSICAL PHASE SPACE
Measure. For a probabilistic treatment a measure in the phase space dΓ
has to be defined. With N particles in d-dimensional space the choice is:
dΓ =
Nd
1 Y dqi dpi
1 −N d
=
h
dq1 . . . dqN d dp1 . . . dpN d .
N ! i=1 h
N!
(4.5)
The normalizing factor 1/N ! removes the degeneracy related to the permutation symmetry of identical particles (details in the next chapter). The
introduction of the Planck constant h = 6.62607 × 10−34 Js is based on the
quasi-classical correspondence of the phase space volume of h to one quantum state.
Ensemble. Calculation of macroscopic quantities from "first principles"
requires solution of the Hamiltonian equations of motion, which is not feasible in a large system, nor is even determination of the initial conditions
for a proper setup of the problem. A statistical approach is used instead.
A large number of different microscopic states give rise to the same macroscopic state. A statistical ensemble consists of different systems ("copies"
of the original microscopic system in various states allowed by conditions
defining the macrostate considered) in these microstates described by the
image points {P j ; j = 1, . . . , n} in the phase space. In the limit n → ∞
the distribution of the image points gives rise to a probability density ̺(P )
normalized as
Z
dΓ ̺(P ) = 1 .
(4.6)
nV
δS
Γ0
Figure 4–2: Derivation of
the continuity equation.
For any function in the phase space
f (P ) = f (q, p) a statistical average may be
then defined as the integral
Z
hf i = dΓ f (P ) ̺(P ) .
(4.7)
Equations of motion. Image points belonging to a statistical ensemble move according to the Hamiltonian equations of motion and thus their number remains the
same at all times. Introduce a velocity field
in the phase space as
V = (q̇, ṗ) .
(4.8)
R
Consider the ensemble measure Γ0 ̺ dΓ of a region Γ0 of the phase space at
the initial time instant. This is the number of image points in Γ0 . During
the evolution of the ensemble the image points move which results in the
change of both the probability density ̺ and the region Γ0 they occupy. The
change of the region brings about a local change of its volume. This is given
by the volume of an oblique cylinder with the basis on a surface element
4.1. PHASE SPACE AND PROBABILITY DENSITY
51
dS of Γ0 and spanned by the trajectories starting at this surface element.
The direction of the axis of the cylinder formed during the movement of the
image points of the surface element is thus given by V and its height during
a short period ∆t by ∆h = V · n∆t, where n is the unit outward normal
vector to the surface element dS. Thus, the total change of the ensemble
measure of the region Γ0 may be written as a sum of contributions from the
change of ̺ and the change of Γ0 as
Z
Z
Z
d
∂̺
dΓ +
̺ dΓ =
V ̺ · ndS .
dt Γ0
∂Γ0
Γ0 ∂t
R
R
Gauss’ theorem yields ∂Γ0 dS · V ̺ = Γ0 dΓ ∇ · (V ̺). Therefore, due to
arbitrariness of Γ0 we arrive at the continuity equation
∂̺
+ ∇ · (V ̺) = 0 ,
∂t
or, in more detail,
∂
∂̺(P, t) X ∂
+
(q̇i ̺) +
(ṗi ̺) = 0 .
∂t
∂qi
∂pi
i
(4.9)
From the Hamilton equations q̇i = ∂H/∂pi , ṗi = −∂H/∂qi it follows that
∂ q̇i
∂ ṗi
+
= 0,
∂qi
∂pi
(4.10)
which – summed over i – yields ∇·V = 0. Substitution of ”incompressibility
condition” (4.10) in continuity equation (4.9) leads to
∂̺ X
∂̺
∂̺
q̇i
+
+ ṗi
= 0,
(4.11)
∂t
∂qi
∂pi
i
or in a shorthand form ∂̺/∂t + V · ∇̺ = 0. Introducing the convective time
derivative
d
dt
≡
=
∂
+V ·∇
∂t
X ∂
∂
∂
q̇i
+
+ ṗi
∂t
∂qi
∂pi
i
(4.12)
we may now cast (4.11) in a compact form known as the Liouville theorem:
d
̺ P (t), t = 0 .
dt
(4.13)
Thus, apart from conservation of the probability measure in the phase
space, the probability density is also conserved on a trajectory.
52
4. CLASSICAL PHASE SPACE
Substitution of the Hamilton equations q̇i = ∂H/∂pi , ṗi = −∂H/∂qi leads
to the Liouville equation
∂̺
i
= L̺ ,
(4.14)
∂t
for the probability density, where L is the Liouville operator
X ∂H ∂
∂H ∂
L = i{H, } ≡ i
−
.
∂qj ∂pj
∂pj ∂qj
j
(4.15)
The Liouville equation (4.14) is not easier to solve than the Hamilton equations of motion, therefore other ideas are needed to construct methods for
calculation of macroscopic properties in systems with large number of particles. What is done in practice is that a stationary solution of the Liouville
equation is sought which leads to the condition
{̺, H} = 0 .
It is also desirable that physically independent subsystems are statistically
independent, which means factorization of the probability density in the
fashion ̺(1, 2) = ̺(1)̺(2), where indices are shorthand for variables corresponding to two such subsystems. From this it follows that ln ̺ should be
an extensive quantity and thus a linear combination of additive integrals
of motion. In mechanical systems there are seven such integrals of motion:
the Hamilton function H, the total momentum P and the total angular momentum L. The last six are usually excluded from the list of integrals of
motion by putting the system in a box and thus only the Hamilton function
is left for the construction of the density function so that ̺ = ̺(H). Different arguments, some of which are presented below, are then used to choose
a suitable and reasonable density function.
4.2 Flow in phase space
Hamiltonian dynamics gives rise to flow of image points in the phase space
resembling the flow of an incompressible fluid.
An energy surface ΓE is a (2N d − 1) dimensional manifold determined
by the condition H(q, p) = E. Its measure or surface area may be expressed
as
Z
Z
ΣE ≡ dΓ δ H(P ) − E = dΓE .
(4.16)
When energy is conserved, image points flow on this surface.
Applicability of the statistical approach depends crucially on the character of the flow in the sense how thoroughly the whole energy surface is
covered. In this respect the conventional classification includes ergodic and
non-ergodic flows (or corresponding systems) illustrated in Fig. 4–4.
Non-ergodic flows. An initial area element∆ΓE of the energy surface
explores only a part of the energy surface ΓE . This is the case, e.g., for
4.2. FLOW IN PHASE SPACE
53
periodic motion and integrable systems allowing complete solution in terms
of action-angle variables.
Figure 4–4: Flow in phase space: (a) non-ergodic, (b) ergodic, not mixing,
(c) ergodic and mixing.
Γ
E + ∆E
E
Ergodix flow. Almost every point
of the surface ΓE approaches any other
point of ΓE arbitrarily close in the
long run. Mathematically this may be
stated as equality of temporal and statistical averages
∆h = ∆E
∇H
f = hf iE
Figure 4–3: Constant energy surfaces in phase space.
(4.17) the time average is
1
T →∞ T
f ≡ lim
(4.17)
for any smooth enough function f (P )
on the energy surface ΓE and almost
all starting points P (t = 0). In relation
Z
0
T
dt f P (t)
and the statistical average on the energy surface
Z
1
dΓE f (P ) .
hf iE ≡
ΣE
(4.18)
(4.19)
The microcanonical ensemble is defined by the density function
̺E (P ) =
1
δ H(P ) − E ,
ΣE
(4.20)
which allows to express expectation values on the energy surface as ensemble averages in the microcanonical ensemble:
Z
hf iE = dΓ f (P ) ̺E (P ) .
54
4. CLASSICAL PHASE SPACE
Mixing flow. This is a special type of ergodic flow in which the image points of a small element dΓE of the energy surface are dispersed in a
nearly uniform distribution over the whole energy surface. For arbitrary
non-stationary density ̺E (P, t) on the energy surface ΓE and any smooth
enough function f (P )
Z
lim hf i ≡ lim
dΓ ̺E (P, t) f (P )
t→∞
t→∞
Z
1
=
dΓ δ H(P ) − E f (P )
ΣE
= hf iE .
(4.21)
Thus, in a mixing flow an arbitrary probability density describing a nonequilibrium state evolves towards the microcanonical ensemble.
Ergodic theory. Ergodic theory analyzes the character of flow in phase
space. For instance, it has been rigorously proved that a two-dimensional
gas of hard discs is ergodic. It is also physically plausible that dynamics of
the usual three-dimensional fluids is ergodic in the framework of classical
mechanics, although no rigorous proof exists. Although ergodic theory is
an exiting branch of mathematics, its value for physics is rather limited.
Statistical mechanics is usually applied to physical systems which rarely
are truly isolated and therefore hardly exhibit non-ergodic behaviour. One
could also say that statistical mechanics deals with ergodic systems only.
4.3 Microcanonical ensemble and entropy
One of the basic problems in statistical physics is to find an ensemble or
density function ̺ giving the correct description of a macroscopic system.
If the energy of a macrosopic ergodic system is fixed, it may be described
– as shown in the preceding section – by the microcanonical ensemble with
the density
1
δ H(P ) − E .
(4.22)
̺E (P ) =
ΣE
The δ function may be technically inconvenient, therefore an alternative
definition may be given as a uniform distribution in an energy shell of thickness ∆E:
̺E,∆E (P ) =
1 θ E + ∆E − H(P ) − θ E − H(P ) .
ZE,∆E
(4.23)
Here, the step function

 1
1/2
θ(x) =

0
,
,
,
x>0
x=0
x<0
(4.24)
4.3. MICROCANONICAL ENSEMBLE AND ENTROPY
55
has been used to confine the probability density to the interval E ≤ H ≤
E + ∆E. In
R this region the density is 1/ZE,∆E , and from the normalization
condition dΓ ̺ = 1 it follows that
Z
ZE,∆E = dΓ [θ(E + ∆E − H) − θ(E − H)] .
(4.25)
The normalizing constant ZE,∆E is the microcanonical statistical sum or
the partition function expressing the number of states in the energy shell of
the phase space (according to the quasiclassical normalization that ∆0 Γ = 1
corresponds to one state). Thus, in the microcanonical ensemble the probability density is distributed completely evenly in the allowed part of the
phase space, i.e. between the ZE,∆E states.
Example 4.1. Calculate the microcanonical partition function for a
system of N free point particles of mass m.
In general, the microcanonical partition function is difficult to calculate.
The ideal gas is one of the rare easily calculable cases. The energy shell
is defined by the inequalities
E≤
N
X
p2i
i=1
2m
= H ≤ E + ∆E ,
√
which
is geometrically a spherical shell with the radii 2mE and
p
2m(E + ∆E) in the 3N dimensional momentum space of the system.
The volume
√ of such a (thin) shell is the surface area of the sphere of radius, say, 2mE inpa 3N dimensional space multiplied by the thickness
of the shell: ∆p ≈ 2m/E∆E/2. The surface area of a sphere of radius
r in a d-dimensional space is
d
Sd =
2π 2 rd−1
,
Γ d2
where Γ(z) is the Gamma function. The coordinate part gives for each
particle the volume V of the box in which the system is enclosed. Thus
(remember the measure) the partition function – with the degeneracy
factor 1/N ! included – is
"
ZE,∆E =
3
3
π 2 V (2mE) 2
h3
#N
∆E
.
N ! Γ 3N
E
2
Entropy. In the statistical theory of Gibbs the density ̺ corresponding
to a macroscopic state may be derived with the aid of a variational principle.
To this end the statistical entropy S[̺] is defined as the following functional
of the density,
Z
S = − dΓ ̺(P ) ln ̺(P ) ,
(4.26)
56
4. CLASSICAL PHASE SPACE
and a variational principle is set requiring the physical density ̺ to maximize the entropy.
For the uniform distribution (4.23) this is
S = ln ZE,∆E = ln W ,
(4.27)
which is the Boltzmann entropy,, where W is the statistical weight of the
macroscopic state i.e. the number of all microscopic states compatible with
the macroscopic boundary conditions.
Derivation of the microcanonical ensemble. Let ∆ΓE be the part of
the phase space with energy in the interval (E, E + ∆E). We seek density
function ̺, which maximizes the statistical entropy (4.26). The variation of
entropy
Z
Z
δS = −
dΓ(δ̺ ln ̺ + ̺ δ ln ̺) = −
dΓ δ̺(ln ̺ + 1)
∆ΓE
∆ΓE
must vanish at an extremum. The constraint
Z
δ1 =
dΓ δ̺ = 0
∆ΓE
forces to look for a conditional extremum. This may be done with the use of
a Lagrange multiplier λ, which leads to the condition
ln ̺ + 1 + λ = 0 ,
with the solution
̺(P ) = ̺0 ,
(P ∈ ∆ΓE ).
(4.28)
Thus, the result is a constant probability density in the energy slice – just
as in the microcanonical ensemble.
Due to linearity of the normalization condition in ̺, the second variation
of the entropy may be calculated as if the variation δρ were completely free,
so that
Z
Z
1
1
∂2
1
δ2 S = −
dΓ (δ̺)2 2 (̺ ln ̺) = −
dΓ (δ̺)2 ≤ 0 ,
2!
∂̺
2 ∆ΓE
̺
∆ΓE
which means that the extremum of S is indeed a maximum.
The normalization condition yields ̺ = 1/ZE,∆E in the energy slice, and
substitution in the expression for entropy yields
Z
1
1
S=−
dΓ ln = ln Z = ln W .
Z
Z
∆ΓE
This is the Boltzmann entropy revisited.
Proof of additivity. Consider a system S consisting of two statistically
uncorrelated subsystems S1 and S2 . Let the corresponding phase spaces be
4.4. PROBLEMS
57
Γ1 and Γ2 . The phase space of the combined system is Γ1+2 = Γ1 ⊗ Γ2 with
the volume element dΓ1+2 = dΓ1 dΓ2 . Due to statistical independence the
probability density is factorized
̺1+2 = ̺1 ̺2 ,
and the normalization assumes the form
Z
Z
Z
dΓ1+2 ̺1+2 = dΓ1 ̺1 dΓ2 ̺2 = 1 .
According to Gibbs’ formula the entropy of the combined system is
Z
S1+2 = − dΓ1+2 ̺1+2 ln ̺1+2
ZZ
= −
dΓ1 dΓ2 ̺1 ̺2 (ln ̺1 + ln ̺2 )
Z
Z
= − dΓ1 ̺1 ln ̺1 − dΓ2 ̺2 ln ̺2 .
This is exactly the desired result:
(4.29)
S1+2 = S1 + S2 .
Thus, entropy is an additive quantity for weakly enough interacting systems. No state of equilibrium in subsystems or between then was assumed.
4.4 Problems
Problem 4.1. Write down the evolution equation for the density function ρ(t, p1 , q1 , p2 , q2 ) of coupled one-dimensional harmonic oscillators
with the Hamilton function
H=
p2
k
p21
+ 2 + (q1 − q2 )2
2m
2m
2
and find the solution with the initial condition
ρ(0, p1 , q1 , p2 , q2 ) = Z −1 exp −β
p2
p21
+ 2
2m
2m
(unnormalized, Z is the normalization constant and β a positive parameter).
Problem 4.2. Calculate the partition function ZE,∆E of the classical
microcanonical density function for
(a) N non-interacting atoms,
(b) N independent harmonic oscillators.
Hint. Volume of a d-dimensional ball of radius R:
Vd =
π d/2 Rd
.
Γ(d/2 + 1)
58
4. CLASSICAL PHASE SPACE
Problem 4.3. Using the microcanonical definition of entropy S =
ln ω(E) calculate the heat capacity of a system of N independent harmonic oscillators.
Hint. Use the microcanonical definition of temperature to find E =
E(T ) first.
5. Quantum-mechanical ensembles
5.1 Density operator and entropy
The complete description of the state of a system in quantum mechanics is
given by the wave function (i.e. the solution of the Schrödinger equation
generated by the Hamilton operator of the system conforming to given initial and boundary conditions) in a suitable Hilbert space. In the case of a
large system (but often not only in that case) this level of description is practically impossible. Since the quantum-mechanical description is inherently
probabilistic, therefore, in contrast to classical mechanics, no new tools are
actually needed for the statistical treatment of systems with large numbers
of degrees of freedom, no new tools. The probabilistic measure required for
the statistical description is provided by the density operator introduced in
quantum mechanics for incomplete description of physical systems.
Ensemble. The notion of a statistical ensemble is similar to that in classical statistics: it is formed by copies of the system in different microstates
corresponding to the same macrostate. A microstate is determined by a
wave function.
Pure state. If the wave function Ψ is known, the density operator is the
projection operator
̺ = |ΨihΨ| .
(5.1)
This is a pure state. It corresponds to an ensemble in which each state is
the same |Ψi (modulo the phase factor). Statistical mechanics of the pure
state reduces to the usual quantum mechanics, in which, e.g., expectation
values are hAi = Tr ̺A = hΨ|A|Ψi.
Mixed state. Basic properties of the density operator in the general
case (”mixed state”) may be inferred by considering a the wave function
Ψ(x, y) of a large system with a weak coupling between subsystems whose
variables are labeled as x and y. Weak coupling means that Htot = H + H ′ ,
where H does not act on the y variables and H ′ on the x variables. Due
to symmetry (antisymmetry) properties imposed on the wave functions the
wave function of the large system does not factorize to a product of wave
functions of the subsystems, however.
In this case the density matrix (density operator in the coordinate basis)
for the x-labeled subsystem may be defined as
Z
̺(x, x′ ) = dy hx, y|ΨihΨ|x′ , yi, .
(5.2)
59
60
5. QUANTUM-MECHANICAL ENSEMBLES
It is immediately seen that the density matrix is hermitian, positive definite and normalized as Tr̺ = 1. It is often convenient to express
the density matrix using the projection operators
to eigenfunctions of
P
Cn (y)hx|ni and ̺(x, x′ ) =
H: let H|ni = En |ni, then hx, y|Ψi =
n
PR
∗
dy Cn (y)Cm
(y)hx|nihm|x′ i from which it follows that
n,m
̺=
XZ
∗
dy Cn (y)Cm
(y)|nihm| .
(5.3)
n,m
The expectation value of an operator A acting on the x-labeled subsystem
only may be calculated as
XZ
∗
dy Cn (y)Cm
hAi = Tr ̺A =
(y)hm|A|ni .
(5.4)
n,m
Properties of the density operator. Obvious requirements for the
density operator are positivity, normalization and self-adjointedness:
̺ = ̺†
hΨ|̺|Ψi ≥ 0,
∀ |Ψi ∈ H
Tr ̺ = 1
(5.5)
Since the operator ̺ is Hermitian, it possesses a complete set of eigenfunctions in the basis of whose
X
̺=
pα |αihα| .
(5.6)
α
Here, the standard normalization
hα|βi = δα,β is implied. The eigenvalues
P
obey 0 ≤ pα ≤ 1 and α pα = 1. An expectation value in this basis assumes
the form
X
pα hα|A|αi .
(5.7)
hAi = Tr ̺A =
α
Equation of motion of the density operator. Taking the time derivative of the density operator (5.2) and using the fact that hx, y|Ψi is a wave
function, i.e. i~∂t hx, y|Ψi = (H + H ′ ) hx, y|Ψi we obtain
i~
∂̺(x, x′ ) X
=
∂t
n,m
Z
∗
dy Cm
(y)hm|x′ i (H + H ′ ) Cn (y)hx|ni
∗
− Cn (y)hx|ni (H + H ′ ) Cm
(y)hm|x′ i , (5.8)
where operators act on functions on functions to the right. Since H ′ acts
only on variables y and is a hermitian operator, the contributions containing H ′ cancel. The operator H is acting on its eigenfunctions in the coordinate representation, therefore Hhx|ni = hx|H|ni and Hhm|x′ i = hm|H|x′ i.
5.1. DENSITY OPERATOR AND ENTROPY
61
Thus
∂̺(x, x′ ) X
=
i~
∂t
n,m
Z
∗
dy Cn (y)Cm
(y) hx|H|nihm|x′ i − hx|nihm|H|x′ i .
In the operator form this the von Neumann equation
i~
d
̺(t) = [H, ̺(t)]
dt
(5.9)
which plays the same role in quantum statistics as the Liouville equation
in classical statistics. It gives a principal possibility to find the density operator as the solution this equation together with suitable initial condition
and normalization.
Stationary ensemble. In a stationary ensemble all expectation values
are time-independent. This implies a time-independent density operator:
̺˙ = 0. Then [̺, H] = 0, which is possible, e.g., if ̺ is a function of H:
̺ = ̺(H). In a stationary ensemble the density operator and the Hamilton
operator also possess common eigenstates. Therefore, in the basis of these
eigenstates |ni the density operator is diagonal
X
|nipn hn|
(5.10)
̺=
n
and the eigenvalue pn may be prescribed the meaning of the probability to
observe the system in the eigenstate |ni (this is a genuine wave function
here, i.e. a solution of the Schrödinger equation with H). The ensemble
expectation values are now
X
hAi = Tr ̺A =
pn hn|A|ni .
(5.11)
n
Here, hn|A|ni is the quantum-mechanical expectation value of the observable A in the quantum state |ni.
It should be noted, however, that the expectation value (5.11) is different from that of in a pure state |Ψi, which
P also may be expressed as a linear
combination of the basis vectors |Ψi = n an |ni. Here, |an |2 is the probability to observe the system in the state |ni in the usual quantum-mechanical
sense. The expectation values in the pure state are
X
X
hΨ|A|Ψi =
|an |2 hn|A|ni +
a∗m an hm|A|ni
(5.12)
n
n6=m
and we see that in the density-operator description (5.11) the interference
terms of the pure ensemble are absent.
Many-particle systems. It is convenient to express wave functions of
many-particle systems as linear combinations of products of one-particle
wave functions of the form
Ψℓ1 (ξ1 )Ψℓ2 (ξ2 ) · · · ΨℓN (ξN ) ,
62
5. QUANTUM-MECHANICAL ENSEMBLES
where ξi = (xi , si ) (si is the spin of the particle) and Ψn are eigenfunctions of the one-particle Hamiltonian enumerated by the quantum number
ℓ. This allows to introduce the occupation number representation for identical particles: the number of occurrences nℓ of one-particle functions of a
particular quantum number ℓ is the number of particles in this state. This
is an illustrative way of speaking only: in a many-particle quantum system
the very notion of one particle in a given state is actually meaningless.
For identical particles symmetry conditions are imposed on the physical wave functions. The particles are either bosons of fermions. The former obey Bose–Einstein statistics and the wave function must be completely symmetric with respect to all permutations of the particle arguments. The latter obey Fermi–Dirac statistics, requiring complete antisymmetry (change of sign) in any permutation of a pair of particle arguments
of the wave functions. These requirements have a profound effect on the
statistical mechanics.
Fock space. From the point of view of statistical physics, the quantum
mechanics of many-particle systems is most conveniently formulated in the
Fock space, which allows to treat systems with variable number of particles.
The Fock space is a direct sum of all properly (anti)symmetrized N -particle
Hilbert spaces. In the coordinate representation the elements of its basis
may be written as row vectors
Φ = (ζ, ψ1 (ξ1 ), ψ2 (ξ1 , ξ2 ), . . . , ψN (ξ1 , . . . , ξN ), . . .) ,
where ζ is a complex number, ξi = (xi , si ) and ψN (ξ1 , . . . , ξN ) is a completely
(anti)symmetric function.
The normalized basis elements for N -particle states may be written as
|{nℓ }i ≡ |n1 , . . . , nℓ , . . .i,
where n1 , . . . , nℓ are the occupation numbers P
of the one-particle states {ℓ} =
{1, 2, 3, . . .}. In an N -particle state obviously ℓ nℓ = N . Let ℓ1 , ℓ2 , . . . , ℓN be
labels of one-particle states. The coordinate representation of a basis vector
is then
rQ
X
ℓ nℓ !
εP hξ1 |ℓ1 i · · · hξN |ℓN i ,
(5.13)
hξ1 , . . . , ξN |{nℓ }i =
N!
P (ξ1 ,...,ξN )
where the summation goes over all N ! permutations of the argument permutations and εP is a sign factor, which for bosons is always 1, whereas
for fermions ±1 according to the parity of the permutation (in the fermionic
case this is a N × N Slater determinant).
The normalization factor in (5.13) may by calculated by the traditional
”box-filling” procedure, in which ”boxes” corresponding to one-particle
states are filled by enumerated balls in all possible ways to count the number of linearly independent terms in the sum on the right-hand side of
5.2. DENSITY OF STATES
63
(5.13). Let us start by putting a ball in the first box. There are N possibilities to do this. For the second ball there are N − 1 possibilities for all
initial N cases, thus N (N −1). After having the required number n1 of balls
in the first box, we notice that we have the counted each different term n1 !
times, since the order of casting the balls does not make any difference.
Restore the correct relative weight of each term by dividing by n1 !. So far
N (N − 1) . . . (N − n1 + 1)/n1 ! different terms with n1 particles have been
obtained. For the second box, the starting number of possibilities is N − n1 .
After completing the filling of all boxes, we arrive at the number of
X
N!
nℓ = N
,
n1 !n2 ! · · · nℓ ! · · ·
ℓ
of linearly independent terms involved in the basis vector (5.13).
Entropy. Statistical entropy related to the ensemble described by ̺ is
defined in analogy with the classical Gibbs entropy as a functional of ̺:
S = −Tr ̺ ln ̺ .
(5.14)
In the basis, in which ̺ is diagonal, equation (5.6), in terms of eigenvalues
pα of ̺ we may write
X
pα ln pα .
(5.15)
S=−
α
The relation between the statistical entropy and the thermodynamic entropy requires separate analysis, that they may be identified is not at all
self-evident.
5.2 Density of states
Let the spectrum and eigenfunctions of the Hamiltonian be
X
En |nihn| .
H|ni = En |ni, H =
(5.16)
n
In a finite volume V the spectrum of H is completely discrete. In the analysis of condensed matter or macroscopic bodies passing to thermodynamic
limit V → ∞, N → ∞ with fixed the number density N/V is eventually
implied leading to practically continuous spectrum.
An important quantity in the analysis of the energy spectrum is the
cumulative distribution function of states:
X
θ(E − En ) .
(5.17)
J(E) =
n
At the point E + 0+ it yields the number of states with energies ≤ E. Since
dθ(x)/dx = δ(x), the density of states may be expressed as
ω(E) =
dJ(E) X
δ(E − En ) .
=
dE
n
(5.18)
64
5. QUANTUM-MECHANICAL ENSEMBLES
The difference J(E + ∆E + 0+ ) − J(E − 0+ ) ≈ ω(E) ∆E is the number of
states with energies in the interval [E, E + ∆E].
In large systems the energy spectrum is dense, due to which a slight
coarse-graining leads to ω(E) and J(E) which are smooth functions of E.
At low temperatures small values of E are important and usually a detailed knowledge of the structure of the spectrum near the ground state is
required.
Example 5.1. Free particle. The Hamilton function is H = p2 /2m. Eigenstates may be conveniently enumerated with the periodic boundary conditions in a cubic box of volume V = L3 . The eigenfunctions are plane waves
1
ψk (r) = √ eik·r ,
V
(5.19)
2π
(nx , ny , nz ) ;
L
with integer nj . The energy of the particle is
(5.20)
where
k=
εk =
p2
~2 k 2
=
.
2m
2m
(5.21)
Density of states in the k space is (L/2π)3 , and with the account of possible
spin degeneracy g = 2S +1 in the limit of large box the ”smoothed” summing
rule follows
Z
Z
X
XZ
V
V
3
(5.22)
d
k
·
·
·
=
g
d3 p · · · ,
··· ⇒
dNk · · · = g
3
3
(2π)
h
σ
kσ
with the last expression in terms of the momentum p = ~k. The ”smoothed”
cumulative distribution function of states is
XZ
p2
dNk θ E −
J1 (E) =
2m
σ
Z p
V 4π 3
V
dp′ p′2 = g 3
p .
(5.23)
= g 3 4π
h
h 3
0
In terms of the energy E = p2 /2m in the continuum limit (V → ∞) we
arrive at
)
J1 (E) = 32 C1 V E 3/2
,
(5.24)
√
ω1 (E) = C1 V E
where the constant C1 is
C1 = 2πg
2m
h2
3/2
.
(5.25)
5.2. DENSITY OF STATES
65
kz
ω 1 (E )
dk
ky
kx
E
Figure 5–1: (a) one-particle states in the wave-vector space, (b) smoothed
density of states.
Example 5.2. Maxwell–Boltzmann gas. Consider now a system of N free
particles in a box of volume V = L3 with periodic boundary conditions. If
quantum-mechanical conditions on occupation numbers are neglected, then
every particle may occupy any momentum state independently of others.
The energy of the system is
E=
N
X
j
=
p2j
,
2m
and the smoothed cumulative
√ distribution function of states is given as the
volume of a ball of radius 2mE in the 3N -dimensional momentum space
√
3N
1 N N π 3N/2 2mE
g V
.
JN (E) =
N!
h3N Γ(3N/2 + 1)
(5.26)
The factor N ! removes approximately the permutation degeneracy of N particles. This is an acceptable approximation, when the occupation numbers
of the one-particle states are small, i.e. in a dilute gas. Indeed, the counting
of the number of states by the volume of a ball in the wave-vector space regards each combination of wave-numbers as the quantum numbers of Q
a separate state. Physical states, however, are linear combinations of N !/ ℓ nℓ !
linearly independent products of one-particle wave functions which means
a huge overcounting of physical states in the approximation we have used.
This overcounting may be removed in a simple way only in the limit, when
the degeneracy is the same for the overwhelming majority of all physical
states, which is the case, when almost all occupation numbers are either 0
or 1. Then the overcounting is N ! fold and we arrive at the ”quasiclassical”
expression (5.26) for the cumulative distribution function of states.
Differentiating with respect to E we obtain the density of states
ωN (E) =
gN V N
N !Γ( 32 N )
2mπ
h2
3N/2
E 3N/2−1 =
(C2 V )N E 3N/2−1
.
N !Γ( 23 N )
(5.27)
66
5. QUANTUM-MECHANICAL ENSEMBLES
5.3 Energy, entropy and temperature
The density operator of a statistical ensemble allows to calculate all properties of the system, e.g. the expectation value of energy and the entropy.
For a connection with thermodynamics the temperature of an equilibrium
state has to be defined. To this end, introduce the microcanonical ensemble, whose density operator may be built requiring the energy of the system be in the interval [E, E + ∆E] and maximizing the statistical entropy
with this constraint. This is most transparently carried out in the energyeigenfunction basis, where the quantities to be varied are functions. Therefore, the result is similar to that of the classical case 1
pn = p(En ) =
1 θ E + ∆E − En − θ E − En .
ZE
(5.28)
The normalization factor, the microcanonical partition function is
ZE = J(E + ∆E) − J(E) ,
i.e. (literally) the number of states with energies in the interval [E, E +∆E].
For small ∆E we may write ZE ≈ ω(E)∆E. For practical purposes
SE = ln ω(E) ,
(5.29)
because only this term is extensive in a large system. The term brought
about by the energy resolution ln ∆E is not extensive and numerically negligible regardless of the macrophysical choice of ∆E.
Definition of temperature. We would like to identify the statistical
entropy with that of the thermodynamics. The guiding relation is the differential form of the first law
dU = dE = T dS − p dV + µ dN
which for constant V, N yields
∂S
∂E
V,N
=
1
.
T
(5.30)
The temperature that has to be prescribed to the equilibrium system described by the microcanonical ensemble depends – according to relations
(5.29) and (5.30) – unambiguously on the density of states corresponding to
the energy E:
1
∂
≡
ln ω(E, V, N ) .
(5.31)
T
∂E
1
The width of the energy window ∆E is omitted here as a parameter because it
turns out to be irrelevant.
5.4. PROBLEMS
67
The density of states ω depends naturally also on the variables V and N
determining the Hilbert space. With the notation
β=
1
T
(5.32)
we arrive at the result β = ∂ ln ω/∂E, i.e. β is, according to Fig. 5–2, the
slope of the curve ln ω(E).
ln ω(E)
α
tan α = β
E
Figure 5–2: Logarithm of the density of states as a function of energy. The
slope of the curve is β = 1/T .
Equilibrium thermodynamics of a macroscopic system may thus be derived from the density of states of the system ω(E, V, N ). This is tantamount to carrying out calculations in the microcanonical ensemble, which
in the thermodynamic sense corresponds to the isolated system. In many
cases, however, calculation of the density of states is very difficult or even
impossible with fixed E and N and other ensembles are better suited for
practical use.
5.4 Problems
Problem 5.1. Consider a physical system described in a 3-dimensional
Hilbert space. Probabilities to observe the system in the states
2
3
1
|ψ1 i = 4 0 5 ,
0
2
3
0
1
|ψ2 i = √ 4 1 5 ,
2
1
2
3
0
1
|ψ3 i = √ 4 −1 5
2
1
are 1/2, 1/3 and 1/6, respectively. Calculate the 3 × 3 density matrix of
the system. Show that Tr ̺2 < Tr ̺ = 1.
Problem 5.2. Calculate the microcanonical density matrix ρ(x, x′ ) of a
one-dimensional free particle described by plane waves in the interval
− L2 ≤ x ≤ L2 with periodic boundary conditions. The energy shell is
defined by the inequalities
2ℏ2 π 2 (N + ∆N )2
2ℏ2 π 2 N 2
≤ En ≤
.
2
mL
mL2
68
5. QUANTUM-MECHANICAL ENSEMBLES
Problem 5.3. A large quantum system consists of N non-identical independent particles each of which may occupy one of two states with
energies 0 and ǫ. Energy eigenstates may thus be enumerated by the
sequences ν = (n1 , n2 , . . . , nN ), where ni = 0 or 1, and
Eν =
N
X
nj ǫ.
j=1
Calculate the cumulative distribution function of states, density of
states, entropy, temperature and heat capacity C = dE/dT .
6. Equilibrium distributions
6.1 Canonical ensemble
To the canonical ensemble corresponds the probability distribution which
maximizes entropy with the constraint that the expectation value of energy
hHi ≡ Tr ̺H = E
(6.1)
is a given constant. This constraint as well as the normalization condition
Tr ̺ = 1 may be taken into account by the Lagrange-multiplier method.
This leads to the variational condition
0 = δ (S − λhHi − λ′ hIi)
= δTr (−̺ ln ̺ − λ̺H − λ′ ̺)
= Tr δ̺(− ln ̺ − I − λH − λ′ I) .
Here, λ, λ′ are the Lagrange multipliers. The last expression in brackets
must vanish. This yields, with a different notation for constants
̺=
1 −βH
e
.
Z
(6.2)
The normalization factor Z is the canonical partition function
Z = Tr e−βH =
X
e−βEn =
n
Z
dE ω(E) e−βE .
(6.3)
The probability distribution (6.3) is the canonical or Gibbs distribution. If
the energy scale is chosen such that the ground-state energy is zero and
unbounded states are possible, then the canonical partition function may be
regarded as the Laplace transform of the microcanonical partition function:
Z=
Z∞
dE ω(E) e−βE
0
In the canonical distribution probabilities of the energy eigenstates |ni are
pn =
1 −βEn
e
.
Z
(6.4)
The canonical distribution of a one-particle system is also called the Boltzmann distribution. If the one-particle energies are εν , then the partition
69
70
6. EQUILIBRIUM DISTRIBUTIONS
function and the probability to observe the state ν are
Z=
X
e−βεν ;
pν =
ν
1 −βεν
e
.
Z
(6.5)
The exponential exp (−βεν ) is the Boltzmann factor. The parameter β has
the meaning of the inverse temperature of the system: β = T −1 .
Entropy and temperature. Since ln ̺ = −βH − ln Z, the entropy
assumes the form
S = −hln ̺i = βE + ln Z ,
(6.6)
where E = hHi = Tr (H e−βH )/Z. The parameter β determines values of
both energy and entropy. To find the temperature, we relate variations of
these quantities as functions of β. Calculate first δZ,
δZ = Tr δ(e−βH ) = −δβ Tr He−βH = −δβ EZ ;
then
δS =
δZ
E δβ + β δE +
Z
= β δE .
According to the definition of the temperature
δE
1
T =
= ,
δS V,N
β
(6.7)
i.e. β = T −1 .
Canonical partition function and free energy. The partition function of a distribution is a central quantity. With its aid all thermal properties of an equilibrium system may be calculated. Differentiating with
respect to β:n we obtain
∂
Z = −Tr e−βH H = −ZhHi ,
∂β
so that the expectation value of the energy is
E=−
∂
∂ ln Z
ln Z = T 2
.
∂β
∂T
(6.8)
Substitution in expression (6.6) for entropy yields entropy solely in terms
of Z:
∂
(T ln Z) .
(6.9)
S=
∂T
Comparison with the thermodynamic definition F = E − T S leads to an
important result [which may be inferred directly from relation (6.6) as well]
F = −T ln Z .
(6.10)
6.1. CANONICAL ENSEMBLE
71
Since Z = Z(T, V, N ), the statistical definition of the free energy in (6.10) is
given in terms of the natural variables. The density operator may now be
written also as
̺ = eβ(F −H) .
(6.11)
is
Fluctuations. The probability distribution of the energy E of the system
P (E) ≡ hδ(H − E)i = Tr ̺ δ(H − E) .
ω (E ) e
(6.12)
−βE
∆E
E(β )
E
Figure 6–1: Probability distribution of energy in the canonical ensemble.
In the canonical ensemble this is
P (E) =
1
ω(E) e−βE .
Z
In a large system energy integrals may be calculated with the aid of
the saddle-point method, since in the weight function ω(E) exp(−βE) =
exp[−βE+S(E)] the argument of the exponential is (implicitly) proportional
to the number of particles N . Consider, e.g., the partition function
Z
Z = dE exp[−βE + ln ω(E)]
The microcanonical entropy is ln ω(E) = S(E), therefore the expanding the
exponential we obtain:
ln ω(E) − βE = ln ω(E) − βE +
∂S(E)
− β (E − E)
∂E
1 ∂ 2 S(E)
(E − E)2 + · · · .
+
2 ∂E 2
The stationarity condition
T (E) ≡
∂E
= β −1 = T ,
∂S(E)
72
6. EQUILIBRIUM DISTRIBUTIONS
renders the microcanonical temperature equal to the canonical one. The
second-order term is negative definite
∂2S
∂
=
∂E 2
∂E
1
1 ∂T
1
=− 2
=− 2
,
T
T ∂E
T CV
so that the partition function is expressed in terms of a convergent integral
Z
1
−βE
2
Z ≈ ω(E) e
(E − E) .
(6.13)
dE exp − 2
2T CV
This relation shows that also microcanonical and canonical entropies coincide at leading order in N . The energy distribution in relation (6.13) is
normal and the variance of energy
p
√
∆E = T 2 CV = O( N ) .
(6.14)
Thus, the relative inaccuracy of the energy is
1
∆E
∝√ ,
E
N
(6.15)
which for a large system is extremely small. For any practical purposes the
canonical and microcanonical ensembles are thus
p equivalent.
More directly the variance of energy ∆E ≡ hH 2 i − hHi2 may be calculated by noting that
hHi = −
TrH e−βH
∂
ln Z =
∂β
Tre−βH
and
∂
∂2
hHi = − 2 ln Z = hHi2 − hH 2 i .
∂β
∂β
Thus
D
E
∂2
2
(∆E)2 ≡ (H − hHi) = − 2 (βF ) = T 2 CV .
∂β
(6.16)
Classical canonical distribution. In classical statistical mechanics
the canonical distribution may be derived in the same way as above. The
canonical distributionmaximizing the Gibbs entropy (4.26) turns out to be
̺(P ) ∝ exp −βH(P ) , where P is a point in the phase space. Thus, the
classical canonical partition function is
Z
Z = dΓ e−βH ,
(6.17)
where the volume element dΓ is given by relation (4.5).
6.1. CANONICAL ENSEMBLE
73
Example 6.1. Canonical partition function of a free point particle.
The Hamilton function is H = p2 /2m. Choose plane waves as the eigenfunctions
1
(6.18)
ψk (r) = √ eik·r ,
V
where
2π
(nx , ny , nz ) ;
L
with integer nj . The energy of the particle is
(6.19)
k=
εk =
~2 k 2
.
2m
(6.20)
In the thermodynamic limit replace
X
nx ,ny ,nz
V
··· ⇒
(2π)3
Z
(6.21)
d3 k · · · ,
to obtain
b
Z1 = Tre−β H =
X
2 2
e−β~
nx ,ny ,nz
k /2m
≈
V
(2π)3
Z
2 2
d3 ke−β~
k /2m
.
(6.22)
Due to rotational symmetry the angular integral in the spherical polar coordinates is taken immediately giving rise to the density of states of the
free particle and an integral calculable as Γ( 21 ):
2πV (2m)3/2
Z1 =
(2π)3 ~3
Z∞
√
V
V (2πmT )3/2
= 3 ,
dε εe−βε =
3
h
λT
(6.23)
0
√
where λT = h/ 2πmT is the thermal de Broglie wavelength which may be
understood as the measure of broadening of the single-particle wave packet
due to thermal motion.
Example 6.2. Canonical partition function of N free point particles. The wave function is a symmetrized or antisymmetrized product of
N plane waves, which thus includes a large number of linearly independent terms. In the canonical ensemble, this imposes restrictions on the
sum over states in the partition function which are difficult to take into
account. Approximately this may be done in the case of small occupation
number, in which case the number of linearly independent terms in the
(anti)symmetric wave function is close to N ! and thus equal for all possible
wave functions. This allows to sum over all wave vector independently and
remove the overcounting introduced this way simply by dividing the result
by N !. Thus
ZN
ZN = 1 .
(6.24)
N!
74
6. EQUILIBRIUM DISTRIBUTIONS
Example 6.3. Canonical density operator of the harmonic oscillator
in the coordinate representation. The unnormalized density operator
is
∞
X
1
b
hx |e−β H |x i =
e−β~ω(n+ 2 ) ψ (x )ψ (x ) ,
1
2
n
1
n
2
n=0
with the eigenfunctions
where q =
p mω
~
mω 1/4 e−q2 /2 H (q)
√ n
ψn (x) =
π~
2n n!
x and Hn (q) are Hermite polynomials
n q2
Hn (q) = (−1) e
d
dq
n
2
−q 2
e
eq
=√
π
Z∞
2
du (−2iu)n e−u
+2iqu
.
−∞
Substitution gives
b |x i =
2
−β H
hx1 |e
r
mω (q12 +q22 )/2
e
π3 ~
Z∞
du
−∞
Z∞
−∞
dv
∞
X
(−2uv)n
n!
n=0
1
Here,
2
× e−β~ω(n+ 2 )−u
+2iq1 u−v 2 +2iq2 v
.
∞
X
(−2uv)n e−β~ωn
= exp −2uve−β~ω .
n!
n=0
The remaining Gaussian integral may be calculated with the aid of the
relation (derivation see below in section 6.4):
Z
p
−1 1 −1
1
Dx C e− 2 xsx+x·y = det (s/2π) e 2 ys y .
In our case s is a 2 × 2 matrix:
2
s=
2e−β~ω
2e−β~ω
2
,
and the vector y = (2iq1 , 2iq2 ). Therefore,
r
e−β~ω/2
mω
b
√
hx1 |e−β H |x2 i =
π~ 1 − e−2β~ω
2
q 2 − 2q1 q2 e−β~ω + q22
q + q22
.
− 1
× exp 1
2
1 − e−2β~ω
With the aid of definitions and transformation rules for hyperbolic functions
this expression may be cast in the more symmetric form
r
mω
b
−β H
hx1 |e
|x2 i =
2π~ sinh β~ω
mω
β~ω
β~ω
2
2
× exp −
(x1 + x2 ) tanh
.
+ (x1 − x2 ) coth
4~
2
2
6.2. GRAND CANONICAL ENSEMBLE
75
The normalization factor is
Tr e
b=
−β H
Z∞
−∞
b
dx hx|e−β H |xi = q
1
β~ω
2 sinh β~ω tanh 2
so that the density matrix is
ρ(x1 , x2 ) =
r
mω
β~ω
tanh
π~
2
β~ω
β~ω
mω
2
2
(x1 + x2 ) tanh
.
+ (x1 − x2 ) coth
× exp −
4~
2
2
6.2 Grand canonical ensemble
The grand canonical ensemble is obtained by maximization of the entropy
S with the following constraints:
b = E = given energy ,
hHi
b i = N = given particle number .
hN
The accented quantities are operators. The Lagrange-multiplier method
leads to the grand canonical distribution
̺G =
1 −β(H−µ
b Nb ) .
e
ZG
(6.25)
Here, the Hamilton operator is the sum
b = H (0) + H (1) + · · · + H (N ) + · · · ,
H
where the operator H (N ) acts only on the Hilbert space of the proper N b on an N particle functions. The action of the particle-number operator N
b is replaced by
particle function is that of the multiplication operator, i.e. N
its eigenvalue N .
The grand canonical partition function is most conveniently written as
the sum over all particle numbers
X
X
(N )
ZG =
eβµN Tr N e−βH
=
z N ZN .
(6.26)
N
N
Here, TrN is the trace in the N -particle Hilbert space and
ZN ≡ Tr N exp −βH (N )
is the canonical partition function of the N -particle system. In relation
(6.26) also the fugacity,
z = eβµ
(6.27)
76
6. EQUILIBRIUM DISTRIBUTIONS
has been introduced.
Grand canonical partition function and grand potential. Derivatives of the partition function 1 ZG = Tr exp[−β(H − µN )] yield expectation
values. A good starting point is the extensive logarithm ln ZG . Thus,
1
∂ ln ZG
=
Tr e−β(H−µN ) βN = βhN i = βN ,
∂µ
ZG
and
1
∂ ln ZG
=−
Tr e−β(H−µN ) (H − µN ) = −hHi + µhN i = −E + µN .
∂β
ZG
Therefore, the expectation values of the particle number and energy are
∂ ln ZG
,
∂µ
(6.28)
∂ ln ZG
∂ ln ZG
+ Tµ
.
∂T
∂µ
(6.29)
N =T
E = T2
With the use of relations (6.28)–(6.29), the entropy S = −hln ̺i now assumes
the form
1
∂
(T ln ZG ) ,
(6.30)
E − µN + ln ZG ⇒
S ≡ −hln ̺i =
T
∂T
Comparison with the thermodynamic definition of the grand potential Ω =
E − T S − µN leads to the identification
(6.31)
Ω = −T ln ZG .
Here, ZG is given in terms of the natural variables T , V and µ of Ω. The
density operator may be written as
(6.32)
̺ = eβ(Ω−H+µN ) .
Fluctuations. Differentiating the partition function twice with respect
to µ we obtain
∂2
Tr e−β(H−µN ) = ZG β 2 hN 2 i .
∂µ2
Combining this result with relation (6.28) we obtain after a straightforward
calculation
(∆N )2
≡
hN 2 i − hN i2 = T 2
= T
1
∂N
.
∂µ
∂ 2 ln ZG
∂µ2
(6.33)
Superfluous accents denoting operators will be omitted. Operators, eigenvalues
and expectation values may be distinguished by the context.
6.3. CONNECTION WITH THERMODYNAMICS
77
Since ln ZG and N are extensive quantities, the relative inaccuracy of the
particle number is
∆N
1
√
,
(6.34)
=O
N
N
which again is very small, for a mole of the order 10−12 .
Thus, particle number, energy and other extensive quantities are determined in practically equal accuracy in all canonical ensembles provided all
relevant response functions remain finite (which is typically not the case
in continuous phase transitions). In many cases, however, it is easier to
calculate the grand canonical partition function than the canonical or microcanonical.
Example 6.4. Grand canonical partition function of a system of free
point particles. The construction is readily accomplished with the use of
the canonical partition function:
ZG =
∞
X
N =0
eβµN
Z1N
= exp Z1 eβµ
N!
and gives rise to the grand potential
Ω = −T ln ZG = −T V eβµ
(2πmT )3/2
.
h3
6.3 Connection with thermodynamics
Consider an adiabatic change in the quantum-mechanical system, i.e. a
change of energy eigenfunctions and eigenvalues which does not affect the
structure of energy levels:
H|ni = En |ni
(H + δH)(|ni + δ|ni) = (En + δEn )(|ni + δ|ni) .
(6.35)
Forming a scalar product of latter equation with hn|, at the linear order in
the variations we obtain
δEn = hn|δH|ni = hδHin .
If the change of the Hamiltonian is due to small changes of parameters
P
δH = i ∂H δxi , then
∂xi
∂H
∂En
=
.
(6.36)
∂xi
∂xi n
If the parameters xi (t) change slowly enough as functions of time t, the
system in eigenstate n remains in that eigenstate without transitions to
other energy eigenstates. This is the adiabatic change in quantum mechanics.
78
6. EQUILIBRIUM DISTRIBUTIONS
Thus, a change in the parameters of the Hamiltonian leads, not unexpectedly, to a change in its eigenvalues. The expectation value of the
Hamilton operator depends, however, also on the probabilities to observe
the system in an energy eigenstate, i.e. on the density operator. Therefore
δhHi = Tr δ̺ H + Tr ̺ δH = Tr δ̺H +
X
δxi Tr ̺
i
∂H
.
∂xi
Define the generalized force
∂H
Fi = −Tr ̺
=−
∂xi
∂H
∂xi
,
conjugate with the generalized displacement δxi . Then
X
Fi δxi .
δhHi = Tr δ̺H −
(6.37)
i
On the other hand, the change of the statistical entropy due to a change in
the density operator is
δS stat = −Tr ln ̺δ̺ ,
since the normalization condition requires Tr δ̺ = 0. Further steps depend
on the choice of the ensemble. In case of the canonical ensemble ln ̺ =
−H/T stat − ln Z and
1
(6.38)
δS stat = stat Tr Hδ̺ .
T
Combining (6.37) and (6.38) we arrive at the relation
X
δE = T stat δS stat −
Fi δxi
(6.39)
i
in the form of the first law for a reversible change δU = T ter δS ter − δW .
Thus, the following identifications maintain the consistency between statistical mechanics and thermodynamics
hHi = E = U = internal energy
T stat = T ter
stat
ter
S
P =S
i Fi δxi = δW = work .
(6.40)
6.4 Thermodynamic fluctuation theory
A large system near equilibrium may be divided in relatively weakly interacting macroscopic subsystems. Then extensive quantities related to subsystems may be defined such that their operators in different subsystems
(at least) approximately commute with each other and the Hamiltonian of
the whole system. Correspondingly, the phase space may be divided in parts
6.4. THERMODYNAMIC FLUCTUATION THEORY
79
(or the Hilbert space to manifolds) characterized, in addition to the total energy E, by the values of these quantities X1 , . . . , Xn . The set of parameters
(E, X1 , . . . , Xn ) determines a (close-to-equilibrium) macrostate.
Let the measure of the macrostate be Σ(E, X1 , . . . , Xn ) (volume of the
corresponding part of the phase space or the number of microstates in the
Hilbert space) and the total measure corresponding the energy E of the
system
X
Σ(E) =
Σ(E, X1 , . . . , Xn ) .
{Xi }
Microstates corresponding to the set E, X1 , . . . , Xn occur at the relative
probability
Σ(E, X1 , . . . , Xn )
f (E, X1 , . . . , Xn ) =
.
Σ(E)
The entropy of the macrostate (E, X1 , . . . , Xn ) is (up to non-extensive terms)
S(E, X1 , . . . , Xn ) = ln Σ(E, X1 , . . . , Xn ) ,
(6.41)
so that its probability is
f (E, X1 , . . . , Xn ) =
1
exp S(E, X1 , . . . , Xn ) .
Σ(E)
(6.42)
The equilibrium values of the parameters Xi0 maximize the entropy. If deviations from the equilibrium values
xi = Xi − Xi0
are small, then a Taylor expansion of the entropy is reasonable
S = S0 −
1
1X
sij xi xj ≡ S 0 − xsx .
2 ij
2
(6.43)
Here, the coefficients are
sij = −
∂2S
∂Xi ∂Xj
0
.
(6.44)
The entropy matrix s is real and symmetric, and for stable equilibrium positive definite. Thus, the variables xi possess a Gaussian probability density
1
f (x) = C e− 2 xsx ,
(6.45)
where C is a normalization factor to be calculated below. To calculate moments of the probability distribution (6.49) the generating function is useful
Z
1
G(y) = Dx C e− 2 xsx ex·y ,
(6.46)
80
where x · y =
6. EQUILIBRIUM DISTRIBUTIONS
P
i
xi yi . Correlation functions are calculated as
Z
1
hxp · · · xr i ≡
Dx C e− 2 xsx xp · · · xr
∂
∂
.
=
···
G(y)
∂yp
∂yr
y=0
(6.47)
To calculate the generating function explicitly, make a shift of the integration variable x → x + a and choose the auxiliary vector a such that the
linear in x term in the exponential disappears: a = s−1 y. This leads to
representation
Z
1
G(y) = e 2 y
T
s−1 y
1
Dx e− 2 xsx .
C
(6.48)
Due to positive
p definiteness of the matrix s the
p following change of variables
is legal z = s/(2π)x. The Jacobian is det s/(2π), therefore
Z
Z
Z
Pn 2
p
−1
1
dz1 · · · dzn e−π i=1 zi .
C
Dx e− 2 xsx = C det s/(2π)
The remaining integral factorizes to an n-fold product of well-known onedimensional integrals
Z∞
2
dz e−πz = 1 .
−∞
The normalized probability density is thus
r
s − 1 xsx
.
e 2
f (x) = det
2π
(6.49)
Note that no explicit dependence on the space dimension n remains here.
Finally, the generating functional assumes the form
1
−1
,
(6.50)
hxi xj i = (s−1 )ij .
(6.51)
G(y) = e 2 ys
y
Calculation of the second derivative yields
Fluctuations in SVN-system. In section 2.9 the deviation of the entropy from the equilibrium value was expressed in terms of deviations of
state variables to analyze stability of the extremum of the entropy. In case
of equilibrium this fluctuation is negative definite and gives the probability
distribution of fluctuations of state variables from their equilibrium values;
according to relation (2.49) the probability density is
1
f = C exp − (∆T ∆S − ∆p ∆V + ∆µ ∆N ) .
(6.52)
2T
6.4. THERMODYNAMIC FLUCTUATION THEORY
81
Fix the particle number for simplicity ∆N = 0 and express the remaining
fluctuations ∆T , ∆S, ∆p and ∆V in terms of the two chosen as independent
variables. Let ∆T and ∆V be the independent variables. The quadratic
form of the entropy fluctuation has been expressed in terms of these variables in Chapter 2.9 in relation (2.50), therefore
1 CV
1
2
2
f (∆T, ∆V ) ∝ exp −
(∆T
)
+
(∆V
)
.
(6.53)
2 T2
V T κT
The elements of the entropy matrix s are thus
sT T =
CV
1
, sV V =
,
T2
V T κT
sT V = sV T = 0 .
Since the matrix s is diagonal, the quadratic expectation values according
to relation (6.51) follow immediately
h(∆T )2 i =
T2
CV
h∆V ∆T i = 0 ,
h(∆V )2 i = V T κT .
(6.54)
The thermodynamic theory of fluctuations and the statistical approach may
lead to seemingly different results. Let us calculate the fluctuation of the
internal energy in the thermodynamic theory. From the expression
∂E
∂p
∂E
∆T +
∆V = CV ∆T + T
− p ∆V
∆E =
∂T V
∂V T
∂T V
with the use of relations (6.54) it follows that
2
∂p
h∆E 2 i = CV T 2 + T
− p T V κT ,
∂T V
which is apparently different from the expression h∆E 2 i = CV T 2 obtained
in the canonical ensemble.
The reason of this discrepancy is that in the canonical ensemble the
volume V is a fixed parameter, whereas in the thermodynamic fluctuation
theory fluctuations of the volume are not restricted. The statistical theory
gives the same result, if an ensemble is used which allows for fluctuations
of the volume (i.e. the isothermal-isobaric ensemble).
Example 6.5. Calculation of the correlation function h∆V ∆pi. The
most straightforward way is to express ∆p as a function of T and V and
use the formulae obtained for fluctuations of the temperature and volume.
Thus,
∂p
∂p
∆T +
∆V ,
∆p =
∂T V
∂V T
wherefrom – with the aid of (6.54) – we obtain
∂p
h∆V 2 i
∂p
h∆V ∆pi =
h∆V ∆T i +
h∆V 2 i = −
= −T .
∂T V
∂V T
V κT
82
6. EQUILIBRIUM DISTRIBUTIONS
Example 6.6. Landau-Placzek formula. Elastic scattering of visible
light in gases (Rayleigh scattering) is brought about by permittivity fluctuations. These are mainly due to density fluctuations (in gases the temperature dependence of the permittivity is negligible). The density fluctuations
may be considered caused by fluctuations of the entropy and density. They
have the important difference that the entropy fluctuations do not propagate but decay due to thermal conduction, whereas the pressure fluctuations propagate at the speed of sound, which gives rise to the Doppler shift
of frequency of the scattered wave. Thus, scattering on the entropy fluctuations brings about scattered light at the frequency of the incident wave,
whereas the pressure fluctuations give rise to the Mandelshtam-Brillouin
doublet with frequencies on both sides of that of the incident wave. Let us
calculate the ratio of the intensity of the doublet (the sum of both components) and the total intensity of the scattered wave.
The total density Itot ∝ h∆ρ2 i. Here, ρ = m/V so that ∆ρ = −(ρ/V )∆V
and Itot ∝ h∆V 2 i. Substitute
∂V
∂V
∆S +
∆p
∆V =
∂S p
∂p S
to obtain
2
h∆V i =
∂V
∂S
2
p
Cp − T
∂V
∂p
,
S
∂p
. The latter term
∂V S
2
in the expression for h∆V i yields thedoublet
contribution. On the other
hand, according to (6.54), h∆V 2 i = −T ∂V
and
∂p T
∂V
−T
∂(V, S) ∂(p, T )
∂(V, S) ∂(p, T )
CV
Idoublet
∂p S
=
=
=
=
.
Itot
∂(p, S) ∂(V, T )
∂(V, T ) ∂(p, S)
Cp
−T ∂V
∂p T
because h∆S 2 i = Cp , h∆S∆pi = 0 and h∆p2 i = −T
This is the Landau-Placzek formula.
6.5 Reversible minimal work and probability
of fluctuations.
The probability of the fluctuation ∆X = X − X 0 of the quantity X may be
calculated also with the use of the notion of reversible minimal work.
To this end, recall that the equilibrium entropy is a monotonically increasing function of the energy of the system. Then the decrease in the
entropy ∆S corresponding to the fluctuation ∆X may be achieved also by
decreasing the energy of the system reversibly in order to follow the the dependence S = S(E) of the equilibrium entropy. This energy is the reversible
6.6. PROBLEMS
83
minimal work R to be exerted to the system to cause the fluctuation ∆X.
Therefore, we may write
S(E − R) ≈ S(E) −
∂S
1
R = S(E) − R ,
∂E
T
which, according to relation (6.42), yields for the probability density of the
fluctuation
R
.
(6.55)
f (∆X) ∝ exp −
T
The nice feature here is that R is expressed in terms of mechanical work or
the like.
Example 6.7. Fluctuation of a string under tension. Consider a string
of length L under tension caused by the longitudinal force F . For a small
transversal displacement y of a string element at the point x the restoring
force is
y
y
F⊥ = F p
+Fp
,
2
2
(L − x)2 + y 2
x +y
which gives rise to the elementary work
i
h p
p
dW = F⊥ dy = d F x2 + y 2 + F (L − x)2 + y 2 .
Thus the total work to move quasistatically the string element to the displacement position is
i
h p
p
W = F F x2 + y 2 + F (L − x)2 + y 2 − L ≈
F y2 L
.
2x(L − x)
This is the minimal reversible work to bring about the displacement of the
string element, therefore the probability density of the corresponding fluctuation is
F y2 L
−
2x(L
− x)
f (y) ∝ e
and the root-mean-square fluctuation ∆y is given by
∆y 2 = hy 2 i − hyi2 =
x(L − x)T
.
FL
6.6 Problems
Problem 6.1. Calculate the canonical partition function for a system
of N identical non-interacting harmonic oscillators with an angular frequency ω. Find the internal energy and and heat capacity. Consider
both classical and quantum-mechanical cases.
84
6. EQUILIBRIUM DISTRIBUTIONS
Problem 6.2. Generalized equipartition theorem. Let the Hamilton function of a classical system be quadratic with respect to all canonical coordinates and momenta ξj :
H=
ν
X
αj ξj 2 ,
j
where αj s are constants. Calculate the canonical partition function and
show that E = 12 νT . Find also the density of states as the inverse
Laplace transform of the canonical partition function with respect to β
(to be treated as a complex variable).
Problem 6.3. A large quantum system consists of N non-identical independent particles each of which may occupy one of two states with
energies 0 and ǫ. Energy eigenstates may thus be enumerated by the
sequences ν = (n1 , n2 , . . . , nN ), where ni = 0 or 1, and
Eν =
N
X
nj ǫ.
j=1
Calculate the canonical partition function, free energy, entropy, and
heat capacity C = dE/dT .
Problem 6.4. Consider a gas of point-like free particles with vanishing
rest mass, i.e. with the one-particle energy ǫ(p) = c|p|. Calculate the
density of states of one particle, calculate then the canonical partition
function of one particle and N particles.
Problem 6.5. Consider a gas of point-like free particles with vanishing
rest mass, i.e. with the one-particle energy ǫ(p) = c|p|. Calculate the
density of states of a system of N such particles.
Hint. The inverse Laplace transform of the canonical partition function
might be useful.
Problem 6.6. Consider a gas of point-like particles with vanishing rest
mass, i.e. with the one-particle energy ǫ(p) = c|p|. Calculate the grand
canonical partition function and the grand potential Ω of the gas and
establish the equation of state. Calculate also the chemical potential
µ = µ(T, n) of the gas.
Problem 6.7. Calculate the electric dipole moment of an ideal gas consisting of linear molecules with a constant dipole moment p0 , when the
gas is in an external electric field E. Use the classical canonical ensemble.
Problem 6.8. The length of the arm of a classical mathematical pendulum is l, the mass attached to it is m and the acceleration of the gravity
of the Earth is g. Determine the fluctuation of the swinging angle of
the pendulum at rest ∆ϕ and also the heat capacity related to small
oscillations of the pendulum.
6.6. PROBLEMS
85
Problem 6.9. With the use of the thermodynamic theory of fluctuations, show that h(∆S)2 i = Cp , h(∆p)2 i = T /V κS , h∆S ∆pi = 0, and
also h∆S ∆V i = V T αp .
Hint. Use S and p as independent variables.
Problem 6.10. In the canonical ensemble, show that
h(E − hEi)3 i =
∂ 2 hEi
.
∂β 2
Problem 6.11. In the isobaric–isothermal ensemble only the average
volume is fixed and the partition function may be written as
Z
Z(T, P ) =
b
dV TrV e−β H(V )−βP V
Z
=
Z
dV e−βP V Zcanonical (T, V ) =
dV e−βP V e−βF (T,V ) .
Show that in this ensemble the energy fluctuation obeys the relation
h(∆E)2 i = T 2 CV − T
∂V
∂p
T
T
∂p
∂T
V
2
−p
.
in accordance with the thermodynamic theory of fluctuations.
7. Ideal equilibrium systems
7.1 Free spin system
In ideal systems there are no interactions between particles. If the solution
of the one-particle problem is known, properties of the ideal many-particle
system may be simply found by taking into account the proper statistics.
The simplest example is the the system of free spins.
States. Consider N particles with spin s = 12 for simplicity. No interactions between the particles. Ignore also spatial motion. This model is fairly
realistic for a crystallic paramagnetic substance, in which at low temperatures site-bound atoms only have the spin degrees of freedom left. Each
particle has two spin states sjz = ± 12 ~. Thus, in the N -particle system there
are 2 × 2 × · · · × 2 = 2N different quantum states. Localization of host atoms
allows to consider spins distinguishable. This means that the statistical
ensembles constructed here are classical, although the very notion of spin
is quantum-mechanical.
Let the total spin be
N
X
Sz ≡
sjz = ~m.
(7.1)
j=1
For simplicity, consider even N , so that the quantum number m may assume the values
1
(7.2)
m = 0, ±1, ±2, · · · , ± N,
2
altogether N + 1 different values. Denote the numbers of spin-up and spindown particles by N + and N − , respectively; then

1

N+ = N + m 

2
(7.3)

1

−
N = N −m 
2
The number of ways W (m) to obtain the total spin ~m is given by the binomial distribution
N!
N!
N
1
W (m) =
= + − = 1
(7.4)
N+
N !N !
N
+
m
! 2N − m !
2
Thus, this is the degeneracy of the state with the total spin Sz = ~m.
86
7.1. FREE SPIN SYSTEM
87
In the limit of large numbers an approximation with the use of the Stirling formula follows
ln W (m)
≈ N ln N − N − N + ln N + + N + − N − ln N − + N −
1
1
1 + 2m/N
= N ln 2 + N ln
− m ln
.
2
1 − 4m2 /N 2
1 − 2m/N
(7.5)
In particular, in the vicinity of the maximum m = 0 we obtain ln W (m) ≈
ln W (0) − N2 m2 . Thus, the binomial distribution follows approximately the
Gaussian normal distribution
2
W (m) ≈ W (0)e−2m /N
(7.6)
√
with the standard deviation ∆m = 12 N . In case of completely random
√
spins the probable values of the total spin (∝ N ) much less than the
largest possible value (∝ N ).
Energy. In a system of non-interacting spins only a coupling to an external field may appear in the Hamilton function. Let B = µ0 H and H = Hez .
The Hamilton function describing the potential energy of the spins is
X
X
b = −µ0
µjz ,
(7.7)
µj · H = −µ0 H
E
j
j
where µj are magnetic moments of the particles.
The magnetic moment of a particle is proportional to its spin
µ = γs .
(7.8)
The gyromagnetic ratio γ is usually different form its classical value γ0 =
q/(2m), where q is the charge of the particle and m its mass. For instance,
for the electron γ ≈ 2γ0 = −e/m, where e is the elementary charge. In the
following the value of the coefficient γ is left unspecified.
Hence, the potential energy of the spin system in the state m is
E = −µ0 γHSz = −εm
(7.9)
with the microscopic energy unit
ε = µ0 ~γH .
(7.10)
Each energy level m has still the degeneracy W (m). Therefore, the density
of states
E
1
,
(7.11)
W −
ω(E) =
|ε|
ε
since according to the definition ω(E) |∆E| = W (m) |∆m|, ja |∆E| = |ε∆m|.
Since in the energy E contains a coupling to the magnetic field such
that the field strength H is an external parameter, the energy should be
interpreted as the magnetic enthalpy.
88
7. IDEAL EQUILIBRIUM SYSTEMS
Mirocanonical ensemble. It is instructive to derive the statistical
mechanics of the spin system first with the aid of the microcanonical ensemble, which is rarely feasible due to difficulties in the calculation of the
density of states ω(E). Denote E0 = 21 εN , so that the total energy obeys
−|E0 | ≤ E ≤ |E0 |. According to relations (7.5), (7.9) and (7.11) the statistical entropy S(E) = ln ω(E) is
E0 − E
E0 2
E
ln
.
(7.12)
S(E) = N ln 2 + ln 2
+
2E0 E0 + E
E0 − E 2
According to the statistical definition of the temperature (5.31) β(E) =
1/T (E) = ∂S/∂E. Differentiation yields
β(E) =
N
E0 − E
ln
.
2E0 E0 + E
(7.13)
Solving for the energy we obtain the result
1
µ0 ~γH
βE0
= − N µ0 ~γH tanh
.
E = −E0 tanh
N
2
2T
(7.14)
Magnetization is the magnetic moment per unit volume. For it we obtain
1 X
N
µ0 ~γH
M=
µj = ez
,
(7.15)
~γ tanh
V j
2V
2T
due to direct proportionality E = −µ0 HV Mz .
Canonical ensemble. Denote the values of an individual spin sjz = ~νj ,
then νj = ± 21 and µjz = ~γνj . Calculation of the trace yields the partition
function


1
1
N
2
2
X
X
X
µjz 
···
exp βµ0 H
ZN =
ν1 =− 21

= 
1
2
X
ν=− 12
νN =− 12
j
N
eβµ0 ~γHν 
= Z1 N ,
where Z1 is the partition function of a single spin.
1
µ0 ~γH
βµ0 ~γH
− 21 βµ0 ~γH
2
+e
= 2 cosh
Z1 = e
.
2T
(7.16)
(7.17)
The partition function yields the free energy, which should be interpreted
as the magnetic Gibbs function :
G(T, H) ≡ −T ln ZN
µ0 ~γH
= −N T ln 2 + ln cosh
.
2T
(7.18)
7.1. FREE SPIN SYSTEM
89
The entropy may be calculated as the temperature derivative of the free
energy:
∂G
µ0 ~γH
S=−
= N ln 2 + ln cosh
∂T H
2T
µ0 ~γH
µ0 ~γH
−
tanh
.
(7.19)
2T
2T
Differentiation with respect to magnetic field yields
X W (m)
∂G
T ∂ZN
−
= µ0 ~γ
e−βεm · m = µ0 ~γhmi = µ0 V Mz .
=
∂H T
ZN ∂H
Z
N
m
Hence, the differential of the free energy is
(7.20)
dG = −S dT − µ0 V M · dH
and the magnetization
M =−
1
µ0 V
∂G
∂H
=
T
N
~γ tanh
2V
µ0 ~γH
2T
,
(7.21)
which is the same as in the microcanonical ensemble.
Susceptibility. According to the definition of the magnetic susceptibility we obtain
2 ∂ G
1
∂M
=−
.
(7.22)
χ=
∂H T
µ0 V ∂H 2 T
From the expression (7.21) of M it follows
2
1
~γ
µ0 N
2
.
χ=
·
V T cosh2 µ0 ~γH
2T
(7.23)
In the limit of weak field this yields Curie’s law
χ=
where the constant is
C=
N
µ0
V
C
,
T
1
~γ
2
(7.24)
2
.
Adiabatic demagnetization. The magnetic properties of a paramagnetic material may be used for cooling. The method easily allows to reach
millikelvin temperatures. The lowest temperatures are reached, however,
with the aid of nuclear demagnetization.
Adiabatic demagnetization is based on the almost complete degeneracy
of the energies of the spin states, due to which large entropy may persist
90
7. IDEAL EQUILIBRIUM SYSTEMS
to extremely low temperatures. Removal of the entropy with the aid of
the magnetic field then lowers the temperature dramatically. In simplified
notation the entropy (7.19) may be cast in the form
S
= ln 2 + ln cosh x − x tanh x ,
N
(7.25)
with the dimensionless variable x
x=
µ0 ~γH
.
2T
(7.26)
Entropy is a monotonically decreasing (increasing) function of x (T ). In the
limit x → ∞ it follows ln cosh x = x − ln 2 + e−2x + · · · , and tanh x =
1 − 2e−2x + · · · . The limit x → 0 is trivial. The asymptotic behaviour of the
entropy is:
S
1
→ ln 2 − x2 ,
x→0
(T → ∞),
N
2
(7.27)
S
→ 2xe−2x ,
x→∞
(T → 0).
N
S/N
H1 < H2
ln 2
1
2
T1
T2
T
Figure 7–1: Adiabatic demagnetization.
In Fig.
7–1 the temperature dependence of the entropy is shown for two different field strengths. The sample is cooled first by some other
effective method in very strong
field (magnetic induction of several teslas). Then the sample
is isolated thermally and the
field strength adiabatically decreased to almost zero. Since
the entropy is a function of the
ratio H/T only, the sample is
cooled in the ratio if the field
strengths
H1
T1
=
.
T2
H2
(7.28)
Negative temperature. The temperature of the spin system may be
rendered negative with the aid of adiabatic demagnetization. The system
is then in a metastable state, as illustrated in Fig. 7–2. In equilibrium in a
non-vanishing field the energy of the systems is negative. If the direction of
the field is adiabatically reversed, occupation numbers of quantum states
cannot change accordingly, although their energies change sign. The energy
of the system becomes positive, and the situation formally corresponds to
an equilibrium system at the negative temperature β ′ < 0. According to
relation (7.12) the entropy S(E) has a maximum at E = 0, around which it
symmetrically decreases to zero. Thus, S(E) is not a monotonically growing
7.2. CLASSICAL IDEAL GAS
91
ω (E) e-βE
ω (E) e-β′E
e-βE
e-β′E
-E 0
E0
E E
E0
-E 0
Figure 7–2: Negative temperature.
function of E. In particular, at E > 0 the derivative of the entropy is negative exhibiting negative temperature. Metastable population inversion
occurs in other systems as well, e.g. in an exited laser. Note that boundedness of the energy spectrum from above is required for these phenomena.
7.2 Classical ideal gas
Maxwell–Boltzmann statistics. In ideal gas interactions between the
molecules are neglected, apart from collisions occuring every now and then.
Their most important effect is the thermalization of the systems, i.e. the
approach to the thermodynamic equilibrium describable by an equilibrium
ensemble. In the ideal gas limit collisions are elastic and practically instantaneous, so that the energetics of a single molecule is completely determined by its properties as a free particle between the collisions. Thus,
thermodynamically a single molecule is a closed system with only a weak
coupling to the rest of the gas.
Hence, the single-particle statistics is determined by the Boltzmann distribution (kappale 6.1)
̺ℓ = hℓ|̺|ℓi =
1 −βεℓ
e
;
Z1
Z1 =
X
e−βεℓ ,
(7.29)
ℓ
where εℓ are energies of the one-particle states ℓ. In a translationalinvariant system of point-like particles the one-particle energies are εℓ =
pℓ 2 /(2m) = 12 mv ℓ 2 . The volume element of the velocity space is d3 v =
m−3 d3 p = (~/m)3 d3 k. The probability distribution of velocity is
mv 2
∝ hk|̺1 |ki,
f (v) = C exp −
2T
with the normalization constant determined by the condition
Z
d3 v f (v) = 1.
(7.30)
(7.31)
92
7. IDEAL EQUILIBRIUM SYSTEMS
This is a Gaussian integral, therefore
Z
mv 2
d v exp −
2T
3
=
r
2πT
m
3
;
and the Maxwell distribution in the velocity space results:
f (v) =
m 3/2
mv 2
exp −
.
2πT
2T
(7.32)
It gives rise to the probability density F (v) of the speed v = |v| through the
normalization condition
Z ∞
Z
Z ∞
3
dv F (v) = d v f (v) = 4π
dv v 2 f (|v|) = 1.
(7.33)
0
0
Thus,
(7.34)
F (v) = 4πv 2 f (v) .
It is illustrated in Fig. 7–3.
F(v)
From the speed distribution the following
vm
v
v2
v
Figure 7–3: Maxwell distribution for speed.
characteristic quantities are readily inferred:

r

2T


vm =
= most probable speed,


m



r

Z ∞
8T
dv vF (v) =
hvi ≡
= mean speed,

πm

0


Z ∞


3T


dv v 2 F (v) = hv 2 i =
= mean square of velocity.
 hv 2 i ≡
m
0
(7.35)
From the Maxwell distribution it also follows that h 12 mvx 2 i = h 21 mvy 2 i =
h 12 mvz 2 i = 12 T . This is an example of the equipartition principle of classical
statistical mechanics.
Example 7.1. Particle flux density of the perfect gas. Calculate the
average number of particles of a Maxwell-Boltzmann gas hitting the wall
7.2. CLASSICAL IDEAL GAS
93
p
Gas
H2
He
N2
O2
CO2
hv 2 i, m/s
1 900
1 350
510
477
407
Root-mean-square velocities of gas molecules at NTP.
of the vessel containing the gas per unit time (giving the particle flux) and
per unit area of the wall (which is the particle flux density).
Let the z axis lie along the outward normal to the wall. Consider particles the z component of the velocity of which is in the interval (vz , vz + dvz )
contained in a straight cylinder perpendicular to the wall with the crosssection area ∆A. The number of these particles hitting the wall during the
time interval ∆ is
∆N = n
Z∞
dvx
Z∞
dvy dvz f (v)vz ∆t∆A
−∞
−∞
where f (v) is the probability density (7.32) of the Maxwell distribution and
n the particle density of the gas. Thus, the contribution of these particles
to the flux density ν is
r
m −mvz2 /2T
∆N
dν =
=n
e
vz dvz
∆A∆t
2πT
and the total particle flux density
r
r
Z∞
m
T
1
−mvz2 /2T
ν=n
= nhvi ,
e
vz dvz = n
2πT
2πm
4
0
where hvi is the mean speed of the particles.
Example 7.2. Angular distribution in effusion. Calculate the expectation value of the cosine of the angle of flight (measured from the outward
normal to the thin wall of the vessel) of molecules of a gas leaking from a
vessel through a fine hole.
The number of molecules moving through the hole per unit area and unit
time in the solid angle determined by the azimuthal angle in the interval
(θ, θ +dθ) (from the direction of the outward normal to the hole) is, by virtue
of an argument similar to that of the preceding example,
Z∞
1
F (v)v cos θ sin θdv dθ ,
ν(θ)dθ = n
2
0
94
7. IDEAL EQUILIBRIUM SYSTEMS
where F (v) is the probability density for the speed (7.33). Dividing by the
flux density of the molecules moving through the hole ν we arrive at the
probability density of the departure angle of the molecules
f (θ) = 2 sin θ cos θ .
Thus,
π
hcos θi = 2
Z2
sin θ cos2 θ dθ =
2
.
3
0
Example 7.3. Maxwell distribution for relative velocity. Consider a
mixture of two perfect gases with molecules of masses m1 , m2 and velocities
v 1 , v 2 , respectively. To calculate the probability density for v = v 1 − v 2 , it
is convenient to introduce the center-of-mass variables through
m1 v 1 + m2 v 2
,
m1 + m2
m2
v,
v1 = V +
m1 + m 2
V =
= v1 − v2
v
v2
=V −
m1
v,
m1 + m 2
(7.36)
(7.37)
with the Jacobi determinant equal to unity. The kinetic energy in these
terms is
1
1
1
1
m1 v12 + m2 v22 = (m1 + m2 )V 2 + µv 2 ,
2
2
2
2
m
m
1
2
where µ = m + m is the reduced mass. The probability density of the rel1
2
ative velocity is thus the Maxwell distribution of a particle with the reduced
mass µ.
Partition function and thermodynamics. Calculate the singleparticle partition function Z1 (β) as the Laplace transform of the singleparticle density of states ω1 (ε) (5.24). The result is
Z1 (β) =
Z
dε ω1 (ε)e−βε = g
X
k
~2 k 2
V √
exp −β
= g 3 2πmT 3 ,
2m
h
where g is the degeneracy factor of internal degrees of freedom. For pointlike particles it is the spin degeneracy, for molecules it is the partition function of the degrees of freedom of the molecule in the center-of-mass frame.
The partition function assumes a suggestive form with the introduction of
a characteristic length scale: the thermal de Broglie wave length
r
h2
λT =
.
(7.38)
2πmT
This quantity gives the spatial spread of the wave packet of the particle
in the typical thermal motion. For the oxygen molecule at NTP, e.g., it is
7.2. CLASSICAL IDEAL GAS
95
λT = 0.187 Å and thus clearly less than the diameter of the molecule. The
partition function is thus
Z1 (β, V ) = g
V
.
λT 3
(7.39)
The canonical partition function of a many-particle system ZN (β, V ) may be
correspondingly expressed as the Laplace transform of the density of states
ωN (E). In the MB gas particles are completely independent, therefore
ZN
1
=
N!
g
X
!N
exp(−βεk )
k
=
1
Z1 N ,
N!
(7.40)
where N ! removes the classical permutation degeneracy of identical particles.
For the free energy FN = −T ln ZN we obtain
F (T, V, N ) = N T
N
3
3
h2
ln
− ln T + ln
− 1 − ln g
V
2
2 2πm
.
(7.41)
Here, the free energy is a thermodynamically extensive quantity. Without
the degeneracy-lifting factor 1/N ! the dependence on the particle number
would have been quite wrong.
The usual equation of state of the perfect gas immediately follows: p =
−∂F/∂V = N T /V . Further, S = −∂F/∂T = −F/T + 3N/2, so that the
internal energy is U = F + T S = 32 N T confirming the result obtained in
chapter 3.1. The heat capacity is determined by the degrees of freedom of
the translation motion, hence f = 3. The explicit expression for the entropy
of the MB gas is
3
3 2πm
5
V
+ ln T + ln 2 + ln g +
.
(7.42)
S = N ln
N
2
2
h
2
With the aid of the Gibbs function G = F + pV = µN and the equation of
state we arrive at the expression for the chemical potential
µ(p, T ) = T
ln p −
5
3
h2
ln T + ln
− ln g .
2
2 2πm
(7.43)
The chemical constant is thus
3
ζ = ln g + ln
2
2πm
h2
" 3/2 #
2πm
= ln g
.
h2
(7.44)
The constant µ0 = 0, because only the kinetic energy of the translational
motion is included in the energy of the molecule.
96
7. IDEAL EQUILIBRIUM SYSTEMS
Grand canonical partition function. Denote the fugacity z = exp βµ
and calculate according to the general rule
X 1
X
N
N
N
zZ1
βµ gV
z ZN =
ZG (T, V, µ) =
. (7.45)
z Z1 = e
= exp e
N!
λT 3
N
N
The grand potential is
Ω(T, V, µ) = −T ln ZG = −T eβµ
gV
.
λT 3
(7.46)
The average number of particles is
P
N z N ZN
Ω
∂Ω
gV
∂ ln ZG
=−
= eβµ 3 = − ,
=
N = PN N
∂
ln
z
∂µ
T
z
Z
λ
N
T
N
and the pressure p = −∂Ω/∂V = −Ω/V , so that we arrive at the ideal gas
equation of state once more.
Validity of the MB gas law. As will be shown later, the MB approximation may be good only if the expectation values of the occupation numbers
nℓ obey for all ℓ the condition
nℓ ≪ 1.
From the single-particle Boltzmann distribution it follows
nℓ = e−β(εℓ −µ) .
(7.47)
The lowest (kinetic) energy is value is 0, therefore eβµ ≪ 1. We have seen
that (for g = 1) exp βµ = N λT 3 /V . With the aid of the mean distance
between the particles (the radius of the ball containing one particle in the
average) ri the condition may be expressed as
λT ≪ ri ,
(7.48)
which allows for a simple interpretation: MB approximation is reliable,
when the wave packets of the particles do not overlap.
Internal degrees
of freedom. Even in monatomic gases the degenerP −βε
ℓ
e
may be thermodynamically important due to the
acy factor g =
ℓ
possible fine and hyperfine structure of atomic energy levels. If there is no
fine or hyperfine structure (practically this means that the angular momentum is zero), then only the usual temperature-independent spin degeneracy
g = 2S + 1 remains.
It should be noted first that only the ground state is of concern for gases,
because the energy gap between the ground state and the first excited state
(without spin-orbit effects) is of the same order as the ionization energy of
the atom. Thus, at temperatures which give rise to appearance of atoms in
excited states in an appreciable amount, the number of ionized atoms is of
the same order and the system ceases to be a monatomic gas.
7.3. DIATOMIC IDEAL GAS
97
Assuming description of the fine structure
in terms of the LS coupling,
P
the factor g may be expressed as g = J (2J + 1) e−βεJ , where the degeneracy with respect to the component of the total momentum J is explicitly
taken into account.
At low temperatures βεJ ≫ 1 and only the lowest order term survives
giving rise to the degeneracy factor g = 2J + 1. At high temperatures βεJ ≪
1 and the exponents are all close to the unity. Therefore, g approaches
the total number of states in the fine structure with given L and S, i.e.
g = (2L + 1)(2S + 1).
Interaction of the electrons with the nuclear spin gives rise to the hyperfine structure, in which the differences between energy levels, however,
are always small compared with the temperature and only give rise to the
degeneracy factor 2I + 1, where I is the nuclear spin. This factor is often
omitted.
7.3 Diatomic ideal gas
Diatomic gases may be homopolar with molecules consisting of two identical atoms like H2 , N2 , O2 etc, or heteropolar like CO, NO, HCl etc. The
additional constraints in the former case have to be taken into account.
When the internuclear distance is close to the equilibrium distance the
energy of the molecule may be expressed as a sum of several independent
terms:
H = H tr + H rot + H vib + H el + H yd .
(7.49)
Here,
• H tr =
p2
= translation energy; m = mass of the molecule.
2m
L2
= energy of rotation; rotation about the center of mass.
2I
I = the moment of inertia of the nucleai, L = the angular momentum
of the molecule (electrons included, this is due to the usual approximation method). The eigenvalues of are the (2ℓ + 1)-fold degenerate
energies
~2
ℓ(ℓ + 1).
2I
1
= energy of vibrations. The electronic energy has
• H vib = ~ωv n
b+
2
a minimum at the equilibrium distance between the nucleai, around
which the harmonic potential is a reasonable approximation. Here, n
b
is an operator with natural numbers 0, 1, 2,. . . as eigenvalues. Energy
levels are non-degenerate.
• H rot =
• H el = electronic energies due to Coulomb interaction between the
electrons in the electric field of the immobile nucleai. Differences be-
98
7. IDEAL EQUILIBRIUM SYSTEMS
4
tween these terms are of the order >
∼ 1 eV ≈ 10 K, so that the electronic term is that of the normal state of the molecule. However, apart
from the case of vanishing molecular spin S = 0 and vanishing angular momentum with respect to the axis connecting nucleai Λ = 0 the
fine structure of electronic terms must be taken into account.
• H yd = energies related to nuclear degrees of freedom. Of interest here
are the nuclear spins, which affect the rotational degrees of freedom
in case of homopolar molecules. In case of heteropolar molecules their
effect is a constant degeneracy factor due to the hyperfine structure.
Denote the nuclear spins by I1 and I2 . Then the number of states of
the hyperfine structure is
gy = (2I1 + 1) (2I2 + 1) .
As in case of monatomic gas, this factor is often omitted which is tantamount to redefining the entropy in such a way that it approaches
the constant value ln gy instead of zero in the limit T → 0.
Near the equilibrium distance of the nucleai there are no significant couplings between the different terms. Therefore, the partition function factorizes to a product of partition functions connected to different degrees of
freedom. Calculate first for a single molecule
Z1 =
p2
~2
~ωv
1
gy (2ℓ + 1) exp −
−
ℓ(ℓ + 1) −
n+
2mT
2IT
T
2
n=0
∞ X
∞
XX
p
ℓ=0
(7.50)
= Z tr Z rot Z vib Z yd .
The factors are
Z
tr
=
X
p
Z rot =
p2
exp −
2mT
∞
X
ℓ=0
Z vib =
V
;
=
λT 3
λT =
Tr
(2ℓ + 1) exp − ℓ(ℓ + 1) ;
T
r
Z yd = gy = (2I1 + 1) (2I2 + 1) .
(7.51)
~2
2I
(7.52)
Tr =
−1
1
Tv
~ωv
n+
= 2 sinh
;
exp −
T
2
2T
n=0
∞
X
h2
2πmT
Tv = ~ωv
(7.53)
(7.54)
Due to the indistinguishability of the molecules, in the partition function of
N the division by the factor N ! has to be introduced (Chapter 5.2), so that
ZN (T, V ) =
1
Z1 (T, V )N .
N!
(7.55)
7.3. DIATOMIC IDEAL GAS
99
This factor is most conveniently related to the traslational motion.
the free energy F = −T ln ZN is split to terms
N
1
3 2πmT
V
F tr = −T ln
+ ln
+
1
,
Z tr
= −N T ln
N!
N
2
h2
(∞
)
X
Tr
rot
(2ℓ + 1) exp − ℓ(ℓ + 1) ,
F = −N T ln
T
ℓ=0
Tv
F vib = N T ln 2 sinh
,
2T
F yd = −N T ln gy .
Then
(7.56)
(7.57)
(7.58)
(7.59)
With the use of relations U = F + T S = F − T ∂F/∂T = −T ∂(F/T )∂T it
is readily seen that the translational degrees of freedom yield the ideal gas
result
3
3
(7.60)
U tr = N T → CVtr = N.
2
2
Since only F tr depends on the volume V , also for the diatomic gas the ideal
gas equation of state follows
p=−
∂F
NT
=
∂V
V
→
2
pV = N T .
(7.61)
Rotation. The temperature parameter related to rotation Tr is always
clearly less than the room temperature. The partition function allows for
analytic result in both limits T ≪ Tr and T ≫ Tr .
r
r
⇒ F rot ≈ −3N T exp − 2T
⇒ U rot ≈
T ≪ Tr : Z rot ≈ 1 + 3 exp − 2T
T
T
r
6N Tr exp − 2T
. This leads to the heat capacity
T
2
2Tr
Tr
exp −
(7.62)
CVrot ≈ 12N
−→ 0.
T
T
T →0
R∞
T ≫ Tr : Z rot ≈ ℓ=0 dℓ (2ℓ + 1) exp − TTr ℓ(ℓ + 1) =
⇒ U rot ≈ N T . In this limit the heat capacity is
CVrot ≈ N .
T
Tr
⇒ F rot ≈ −N T ln TTr
(7.63)
In the limit T ≫ Tr the molecule thus has two degrees of freedom: rotations about the x and y axes. Approximative calculation between the limits
reveals a weak maximum of CVrot , see Fig. 7–4.
Vibration. The temperature parameter of vibrations Tv is usually much
higher than the room temperature. In the limit T ≪ Tv on F vib ≈ 12 N Tv −
N T exp − TTv . The heat capacity becomes
2
Tv
Tv
vib
2
2
exp −
.
(7.64)
CV = −T ∂ F/∂T −→ N
T
T
T →0
100
7. IDEAL EQUILIBRIUM SYSTEMS
Gas
H2
N2
NO
O2
HCl
Cl2
Tr (K)
85.4
2.9
2.4
2.1
15.2
0.36
Tv (K)
6100
3340
2690
2230
4140
Parameters of diatomic gases
In the limit T ≫ Tv in turn F vib ≈ N T ln
rise to the heat capacity
CVvib ≈ N .
Tv
T
⇒ U vib ≈ N T , which gives
(7.65)
Here we see two effective degrees of freedom as well, viz. translational
motion and potential energy. Vibrational degrees of freedom become excited
at relatively high temperatures; at low temperatures the thermal energy of
order T is insufficient for this. The temperature dependence of the heat
capacity is depicted in Fig. 7–4.
CV/N
7/2
5/2
3/2
huoneenlämpötila
Tr
ionisaatio,
dissosiaatio ym.
Tv
T
Figure 7–4: Heat capacity of diatomic gas.
Rotation of homopolar molecule. The rotational partition function
used above is valid for heteropolar molecules only. In case of identical nucleai symmetry requirements imposed on the nuclear wave function have
to be taken into account. Consider the simple (and practically most important) example of H2 . The spin of the nucleus ia I = 12 . Thus, the total spin
of the protons of H2 (I1 = I2 = 12 ) may be 0 or 1; the two cases are:
Ortohydrogen: I = 1, Iz = −1, 0, 1. Triplet, symmetric spin wave
function.
Parahydrogen: I = 0, Iz = 0. Singlet, antisymmetric spin wave function.
Since protons are identical fermions, their wave function must be antisymmetric. Permutation of the coordinate variables of the protons corresponds
7.4. STATISTICS OF BOSONS AND FERMIONS
101
to the change of sign of the relative position vector r = r 1 − r 2 . With respect to this operation the parity of rotation states (described by the spherical harmonics Yℓm ) follows the parity of the quantum number ℓ. Thus, the
rules of calculation are:
X
Tr
(2ℓ + 1) exp − ℓ(ℓ + 1) ,
Ortohydrogen, I=1: Zorto =
T
ℓ=1,3,...
X
Tr
(2ℓ + 1) exp − ℓ(ℓ + 1) ,
Parahydrogen, I=0: Zpara =
T
ℓ=0,2,...
The partition function of the equilibrium system is thus
Z rot−yd = 3Zorto + Zpara .
(7.66)
At high temperature molecular collisions lead to frequent enough conversions between the orto and parastates to bring about thermodynamic equilibrium. Then the partition function (7.66) is applicable. For T ≫ Tr we see
that Zorto = Zpara and thus all four spin states are equally probable.
At low temperature T ≪ Tr the parahydrogen is dominant: Zpara ≫
Zorto . In may happen, however, that in the cooling the orto-para conversion
does not occur often enough, and the gas stays as a metastable mixture of
orto and paragases with the spin populations in the ration 3:1. The only
example of this phenomenon in practice is the hydrogen gas. Instead of the
equilibrium partition function (7.66) for the N -particle system the partition
function
rot−yd
ZN,meta
= Zorto 3N/4 Zpara N/4
(7.67)
must be used, since the numbers of the orto and paramolecules are 3N/4
and N/4, respectively. Due to factorization separate additive contributions
to the internal energy follow:
rot−yd
=
Umeta
3
1
Uorto + Upara ;
4
4
with the corresponding heat capacity
rot−yd
CV,meta
=
1
3
Corto + Cpara .
4
4
(7.68)
This is an example of non-ergodic behaviour of a macroscopic system, when
the approach to the global thermodynamic equilibrium is hindered by dynamic reasons or the relaxation time to arrive at equilibrium is too long.
7.4 Statistics of bosons and fermions
Bose–Einstein and Fermi–Dirac statistics. Relativistic quantum
mechanics shows that there is a deep connection between the spin os a
particle and its statistics. For identical particles with an integer spin
s = 0, 1, 2, . . . the many-particle wave function must be symmetric under
permutations. In case of a half-integer spin s = 21 , 32 , 25 , . . . an antisymmetric
102
7. IDEAL EQUILIBRIUM SYSTEMS
wave function is required. In the former case the particles are bosons, obeying Bose–Einstein statistics; in the latter fermions, obeying Fermi–Dirac
statistics. Further, the abbreviations BE- and FD-statistics will be used.
Enumerate the one-particle states with the label ℓ. For non-interacting
particles many-particle wave functions are constructed as symmetrized
(bosons) or antisymmetrized (fermions) products of one-particle wave functions. Due to the (anti)symmetrization it does not matter which arguments
are prescribed to each one-particle wave function initially. What matters is
the number of wave functions corresponding to a given state in the set of
one-particle functions with the aid of which the many-particle wave function is constructed. This set is thus unambiguously described by the occupation numbers nℓ of the one-particle states. The distribution of particles to
these states is illustrated in Fig. 7–5. By construction of the wave function,
to each set of occupation numbers n1 , n2 , n3 . . . , nℓ , it corresponds exactly
one (and thus non-degenerate) many-particle wave function. Therefore,
summation over all microstates in the partition function is tantamount to
summing over all possible occupation numbers of one-particle states, which
are 0 and 1 for fermions and any non-negative integer for bosons.
ε
ε
ε3
ε4
ε1
ε0
ε1
ε0
Figure 7–5: Occupation scheme of one-particle states for (a) bosons, and (b)
fermions.
The energy eigenvalue in the corresponding many-particle basis state
|{nℓ }i = | n1 , n2 , n3 . . . , nℓ , i of the Fock space is
E=
X
εℓ nℓ .
(7.69)
ℓ
Only the grand canonical partition function may be readily calculated.
Since no constraints are imposed on the occupation numbers in this case,
all combinations of allowed one-particle state occupations are included in
the partition function. This means that the sum over the occupation numbers of each state ℓ is independent of others. For a system of bosons we
7.4. STATISTICS OF BOSONS AND FERMIONS
103
obtain
ZG,BE =
X
{nℓ }
=
"
exp −β
"∞
Y X
ℓ
e
X
ℓ
#
nℓ (εℓ − µ) =
−βn(εℓ −µ)
n=0
#
=
Y
ℓ
∞
∞
X
X
n1 =0 n2 =0
···
1
1−
e−β(εℓ −µ)
Y
ℓ
exp [−βnℓ (εℓ − µ)]
(7.70)
,
where µ is the chemical potential. In a fermionic system the Pauli exclusion
rule allows for no more than a single-particle occupancy, therefore
"
#
" 1
#
1
1
X
X
X
Y X
−βn(εℓ −µ)
ZG,F D =
· · · exp −β
nℓ (εℓ − µ) =
e
n1 =0 n2 =0
=
Yh
1+e
−β(εℓ −µ)
ℓ
i
ℓ
ℓ
n=0
(7.71)
.
In either case the probability in the ensemble of a many-particle quantum
state |{nℓ }i is
#
"
X
1
nℓ (εℓ − µ) .
(7.72)
exp −β
P ({nℓ }) =
ZG
ℓ
From now on both the bosonic and fermionic results are exposed parallelly,
because they differ in a couple of signs only. In the formulae the upper
signs refer to the boson and lower signs to the fermion statistics. The grand
canonical partition function gives rise to the grand potential
BE
FD
Ω(T, V, µ) = ±T
X
ℓ
n
o
ln 1 ∓ e−β(εℓ −µ) ,
(7.73)
which depends, apart from the explicitly indicated variables T, V, µ, on the
one-particle energy spectrum εℓ . The derivative with respect to εℓ is the
expectation value of the occupation number
∂Ω
∂εℓ
∂
ln ZG
∂εℓ
#
"
X
1 X
nk (εk − µ) = hnℓ i = nℓ
nℓ exp −β
ZG
= −T
=
(7.74)
k
{nk }
according to the definition of the probability density (7.72). Calculation of
this derivative from the expression (7.73) for Ω yields
BE
FD
nℓ =
1
eβ(εℓ −µ)
∓1
.
(7.75)
The occupation number of a Fermi gas fulfils the condition 0 ≤ nℓ ≤ 1.
Distribution of the occupation numbers heavily depend on the chemical
potential µ. With the usual normalization of the one-particle energy levels
104
7. IDEAL EQUILIBRIUM SYSTEMS
with vanishing ground-state energy ε0 = 0 in an ideal boson system µ ≤ 0.
The chemical potential of a fermion system may have both signs, but it is
often positive and even µ ≫ T , and then the degenerate fermion system
exhibits strong quantum effects.
Thermodynamic variables. According to section 2.6 the differential of
the grand potential is dΩ = −S dT − p dV − N dµ. Thus, the entropy is
i 1X
X h
∂Ω
S=−
ln 1 ∓ e−β(εℓ −µ) +
=∓
nℓ (εℓ − µ) .
∂T µ
T
ℓ
ℓ
This result may be written completely in terms of the occupation numbers.
Since
1 ± nℓ
1
± 1 ⇒ β (εℓ − µ) = ln
,
eβ(εℓ −µ) =
nℓ
nℓ
the entropy assumes the form
X
BE
{± (1 ± nℓ ) ln (1 ± nℓ ) − nℓ ln nℓ } .
S=
(7.76)
FD
ℓ
The average particle number N and the energy E (= the internal energy
U ) may be obtained from the thermodynamic relations N = −∂Ω/∂µ and
E = Ω + T S + µN = Ω − T (∂Ω/∂T ) − µ(∂Ω/∂µ), or from the definition of
expectation values as
X
N =N =
nℓ ,
(7.77)
E=
X
ℓ
(7.78)
n ℓ εℓ .
ℓ
Further results depend on the structure of the one-particle spectrum.
Translation-invariant ideal quantum gas. In translation-invariant
case plane waves in a box may be used as the spatial wave functions. For
non-relativistic particles the spectrum is εℓ = p2 /2m and in the thermodynamic limit the sum over one-particle states may be replaced by the integral
(see section 5.2)
Z ∞
X
X
··· =
dε ω1 (ε) · · · ,
(7.79)
··· = g
ℓ
where ω1 (ε) = C1 V
√
0
k
ε and
C1 = 2πg
2m
h2
32
.
(7.80)
Here, g is the spin degeneracy factor (or the partition function of internal
degrees of freedom). The particle number, energy and grand potential then
are, according to relations (7.77), (7.78) and (7.73),
Z ∞
√
1
BE
N
=
C
V
dε ε β(ε−µ)
,
(7.81)
1
FD
e
∓1
0
7.4. STATISTICS OF BOSONS AND FERMIONS
BE
FD
BE
FD
E = C1 V
Z
∞
dε
0
Ω = ±C1 V T
Z
∞
dε
√
0
105
ε3/2
eβ(ε−µ)
∓1
,
h
i
ε ln 1 ∓ e−β(ε−µ) .
(7.82)
(7.83)
Integration by parts in the last relation and comparison with (8.4) allows
to relate the grand potential and the internal energy as
2
Ω=− E,
3
(7.84)
which immediately gives rise to the equation of state in the form
pV =
2
E.
3
(7.85)
Here, the energy is expressed as a function of V , T and µ. Therefore, to
arrive at the equation of state in the usual variables p = p(N/V, T ), the
chemical potential µ = µ(N/V, T ) has to be solved from the ”normalization
condition” (7.81).
For small occupation numbers nℓ ≈ e−β(εℓ −µ) ≪ 1 the grand potential
may be calculated explicitly with the result
Ω ≈ −eµ/T gV T 5/2
(2πm)3/2
,
h3
which is exactly the same expression as that was obtained for the MB gas
(7.46) and confirms most directly and irrevocably the correctness of the
counting of states with the aid of the normalization factor N ! for the MB
gas.
Adiabatic equation of state of ideal quantum gas. With the aid of
the change of variables ε = T z it follows from the integral representations
(7.81) - (7.83) that
µ
,
(7.86)
Ω = −pV = V T 5/2 f
T
µ
∂Ω
,
(7.87)
= V T 3/2 g
S=−
∂T
T
∂Ω
µ
N =−
,
(7.88)
= −V T 3/2 f ′
∂µ
T
where the explicit form of the functions f and g is unimportant, although
integral representations may be constructed in an obvious way, e.g., from
relation (7.83).
Consider fixed N and a reversible adiabatic process. Then the entropy
S must be constant as well. From relations (7.87) and (7.88) it then follows
that the ratio S/N is a function of the ratio µ/T only. Hence, in a reversible
adiabatic process the ratio µ/T is also a constant. From equation (7.86) we
106
7. IDEAL EQUILIBRIUM SYSTEMS
then infer that in the adiabatic process of a quantum ideal gas the expression pT −5/2 remains constant and thus arrive at the adiabatic equations of
state for translation-invariant ideal BE and FE gas:
−5/2
pT −5/2 = p0 T0
,
3/2
pV 3/2 = p0 V0
3/2
V T 3/2 = V0 T0
,
.
Example 7.4. Fluctuation of the occupation number and the particle number of boson gas. Let us calculate ∆n2ℓ = h(nℓ − nℓ )2 i in a boson
gas. By definition
P
P
ℓ nℓ (εℓ − µ)]
{nℓ } nℓ exp [−β
P
.
(7.89)
nℓ = P
ℓ nℓ (εℓ − µ)]
{nℓ } exp [−β
Differentiating with respect to εℓ we obtain
∂nℓ
= β n2ℓ − hn2ℓ i .
∂εℓ
On the other hand we have seen that
nℓ =
1
,
eβ(εℓ −µ) − 1
(7.90)
whose derivative is
)
(
1
1
∂nℓ
2
= −β
+
2 = −β nℓ + nℓ .
β(ε
−µ)
∂εℓ
eβ(εℓ −µ) − 1
e ℓ
−1
Thus, the fluctuation of the occupation number is given by the relation
h(nℓ − nℓ )2 i = nℓ + n2ℓ .
Fluctuation of the particle number is
!2 +
*
X
X
2
2
∆N = h N − N i =
[hni nj i − ni nj ] .
(ni − ni
i,j
i
To calculate the correlation function of the occupation number, differentiate
(7.89) with respect to εm :
∂nℓ
= β (nℓ nm − hnℓ nm i) .
∂εm
On the other hand, the derivative of (7.90) yields
therefore
∂nℓ
= −βδℓm nℓ + n2ℓ ,
∂εm
h(ni − ni ) (nj − nj )i = δij ni + n2i
and the fluctuation of the particle number obeys
D
2 E X
nℓ + n2ℓ .
=
N −N
ℓ
7.5. PROBLEMS
107
7.5 Problems
Problem 7.1. Show that for the heat capacity of a system of noninteracting magnetic moments with the total angular momentum J the
relation
CH ≡ T
∂S
∂T
=
H
J(J + 1)
V µ0 H 2
χH = N
T
3
µB gB
T
2
holds in the weak-field limit (µB gB ≪ T ). Here, µB is the Bohr magneton and g the Lande factor.
Problem 7.2. A paramagnet in one dimension can be modelled as a
linear chain of N + 1 spins. Each spin interacts with its neighbours
in such a way that the energy is E = nǫ, where n is the number of
domain walls separating regions of up spins from regions of down spins.
Calculate the entropy S(E) with the use of the statistical weight of the
macrostate with the energy E = nǫ, and show that in the limit of both
n and N large the energy may be expressed as
E=
Nǫ
.
eǫ/T + 1
Problem 7.3. Consider the model of one-dimensional paramagnet as a
linear chain of N + 1 spins interacting in such a way that the energy is
E = nǫ, where n is the number of domain walls separating regions of
up spins from regions of down spins. Calculate the partition function,
free energy and entropy of this system in the canonical ensemble.
Problem 7.4. A monatomic gas at the temperature T is contained in a
vessel from which it leaks through a fine hole. Show that the average
kinetic energy of the molecules leaving through the hole is 2T .
Problem 7.5. In a thin-walled vessel of volume V there are N0
molecules of a perfect gas. At the time instant t = 0 the gas begins
to leak out through a tiny hole of area A. Assuming that the outside
pressure is negligible, calculate the number of molecules in the vessel
as a function of time t.
Problem 7.6. Calculate hv 2 i and the average speed h|v|i in a gas whose
center of mass is moving at the velocity V , i.e. the velocity distribution
is
1
2
f (v) ∝ exp −
m(v − V ) .
2T
Problem 7.7.
a) From spectroscopy it is known that the nitrogen molecule has vibrational excited states with energy levels En = ~ω(n + 12 ). If the
level separation ~ω = 0.3 eV, what is then the ratio of the number
of molecules in the first excited state (n = 1) and in the ground
state (n = 0), when the gas is in thermodynamic equilibrium at
the temperature 1000◦ K?
108
7. IDEAL EQUILIBRIUM SYSTEMS
b) The energy gap between the ground state and the first excited
state of a helium atom is 19.82 eV. Assuming nondegenerate
ground state and degenerate excited state with g = 3 estimate
the relative frequency to find atoms in these states in a gas at the
temperature 10 000 K?
Problem 7.8. Derive the following results for the fluctuations of a perfect FD gas:
h(nl − nl )2 i
=
h(N − N ) i
=
2
nl (1 − nl )
X
l
nl (1 − nl ) .
8. Bosonic systems
8.1 Bose gas and Bose condensation
Perfect Bose-Einstein gas. In case of translational invariant ideal
boson system with the non-relativistic dispersion law εℓ = p2 /2m in the
thermodynamic limit the sum over one-particle states may be replaced by
the integral
Z ∞
X
X
··· =
dε ω1 (ε) · · · ,
(8.1)
··· = g
ℓ
√
where ω1 (ε) = C1 V ε and
0
k
C1 = 2πg
2m
h2
32
.
(8.2)
The particle number, energy and the grand potential are then
Z ∞
√
1
,
(8.3)
N = C1 V
dε ε β(ε−µ)
e
−1
0
Z ∞
ε3/2
,
(8.4)
E = C1 V
dε β(ε−µ)
e
−1
0
Z ∞
Z ∞
h
i
√
2
ε3/2
dε ε ln 1 − e−β(ε−µ) = − C1 V
Ω = C1 V T
dε β(ε−µ)
. (8.5)
3
e
−1
0
0
In the limit of dilute gas (n = N/V → 0, when also z = eβµ → 0) these
quantities coincide with those of the Maxwell–Boltzmann ideal gas so that
in this region the ideal Bose gas obeys the equation of state of the classical
ideal gas.
Corrections to the MB limit may be conveniently obtained, when the
functions (8.3), (8.4) and (8.5) are expressed as series in the fugacity z.
Consider, for instance, the integral for the grand potential:
Z ∞
Z ∞
ε3/2
2
ε3/2 ze−βε
2
dε βε −1
= − C1 V
dε
.
Ω = − C1 V
3
e z −1
3
1 − e−βε z
0
0
The denominator of the integrand gives rise to geometric series, and having changed the order of integration and summation we arrive at readily
calculable integrals of the form
Z ∞
T 5/2 Γ(5/2)
.
dε ε3/2 e−βε(n+1) =
(n + 1)5/2
0
109
110
8. BOSONIC SYSTEMS
We thus obtain
2
Ω = − C1 Γ
3
∞
X
5
zn
V zT 5/2
.
2
(n + 1)5/2
n=0
The series here is one definition of a less known special function, the polylogarithm Liν :
∞
X
zn
Liν (z) =
.
(8.6)
nν
n=1
From this series of unit radius of convergence we immediately see the connection with the Riemann ζ function:
(8.7)
Liν (1) = ζ(ν) .
From the calculation of the integral for the grand potential the following
generalization is readily inferred:
Liν (z) =
1
Γ(ν)
Z
∞
0
dt
tν−1
,
−1
et z −1
(8.8)
which allows for analytic continuation to values Re z < 1. The polylogarithm Liν (z) has a branching point at z = 1, and the corresponding branch
cut is usually put on the real axis from z = 1 to z = ∞.
Thus, the fugacity expansion of the grand potential may be compactly
written as
5
2
V T 5/2 Li 52 (z) .
(8.9)
Ω = − C1 Γ
3
2
Similarly
3
V T 3/2 Li 32 (z) .
N = C1 Γ
2
(8.10)
Quantum corrections to the MB limit of the boson gas may be expressed
in more transparent variables by solving for the chemical potential µ =
µ(n, T ) from (8.10) and substituting in (8.9) to write the grand potential as
a function of the volume, particle density and the temperature. As a matter
of fact, this is the procedure for construction of the virial expansion for the
BE gas, since Ω = −pV .
Bose–Einstein condensation. In the limit of dense matter properties
of the ideal BE gas differ dramatically from those of the classical ideal gas.
Since in stable matter ∂N /∂µ > 0 and µ ≤ 0, the particle number N at given
temperature approaches its maximum value in the limit µ → 0− . Denote
this maximum value N1 (T ):
N1 (T ) = C1 V
Z
0
∞
√
ε
= C1 V T 3/2 Γ
dε βε
e −1
√ 3
3
3
π
ζ
=
ζ
C1 V T 3/2 ,
2
2
2
2
8.1. BOSE GAS AND BOSE CONDENSATION
111
where ζ(1.5) ≈ 2.612. The maximum particle density is thus
√ N1 (T )
π
3
n1 (T ) =
= AT 3/2 ; A =
ζ
C1 .
V
2
2
(8.11)
For fixed particle density n direct calculation yields
∂µ
∂T
n
=−
1
T
Z
∞
ε1/2 (ε − µ)eβ(ε−µ)
(eβ(ε−µ) − 1)2
0
< 0.
Z ∞
ε1/2 eβ(ε−µ)
dε β(ε−µ)
(e
− 1)2
0
dε
revealing that the condition n < n1 (T ) may be fullfilled only above the
3/2
critical temperature Tc determined by the condition ATc = n as
h2
Tc =
2πm
n
gζ
3
2
! 32
.
(8.12)
When the temperature is further lowered, the chemical potential cannot
grow any more. In the region T < Tc the Bose condensation takes place
meaning that a macroscopic portion of the particles occupies the lowestenergy one-particle state ε0 = 0. In this region in the sum over oneparticle states (8.1) the continuum approximation becomes inapplicable.
The lowest-energy state must be left outside the integration as a separate
term. The correct rule is
Z ∞
X
dε ω1 (ε) · · · .
(8.13)
· · · = (term ℓ = 0) +
0
ℓ
In a system quantized in a finite volume the energy levels εℓ are discrete
and differences between low-lying levels ∝ V −2/3 . When µ approaches the
energy ε0 = 0 from below, the occupation number
n0 =
1
e−βµ
−1
⇒ N0 (T )
becomes arbitrarily large. The continuum integral (8.3) represents – up to
extensive terms – correctly the number of the excited particles (ℓ 6= 0), so
that in the region T < Tc on N = N0 + AV T 3/2 . With the use of the result
(8.11) we obtain
23
T
N1
=n
V
Tc
"
(8.14)
32 #
N0
T
=n 1−
V
Tc
These functions are illustrated in Fig. 8–1.
Bose–Einstein gas is an ex-
112
8. BOSONIC SYSTEMS
N
N1
N
0
µ (< 0)
N0
T
µ
Figure 8–1: Particle numbers and chemical potential of ideal Bose gas as
functions of temperature in a fixed volume.
ceptional example of a system, in which the grand canonical ensemble is
not statistically equivalent to the canonical ensemble. In the condensate
phase the grand-canonical fluctuation of the particle number is macroscopic
∆N/N ∝ 1. This is clearly unphysical, so that in principle it would be more
correct to use the canonical ensemble. This may practically effected – up
to extensive terms – by putting the chemical potential equal to zero in the
mean occupation number of the excited states and replacing thereafter the
sum over excited states by the continuum integral. Direct contributions of
the condensate drop out from the thermodynamic potentials. For instance,
the internal energy is determined by the excited particles only
Z ∞
Z ∞
5
5
x3/2
ε3/2
= C1 V T 5/2
= C1 V T 5/2 Γ
ζ
.
dx x
E = C1 V
dε βε
e
−
1
e
−
1
2
2
0
0
From here the heat capacity immediately follows
5
5
∂E
5
3/2
= C1 V T Γ
ζ
.
CV =
∂T V
2
2
2
Integration of the expression for the heat capacity as the derivative of the
entropy allows, together with the normalization due to the third law, to find
the entropy as well:
∂E
5
3
5
3/2
S=
ζ
.
= C1 V T Γ
∂T V
5
2
2
Finally, the equation of states follows from the observation that in the extensive approximation F = Ω = − 32 E and
∂F
5
5
2
p=−
ζ
.
= C1 T 5/2 Γ
∂T V
3
2
2
The pressure is thus independent of the volume! Physically this means that
any change in the volume is accompanied by such a change in the number of
8.1. BOSE GAS AND BOSE CONDENSATION
113
particles in the condensate that the pressure remains the same (at constant
temperature). For instance, an isothermal compression would lead to the
growth of N0 , but N1 /V would remain constant together as the pressure
p. There is no energy, entropy or pressure related to the particles in the
condensate state ε0 = 0. The phase diagrams of the matter in the Vp and
Tp planes are depicted in Fig. 8–2. Due to infinite compressibility a part of
the Tp plane is completely excluded.
p
normaali faasi
p
T3 > T2 > T1
poissuljettu alue
kondensoitunut
normaali faasi
kondensoitunut faasi
V
T
Figure 8–2: Phase diagrams of the ideal Bose gas.
Bose condensation in alkali vapours. The Bose-Einstein condensation was experimentally observed in 1995 in alkali vapours (87 Rb, 23 Na,
7
Li) laser cooled below microkelvin temperatures. The cooling took place in
magneto-optical traps, in which the potential energy of the magnetic moments of the atoms in a radio-frequency alternating magnetic field may be
described by a harmonic potential
V (x, y, z) =
1
1
1
k1 x2 + k2 y 2 + k3 z 2 .
2
2
2
(8.15)
The system is thus inhomogeneous and the Bose condensation shows not
only in the momentum space but also in the real space.
The genuine Bose condensation requires a continuum spectrum of the
one-particle states starting from the ground state. In the boson gas in a
box this is achieved in the usual thermodynamic limit: N, V → ∞ with
constant N/V . In the harmonic potential the corresponding limit is N →
∞, ωi → 0 with N ω1 ω2 ω3 fixed, i.e. the harmonic potential becomes flat.
However, differences in the approach to the flat full space lead to physical
results different from those obtained in the thermodynamic limit of a gas
initially enclosed in a rectangular box, which emphasizes the sensitivity
of the thermodynamic limit to the way it is accomplished. For instance,
relations (8.14) are replaced by
114
8. BOSONIC SYSTEMS
N1 = N
"
T
Tc
3
N0 = N 1 −
T
Tc
(8.16)
3 #
with different powers of the temperature.
To see this, let us calculate the density of states of a boson in the harmonic potential (8.15). The eigenvalues of the Hamilton operator are given
by
1
1
1
εn1 ,n2 ,n3 = ~ω1 n1 +
+ ~ω2 n2 +
+ ~ω3 n3 +
,
(8.17)
2
2
2
where the ni = 0, 1, 2 . . . are integers labeling the eigenfunctions. Expectation values are calculated over all values of ni weighed by the mean occupation number. In the thermodynamic limit it would be desirable to arrive
at an integral over the one-particle energies. Let us first introduce the variables
u = ~ω1 n1 ,
v = ~ω2 n2 ,
w = ~ω3 n3 ,
for which the difference between adjacent values ~ωi becomes small in the
limit ωi → 0, thus allowing for representation of the sum over ni as an
integral sum:
X
n1 ,n2 ,n3
Z∞ Z∞ Z∞
X
1
1
∆u∆v∆w −→ 3
du dv dw .
= 3
~ ω1 ω2 ω3 u,v,w
~ ω1 ω2 ω3
0
0
0
The average number of particles, for instance, may be expressed as
N=
X
n1 ,n2
g
gn (εn1 ,n2 ,n3 ) −→ 3
~
ω
1 ω2 ω3
,n
3
Z∞ Z∞ Z∞
du dv dw
0
0
1
eβ(u+v+w+ε0 −µ)
0
−1
,
(8.18)
where ε0 = ~2 (ω1 + ω2 + ω3 ) is the energy of zero-point oscillations. This
relation suggests that the thermodynamic limit – in analogy with the gas
in the box – corresponds to N → ∞, ωi → 0 such that the product N ω1 ω2 ω3
is kept constant.
Introduce then the variables u = x2 , v = y 2 , w = z 2 in which it is evident
that the the integral over one-particle excitation energies may be obtained
in the spherical polar coordinates, since the mean occupation number depends on x2 + y 2 + z 2 = r2 only:
8g
N= 3
~ ω1 ω2 ω3
Z∞ Z∞ Z∞
dx dy dz
0
0
0
xyz
.
eβ(x2 +y2 +z2 +ε0 −µ) − 1
8.2. BLACK BODY RADIATION
115
After calculation of the angular integrals and introduction of the excitation energy ε′ = r2 as the new radial integration variable we arrive at the
expression
Z∞
g
1
N= 3
(8.19)
dε′ ε′2 β(ε′ +ε −µ)
0
2~ ω1 ω2 ω3
e
−1
0
showing that the density of states of the BE gas in the harmonic potential
well is
2
g (ε − ε0 )
.
(8.20)
ω1 (ε) = 3
2~ ω1 ω2 ω3
The condensation temperature is now determined by the condition µ = ε0
as
1
N ω 1 ω2 ω3 3
.
(8.21)
Tc = ~
gζ(3)
As for the gas in the box, from relations (8.19), (8.21) the temperature dependence of the number of particles is inferred in the form (8.16) quoted
above. The thermodynamics of the BE gas in the harmonic well below Tc is
also determined by particles above the condensate only. For instance, the
internal energy assumes the form
g
E = N0 ε 0 + 3
2~ ω1 ω2 ω3
Z∞
ε′ + ε 0
π4 T 4
dε′ ε′2 βε′
= N ε0 +
.
e −1
30~3 ω1 ω2 ω3
(8.22)
0
8.2 Black body radiation
Quantization of the energy of electromagnetic radiation. Consider a free electromagnetic field with vanishing scalar potential φ = 0 and
the vector potential A subject to the transversality condition ∇ · A = 0.
Then the fields are
E=−
∂A
,
∂t
B = ∇ × A,
(8.23)
and the Maxwell equations give rise to the wave equation for A
∇2 A −
1 ∂2A
= 0.
c2 ∂t2
Put the system in a box of volume V and express the vector potential as the
linear combination of plane waves (normalized to unity in the volume V )
X e−iωt+ik·r
eiωt−ik·r
√
,
(8.24)
+ a∗k √
ak
A=
V
V
k
where k · ak = 0 and ak ∝ e−iωt with the dispersion law ω(k) = c|k|. The
sum in (8.24) is taken over all integers labeling the components of the wave
116
8. BOSONIC SYSTEMS
1 , 2πn2 , 2πn3 . Introduce the canonical
vector in the box, where k = 2πn
L
L
L
x
variables
y
z
ε0 ak e−iωt + a∗k eiωt ,
√
P k = Q̇k = −iω ε0 ak e−iωt − a∗k eiωt .
Qk =
√
(8.25)
Expressing the coefficients of the Fourier expansion of the vector potential
in terms of the canonical variables
1
Pk
−iωt
ak e
= √
Qk + i
2 ε0
ω
1
P
k
a∗k eiωt = √
,
(8.26)
Qk − i
2 ε0
ω
substituting in the expression for the Hamilton function of the electromagnetic field becomes
Z
ε0 E 2
B2
H = d3 r
+
,
2
2µ0
resolving the wave-vector sums with the aid of the identity
Z
′
d3 reir·(k−k ) = V δn1 n′1 δn2 n′2 δn3 n′3
and the transversality condition k · ak = 0 we arrive at the the Hamilton
function of the electromagnetic field in the form of the sum of Hamilton
functions of independent harmonic oscillators:
Z
B2
1X
ε0 E 2
P 2k + ω 2 Q2k .
+
=
H = d3 r
2
2µ0
2
k
Due to transversality k·ak = 0 the vectors P k and Qk have only two components corresponding the two independent polarization modes. Therefore, in
the box the Hamilton function of the electromagnetic field in the canonical
variables may be written as
1X 2
H=
Pk,λ + ω 2 Q2k,λ ,
(8.27)
2
k,λ
where λ labels the polarization components. This representation suggests
quantization with the introduction of the momentum and coordinate operators with the standard commutation rules
i
h
b k′ ,λ′ = −iℏδλ,λ′ δk,k′ .
Pbk,λ , Q
(8.28)
Due to the orthogonality of plane waves with different wave vectors the
Hamilton operator assumes exactly the same form as the Hamilton function
with canonical variables replaced by operators:
X
2
b2
b =1
Pbk,λ
+ ω2 Q
(8.29)
H
k,λ .
2
k,λ
8.2. BLACK BODY RADIATION
117
The energy eigenvalues are thus
E=
X
k,λ
1
ℏω nk,λ +
,
2
(8.30)
where nk,λ = 0, 1, 2, . . . for each set k, λ. Similarly, the expression for the
momentum of the electromagnetic field
Z
1
P =
d3 rE × B
µ0 c
in terms of the canonical variables (8.25) gives rise to the momentum operator in the form
X k 2
b =1
b 2k,λ ,
P
(8.31)
Pbk,λ
+ ω2 Q
2
ck
k,λ
whose eigenvalues are also expressed with the aid of the occupation numbers nk,λ as
X 1
ℏk nk,λ +
P =
.
(8.32)
2
k,λ
The representations (8.30) and (8.32) allow for the interpretation of the excitations of the quantized electromagnetic field as photons of energy ε = ℏω
and momentum p = ℏk with the dispersion law ε = cp. The quantity nk,λ
then is the number of photons of frequency ω, direction of propagation k/k
and polarization state λ. Note that this involves a new physical interpretation of the excited states of the harmonic oscillator, each of which now
represents nk,λ particles with the energy ε = ℏω, momentum p = ℏk and
the polarization vector eλ .
Since the occupation number nk,λ = 0, 1, 2, . . . may be any non-negative
integer, the photons obey Bose-Einstein statistics. The polarization state is
a degree of freedom corresponding to spin, whose value is actually 1, but
the longitudinal component is absent in the radiation field and the spin degeneracy factor is g = 2. In the description of radiation the zero-frequency
energy and momentum are usually subtracted from expressions (8.30) and
(8.32) and deferred to the definition of the ”vacuum”.
The energy of photons may be arbitrarily small, therefore it is not possible to fix their number or even its average as the condition for the construction of a statistical ensemble. Rather, the number of photons is determined
as an equilibrium condition for the free energy as ∂F = µ = 0. Therefore,
∂N
the mean ”occupation number” of the photons corresponding to an oscillator
mode is
1
n(ω) = β~ω
.
(8.33)
e
−1
This is the Planck distribution.. The density of states is often determined
with respect to the angular frequency. In the thermodynamic limit the sum
118
8. BOSONIC SYSTEMS
over wave vector components is first replaced by the integral over the wavevector space with account of the isotropy of the mean occupation number
dNk = g(L/2π)3 d3 k = gV /(2π 2 ) k 2 dk . When the condition dNk = dNω =
f (ω)dω is then imposed, the density of states with respect to ω is found as:
Z ∞
X
ω2
dω f (ω) · · · ;
f (ω)dω = V 2 3 dω
··· =
(8.34)
π c
0
ℓ
The full energy density of the radiation field now assumes the form
Z ∞
E
E(T ) ≡
=
dω E(ω, T ) ,
(8.35)
V
0
where the spectral energy density E(ω, T ) is given by the Planck radiation
law
~ω 3
.
E(ω, T ) = 2 3 β~ω
(8.36)
π c (e
− 1)
ε(ω,T)
ω max
ω max
ω max
T3
T2
T1
ω
Figure 8–3: Spectral energy density of the radiation field according to
Planck’s radiation law.
In Fig. 8–3 the form of the intensity distribution of the radiation according to Planck’s law is depicted. In the limit of large frequency or low
temperature this law tends to Wien’s law
E(ω, T ) ∝ ν 3 exp(−hν/T ),
where ν = ω/2π and h is a fitting parameter. In the limit of small frequency
the Rayleigh–Jeans law is recovered
E(ω, T ) ∝ ω 2 T ,
which corresponds to the classical equipartition theorem. In Planck’s radiation law the frequency of the maximum energy density ωmax ∝ T thus
explaining Wien’s law for the peak of the spectrum.
8.2. BLACK BODY RADIATION
119
Integration over the frequency in Planck’s radiation law yields
Z ∞
Z ∞
x3
~ω 3
T4
dx x
dω 2 3 β~ω
= 2 3 3
.
E(T ) =
π c (e
− 1)
π c ~ 0
e −1
0
The remaining integral is of the familiar type, thus
Z ∞
x3
π4
dx x
= Γ(4)ζ(4) =
e −1
15
0
leading to the energy density
E(T ) =
4 4
σT ,
c
(8.37)
where σ is the Stefan–Boltzmann constant
σ=
π2
.
60~3 c2
(8.38)
In the ordinary units σ = π 2 kB 4 /(60~3 c2 ) ≈ 5.671 × 10−8 W/(m2 K4 ).
Thermodynamics. Thermodynamics of the photon gas may be deduced
from the expression for the energy density (8.37), since the internal energy
U = EV , i.e.
4σ
U=
V T 4.
(8.39)
c
Therefore, the heat capacity is
16 3
∂S
∂U
=
.
(8.40)
σT = T
CV =
∂T V
c
∂T V
Integrating the last equation with the account of the condition S → 0, T → 0
we arrive at the expression for the entropy
S=
16σ
V T3 .
3c
(8.41)
Therefore, the free energy F = U − T S is
F =−
4σ
V T4 .
3c
(8.42)
Calculation of the pressure p = −∂F/∂V yields the equation of state as
p=
1
4σ 4
T = E(T ).
3c
3
(8.43)
It is worth noting that the pressure is independent of the volume leading to
infinite isothermal compressibility as in the Bose condensate.
120
8. BOSONIC SYSTEMS
It should be borne in mind that as a many-particle system photons constitute an extremely ideal gas, because even scattering events between photons are very rare. Therefore, a pure photon gas is a non-ergodic system
that cannot thermalize due to intrinsic dynamics. If the photon gas is in
equilibrium, which strictly speaking hardly ever is the case, then this is
solely due to interaction with a suitable environment, a black body,. The
black body is an equilibrium system acting as a heat bath by emitting and
absorbing photons without reflection, isotropically in spatial directions and
within a continuous spectrum.
I
cτ
θ
A
Figure 8–4: Black-body radiation and absorption.
Emission and absorption of a black body. A black body may be modelled as a hole in the wall of a hollow container with isotropic black-body
radiation in it. Then the radiation power in a fixed direction may be calculated with the aid of the geometric construction depicted in Fig. 8–4b. During the time τ the radiation in the direction of the angle θ from the outward
normal of the surface comes from the region, whose depth is ℓ(θ) = cτ cos θ
and the volume Aℓ(θ). The total energy of the photons propagating from
this region in the solid angle element dΩ in the direction θ is therefore
E(T )Acτ cos θ
dΩ
,
4π
since of all the photons in a volume element the part dΩ/(4π) is propagating
in the required direction.
The radiance L is the radiation power in a given direction in a unit solid
angle per visible surface area of the radiating body. Since in the direction θ
the visible emitting area is A⊥ = A cos θ, the radiance of the black body is
(to obtain power, divide energy by τ )
L=
c E(T )
.
4π
Since L is independent of direction (i.e., the black-body radiation is diffuse),
the radiating surface appears equally bright from any oblique direction as
from the direction of the normal.
Radiant excitance M is the radiant power to the half-space above the
surface per surface area. Thus, it may be written as the integral of the
8.3. LATTICE VIBRATIONS
121
radiance multiplied by the ratio cos θ = A⊥ /A over the corresponding solid
angle:
Zπ/2
Zπ/2
Z2π
1
c E(T )
Mb =
dθ sin θ dϕL cos θ = 2π
dθ sin θ cos θ = E(T )c . (8.44)
4π
4
0
0
0
Thus, black-body radiation obeys Lambert’s law
M = πL,˛.
Due to relation (8.37), the Stefan–Boltzmann law for the radiant excitance
Mb = σT 4
(8.45)
follows from equation (8.44).
A black body absorbs all the incident radiation, emits diffuse radiation
and its spectral excitance (or spectral radiance) is a universal function depending on the temperature and the angular frequency of the radiation
only.
Irradiance E is the radiant power incident to the surface per surface
area. It may be absorbed (denote the corresponding portion by Ea ), reflected
(Er )or transmitted (Et ). Thus, radiant properties of the surface are characterized by the (spectral) absorption ratio (Ea (ω) = α(ω)E(ω)) Ea = αE, reflection ratio Er = ρE and transmission ratio Et = τ E. Energy conservation
implies that α + ρ + τ = 1. For a black body α = 1, ρ = τ = 0.
Emission of a generic radiating body is characterized by its emissivity
(or spectral emissivity) ǫ:
M (ω, T ) = ǫ(ω, T )Mb (ω, T ) ,
where M (ω, T ) is the excitance of the body at hand. The spectral emissivity
of a grey body is independent of the frequency.
If the body is at equilibrium with the radiation field, then Kirchhoff’s
radiation law
ǫ(ω, T ) = α(ω, T )
holds.
8.3 Lattice vibrations
The relative motion of nucleai in molecules and solids is so slow in comparison with the motion of electrons that these degrees of freedom may
be separated in the Born–Oppenheimer or adiabatic approximation. The
system of nucleai is then described the Hamilton operator on which the influence of electrons appears as a potential energy depending on the nuclear
configuration
N
X
pj 2
+ V (r 1 , r 2 , . . . , r N ) ,
(8.46)
H=
2Mj
j=1
122
8. BOSONIC SYSTEMS
where Mj are the masses of the nucleai (atoms). Denote the equilibrium
positions by Rj and displacements thereof uj = r j − Rj . Expanding with
respect to the latter we obtain
V = V0 +
X
Vjx ujx +
jx
1 X xy
Vjk ujx uky + · · · ,
2!
jk,xy
because the linear in displacements terms vanish due to the equilibrium
condition.
In the harmonic approximation the Hamilton function is
H=
N
X
pj 2
1 X X xy
Vjk ujx uky .
+
2Mj
2!
xy
j=1
(8.47)
jk
A Hamilton function of this form may always be diagonalized with the aid of
a canonical transformation which leads to the representation in the normal
coordinates as a system of independent harmonic oscillators:
H=
1X
Pℓ 2 + ωℓ 2 Qℓ 2 ,
2
(8.48)
ℓ
where the angular frequencies ωℓ are obtained from the characteristic equation
h
i
xy
det Vjk
− ω 2 Mj δjk δxy = 0 .
In the quantized theory the canonical variables Pℓ , Qℓ are replaced by operators obeying the usual commutation rules (8.28) so that the Hamilton
b ℓ in (8.48). Thus, the
operator is obtained by the substitution Pℓ , Qℓ → Pbℓ , Q
energy eigenvalues may be written as
X
1
(8.49)
E=
ℏωℓ nℓ +
2
ℓ
giving rise, as in the case of electromagnetic radiation, to the interpretation
of the excited states of the lattice vibrations as a system of particle-like
entities, phonons.
Exact solution. The set of harmonic oscillators (8.48) with the energy
eigenvalues (8.49) gives rise – in analogy with the quantized electromagnetic field – to statistical description in terms of grand canonical ensemble
with vanishing chemical potential. Thus, F = Ω and we arrive at the formally exact relation
X
(8.50)
ln 1 − e−βℏωℓ .
F =T
ℓ
This expression gives rise to a simple analytic form in the limits βℏωℓ ≫ 1
and βℏωℓ ≪ 1.
8.3. LATTICE VIBRATIONS
123
In a box with periodic boundary conditions the wave functions may be
labeled by the components of the wave vector k and the polarization index
λ. Then (8.49) is replaced by
X
1
ℏωλ (k) nλ (k) +
E=
.
(8.51)
2
k,λ
In a lattice no continuum limit is taken, so the number of components of k
remains finite and equal to the number N of elementary cells in the specimen. Roughly speaking, the elementary cell is the smallest volume element
of the solid whose periodic continuation to fill the space reproduces the lattice at hand. The elementary cell does not always convey all the symmetry
properties of the lattice, therefore it is often replaced by the Wigner-Seitz
cell, which obeys the lattice symmetries. The corresponding entity in the
wave-vector space is the first Brillouin zone.
The number of components of the wave vector is usually restricted to
the value N by requiring that all k belong to the first Brillouin zone in the
wave-vector space. It should be borne in mind, although it is not used here,
that the components of k are not necessarily orthogonal. The elementary
cell may contain several atoms. Let the number of atoms in the elementary cell be ν, then there are, in general, 3ν polarization states of the lattice
vibrations. Due to translational invariance, there are always three polarization states with the dispersion law ω ∝ k. These are sound waves or
acoustic modes of lattice vibrations. If the lattice is not simple, i.e. ν > 1,
then optical modes of lattice vibrations appear whose frequency has finite
limit, when k → 0. Fig. 8–5 shows typical dispersion laws for a lattice with
two atoms in the elementary cell.
ω(k)
optiset
akustiset
k
0
Figure 8–5: Dispersion curves of phonons.
At low temperatures (βℏωℓ ≫ 1) the exponential cuts off the largeenergy vibrations. The vibration modes with the lowest energy are sound
waves. In an isotropic substance – which is assumed for simplicity – the
sound waves have three independent vibration modes: one longitudinal and
124
8. BOSONIC SYSTEMS
two transversal modes, for which the dispersion relations may be written
as
ωt = ct k ,
ωl = cl k ,
(8.52)
which allow for transformation from the wave-vector sum to wave-vector
integral in a fashion similar to that used for the photon gas. Since the
speed of sound for the transversal components is the same, the energies
have a twofold degeneracy with respect to polarization and we obtain, at
T → 0:
TV
F =
2π 2
Z∞
dk k 2 2 ln 1 − e−βℏωt + ln 1 − e−βℏωl
0
TV
=
2π 2
2
1
+ 3
c3t
cl
Z∞
dω ω 2 ln 1 − e−βℏω
0
=
3T V
2π 2 c3
Z∞
dω ω 2 ln 1 − e−βℏω , (8.53)
0
where c is the geometric-like mean speed of sound. We see that the free
energy is formally the same as for the photon gas, therefore the substitution
c → c and g = 3 immediately yields the basic thermodynamic quantities of
the phonon gas:
F − F0 = −
π2 V T 4
,
30(ℏc)3
S=
2π 2 V T 3
,
15(ℏc)3
CV =
2π 2 V T 3
.
5(ℏc)3
(8.54)
Here, F0 is the free energy of zero-mode vibrations, which is independent
of the temperature. The physically most important conclusion is the temperature dependence CV ∝ T 3 of lattice vibrations, so that the vibrational
contribution at low temperatures vanishes faster than the electronic contribution.
At high temperatures (βℏωℓ ≪ 1) we approximate 1 − e−βℏωℓ ≈ βℏωℓ and
write
3N ν
X
ℏω
F = F0 +
ln βℏωℓ = F0 + ln
,
(8.55)
T
ℓ
where ω is the geometric mean frequency and 3N ν is the number of vibration modes (N is the number of elementary cells in the lattice and ν the
number of atoms in the elementary cell). Relation (8.55) gives rise the the
heat capacity
CV = 3N ν ,
ℏωℓ ≪ T
in accordance with the Dulong-Petit law conforming to the classical
equipartition theorem (the average energy of each vibration mode is T ).
Debye model. A fairly realistic analytic approximation to the lattice
heat capacity of solids is provided by the Debye model, whose basic assumptions are:
8.3. LATTICE VIBRATIONS
CV/N
125
Dulong-Petit
3
Debye
Einstein
θ
T
Figure 8–6: Lattice heat capacity of a solid.
• Take into account only acoustic modes: one longitudinal and two
transversal modes.
• Assume linear dispersion relations:
(
ωl (k) = cl k
(8.56)
ωt (k) = ct k
• Cut off the spectra at the Debye frequency ωD = θD /~ (where θD is the
Debye temperature) chosen such that the number of vibrational modes
is 3N ν.
Density of states in each mode is
dNj =
L
2π
3
4πk 2 dk =
V
ω 2 dω ,
2π 2 c3j
so that the total density of states is
2
1
V
+ 3 ω 2 dω .
dN =
2π 2 c3t
cl
Z ωD
dN , yielding
The total number of modes is 3N ν =
ω=0
3
ωD
Nν
=
18π 2
V
2
1
+ 3
c3t
cl
−1
.
(8.57)
The density of states is thus
dN (ω) =
9N ν 2
3 ω dω.
ωD
(ω < ωD )
(8.58)
126
8. BOSONIC SYSTEMS
Substance
Li
Na
K
Be
Mg
Ca
* diamond
θD (K)
400
150
100
1000
318
230
Substance
B
Al
Ga
C*
Si
Ge
θD (K)
1250
394
240
1860
625
360
Substance
Cu
Ag
Au
Zn
Cr
Fe
θD (K)
315
215
170
234
460
420
Debye temperatures of some substances
In the Debye model the free energy is
9N ν
F = F0 +
ωD 3
ZωD
dω ω 2 ln 1 − e−βℏω .
(8.59)
0
Stretching integration variable we obtain
F = F0 + 9N νT
T
θD
3 θZD /T
dz z 2 ln 1 − e−z .
(8.60)
0
From this expression the internal energy follows in the form
E = F + T S = E0 + 3N νT
T
θD
3 θZD /T
0
z 3 dz
= E0 + 3N νT D
ez − 1
T
θD
, (8.61)
where the last equality defines the Debye function D(x). In the Debye model
12π 4
CV −→ N
5
T →0
T
θD
3
.
(8.62)
which is an exact result, because the model is tailored to exact for lowenergy phonons. In Fig. 8–6 the specific heat of the Debye model is compared with results of other models. The best fit is given by the Debye model.
For solids with complex elementary cells the agreement with the experiment is worse.
8.4. PROBLEMS
127
8.4 Problems
Problem 8.1. Derive the following expressions for the fugacity, chemical potential and internal energy of the ideal Bose gas:
η2
+
23/2
z
=
η−
µ
=
T ln η −
E
=
1
1
− 3/2
4
3
η
23/2
+3
η
3
N T 1 − 5/2 − 2
2
2
η 3 + O(η 4 )
1
1
− 5/2
24
3
1
35/2
−
1
24
η 2 + O(η 3 )
η 2 + O(η 3 )
where η = N λ3T /gV .
Problem 8.2. Consider an ideal BE gas in the harmonic potential
1
1
1
V (x, y, z) = k1 x2 + k2 y 2 + k3 z 2 .
2
2
2
This is the effective potential energy of alkali atoms in the experimental
observations of the BE condensation in alkali vapours. The BE condensation takes place in the limit ki → 0, N → ∞, which replaces the
usual thermodynamic limit. In this limit, the density of states of nonrelativistic boson gas may be written as
ε2
dN
,
= 3
dε
2~ ω1 ω2 ω3
where ωi are the angular frequencies of the harmonic oscillations in the
potential V .
Calculate the internal energy U , heat capacity CN and entropy S of
this gas below the condensation temperature, and find also the equation
of state.
Problem 8.3. Is the BE condensation possible in a two-dimensional
perfect boson gas? Consider the gas both in a box and harmonic potential well.
Problem 8.4. Calculate the radiant energy of the Sun emitted in the
microwave band of width 1,0 MHz centered at the wavelength 3,0 cm.
Consider the Sun a black body at the temperature 5800 K.
Problem 8.5. Show that in an adiabatic expansion of isolated photon
gas the wavelengths of the photons grow proportionally to the diameter
of the space they occupy. What was the radius of the Universe in the
end of the radiation era (i.e. when the ions formed atoms and radiation
and matter decoupled), if it is now about 15 · 109 light years?
Problem 8.6. Show that, according to the Debye theory, the heat capacity of lattice vibrations at T ≪ θD (=Debye temperature) is
CV =
and at T ≫ θD
"
12π 4
N
5
1
CV = 3N 1 −
20
T
θD
θD
T
3
2
,
#
+ ··· .
9. Fermionic systems
Perfect Fermi gas. The ideal translation invariant Fermi gas is a realistic starting point for many weakly interacting fermionic subtances, e.g.
the system of conduction electrons of a metal. The non-relativistic density
of states in a three-dimensional space is
ω1 (ε) = V 2πg
2m
h2
23
√
(9.1)
ε.
The spin degeneracy is now g = 2. Since the mean occupation number is
n(ε) =
1
eβ(ε−µ)
+1
(9.2)
,
the particle and energy densities are
N
= 4π
V
E
= 4π
V
2m
h2
2m
h2
23 Z
dε
0
23 Z
√
∞
∞
dε
0
ε
eβ(ε−µ)
(9.3)
+1
ε3/2
eβ(ε−µ) + 1
.
(9.4)
They may be expressed in terms of the polylogarithm as
N
= −2
V
2πmT
h2
3Ω
E
=−
= −3
V
2V
32
Li 23 (−z) ,
(9.5)
32
(9.6)
2πm
h2
5
T 2 Li 25 (−z) ,
where z = eβµ is the fugacity.
9.1 Conduction electrons in metals
Degenerate Fermi gas. The most important Fermi systems are far
from the Maxwell-Boltzmann limit: they are dense and heavily degenerate,
since µ > 0 and T /µ ≪ 1. The average occupation number (9.2) is then
nearly a step function and its derivative the δ function with the minus sign.
(see Fig. 9–1).
128
9.1. CONDUCTION ELECTRONS IN METALS
129
(∆n)2 = – T dn
dε
1/4
n
1
T
0
ε
µ
µ
ε
Figure 9–1: Average occupation number of the degenerate Fermi gas.
Completely degenerate Fermi gas (T = 0). In this limit the mean
occupation number is the step function n(ε) = θ(µ − ε). The chemical potential at T = 0 is called the Fermi energy (which in the adopted system of
units is the Fermi temperature as well)
µ = εF =
~2 kF 2
;
2m
(9.7)
where ~kF = pF is the Fermi momentum. In the momentum space all states
up to the Fermi momentum are occupied. In the free Fermi gas the Fermi
surface is a spherical surface of radius kF in the wave-vector space dividing
the occupied and empty states. The particle density is calculated from the
normalization condition
N = 2πgV
2m
h2
3/2 ZεF
3/2
√
2m
4
3/2
dε ε = πgV
εF
3
h2
(9.8)
0
as
n=
gkF 3
6π 2
⇒
kF 3
,
3π 2
(9.9)
where the latter form applies to spin- 21 particles like electrons. The energy
per particle is
R1
dx x3/2
3
= εF .
ε = εF R01
5
dx x1/2
0
From the general non-relativistic relation
pV =
2
E.
3
(9.10)
the equation of state follows in the form
p=
g ℏ2
15π 2 2m
5/3
6π 2
n
.
g
(9.11)
The density of the electron gas of metals is so high that the ordinary room
temperature T is always much lower than the Fermi temperature (or the
130
9. FERMIONIC SYSTEMS
degeneration temperature, T ≪ TF = εF . It should be borne in mind, however, that this is not the case for conduction electrons in insulators and
semiconductors.
Most properties of the degenerate electron gas may be calculated in the
limit of zero temperature. The parameters ri and rs describing the density
of the electron gas are defined as
V
1
4
= = πri 3 ;
N
n
3
rs =
ri
,
a0
(9.12)
where a0 is the first Bohr radius
a0 =
In metals 1.9 <
∼ rs
below.
<
∼
Metal
Be
Al
Ga
Zn
Mg
Cu
Li
Na
K
Rb
Cs
4πε0 ~2
= 0.529 Å.
me2
5.6. Some properties of metals are quoted in the table
rs
1.88
2.07
2.19
2.30
2.65
2.67
3.25
3.93
4.86
5.20
5.63
Valence
2
3
3
2
2
1
1
1
1
1
1
n (1028 m−3 )
24.3
18.2
15.4
13.3
8.67
8.47
4.70
2.66
1.41
1.15
0.904
TF (eV)
14.2
11.7
10.5
9.48
7.14
7.03
4.75
3.25
2.12
1.85
1.58
Parameters of metals
The low-temperature asymptotic behaviour of integrals like (9.3) and
(9.4) whose integrand contains the mean occupation number may obtained
in the following way. Introduce in the generic integral a suitable change of
variable
Z∞
Z∞
f (µ + T z) dz
f (ε) dε
I=
=T
,
ez + 1
eβ(ε−µ) + 1
0
−µ
T
and divide the integration interval at the origin with the subsequent change
of sign of the integration variable to obtain
µ
ZT
Z∞
f (µ − T z) dz
f (µ + T z) dz
+
T
.
I=T
ez + 1
e−z + 1
0
0
(9.13)
9.1. CONDUCTION ELECTRONS IN METALS
131
In the latter integral the identity
1
1
=1− z
e−z + 1
e +1
allows to single out a term similar to the first integral in decomposition
(9.13) so that
µ
I=T
ZT
0
µ
ZT
Z∞
f (µ − T z) dz
f (µ + T z) dz
−T
.
f (µ − T z) dz + T
ez + 1
ez + 1
(9.14)
0
0
So far no approximations have been made. Further we use the observation that in the limit T → 0 the upper limit of the last integral in relation
(9.15) grows without limit (because µ has the finite limit εF ), and – with the
exponential accuracy of order ∼ e−µ/T – we send this limit to infinity. Thus
µ
I≈T
ZT
0
Z∞
[f (µ + T z) − f (µ − T z)] dz
f (µ − T z) dz + T
ez + 1
0
The change of variable µ − T z = ε in the first integral together with the
Taylor expansion in T z of the numerator of the integrand in the second
yield
i
h
Zµ
Z∞ 2T zf ′ (µ) + 1 (T z)3 f ′′′ (µ) dz
3
+ ...
(9.15)
I = f (ε) dε + T
ez + 1
0
0
Integrals appearing here may be calculated in a fashion analogous to that
used in the calculation of the similar bosonic integral:
Z∞
Z∞
Z∞
∞
X
z x−1 e−z
z x−1
x−1
= dz
=
dz
z
(−1)n e−z(n+1)
dz z
e +1
1 + e−z
n=0
0
0
0
∞
X
n+1
(−1)
=
nx
n=1
Z∞
∞
X
(−1)n+1
.
dz z x−1 e−z = Γ(x)
nx
n=1
(9.16)
0
Here, the alternating series does not give the anticipated ζ function directly.
The following trick, however, allows to reorganize it to a difference of easily
identifiable positive-term series:
∞
∞
∞
X
X
X
(−1)n+1
1
1
=
−
=
x
x
x
n
(2k
+
1)
(2l)
n=1
k=0
l=1
∞
X
k=0
∞
∞
l=1
l=1
X 1
X 1
1
+
−
2
(2k + 1)x
(2l)x
(2l)x
132
9. FERMIONIC SYSTEMS
Combination of the two first series on the right-hand side is the desired
series of Riemann’s ζ function, so is the third series from the terms of which
the common factor 2−x is easily extracted, thus we arrive at the conclusion
Z∞
dz
0
z x−1
= Γ(x)ζ(x) 1 − 21−x .
z
e +1
(9.17)
Taking into account the numerical results
ζ(2) =
π2
;
6
ζ(4) =
π4
90
we arrive at the low-temperature expansion for the typical Fermi-gas integral (the Sommerfeld expansion)
Z∞
dε
0
f (ε)
≈
β(ε−µ)
e
+1
Zµ
7π 4 4 ′′′
π2 2 ′
T f (µ) +
T f (µ) + · · ·
dεf (ε) +
6
360
(9.18)
0
Heat capacity. The particle density n = N/V of the electron gas in metals is fixed by the charge neutrality. Expressing the same particle density
at low temperatures with the aid of the Sommerfeld expansion of the normalization condition (9.3) on one hand and at zero temperature in terms of
εF [see relation (9.8)] on the other we arrive at the relation
√
Z ∞
2
π2 2 1
2
ε
dε β(ε−µ)
= µ3/2 +
T √ + · · · = εF 3/2 .
3
12
µ
3
e
+
1
0
The solution is the low-temperature expansion of the chemical potential:
#
"
2
π2 T
+ ··· .
(9.19)
µ(T ) = εF 1 −
12 TF
To calculate the energy according to (9.4) the following integral is needed
Z∞
dε
ε3/2
eβ(ε−µ)
+1
=
0
=
2 5/2 π 2 2 √
µ +
T µ + ···
5
4
"
2 #
5π 2 T
2 5/2
εF
1+
+ ··· .
5
12 TF
Here, expansion (9.18) has been used first, after which the expression of µ
from equation (9.19) has been substituted. The energy per particle is now
ε(T ) =
π2 T 2
3
εF +
.
5
4 TF
(9.20)
9.1. CONDUCTION ELECTRONS IN METALS
133
Further, the heat capacity is
CV =
∂ (N ε)
π2 T
=N
.
∂T
2 TF
(9.21)
The heat capacity of the electron gas vanishes as a linear function of T at
low temperatures. Numerically it is small, however, because the degeneracy temperature TF is very high.
The heat capacity of lattice vibrations of a solid vanishes as the third
power of the temperature at low temperatures (8.62), therefore in metals
the heat capacity of the electron gas becomes dominant at low temperatures. The crossover takes place at temperatures of the order of a few
kelvins.
Electrons and holes in semiconductors. In the band theory of electrical conductivity of a crystalline solid the periodic structure of positive
ions brings about the band structure of one-particle electron states in the
crystal. The bands – densely filled by one-particle states – are separated
by relatively wide gaps in the electron energy spectrum. From the point of
view of electrical conductivity, the two highest-energy bands with occupied
states are of particular interest. At T = 0 the valence band is completely
filled, whereas the conduction band above it is partly filled in conductors
and empty in insulators and semiconductors, the difference between the
latter being in the magnitude of the energy gap between the bottom of the
conduction band and the top of the valence band.
The electron states within bands correspond to almost freely moving
electrons with a rather complicated dispersion relation, though. Near the
bottom of the conduction band, for instance, the dispersion relation – with
the one-particle energy measured from the bottom of the band – is typically
that of a non-relativistic electron with an effective mass me of the order
of the physical mass of the electron. Near the top of the conduction band
the dispersion relation gives rise to the description of conductivity in terms
of moving empty electron states called holes, which behave as positively
charged particles with an effective mass mh of the order of the physical
2 2
electron mass and the dispersion relation εh (k) ≈ − ~2mk , where the origin
h
of the energy scale is at the top of the valence band, hence the negative
energy of the hole.
Since the hole is an electron absent at a state in the valence band, the
mean occupation number of a hole state is the mean occupation number of
the corresponding electron state subtracted from the maximum occupation
number one:
1
1
= β(−ε +µ)
.
(9.22)
nh (ε) = 1 − β(ε −µ)
h
e h
+1
e
+1
Thus, the chemical potential of holes is the chemical potential of electrons
with the minus sign.
From the point of view of the use of the perfect Fermi gas in the description of the electron gas in insulators and semiconductors it should be
134
9. FERMIONIC SYSTEMS
noted that the number density and thus the Fermi temperature of currentcarrying electrons in these materials is much less than in metals. In most
cases the electron gas is actually far from degeneracy and rather accurately
described by the Maxwell-Boltzmann distribution.
Example 9.1. Chemical potential in a semiconductor. Consider the
model of an intrinsic semiconductor corresponding to Fig. 9–2. Assume
that electrons and holes behave as free particles of masses me and mh ,
respectively, and that Eg ≫ T . Show that the
densities of electrons
particle
E
and holes obey the relation ne = nh ∝ exp − 2Tg , calculate the coefficient
of proportionality and the chemical potential at low temperatures. Here,
Eg is the energy gap between the conduction and valence bands.
elektronitilat
Eg
aukkotilat
Figure 9–2: Creation of a hole in the valence band through the excitation
of an electron to the conduction band.
In the MB approximation the number of electrons in the conduction
band may be calculated as
3 Z∞
Z∞
p
2me 2
−β(ε−µ)
dε ε − Eg e−β(ε−µ) ,
Ne = dε ω1e (ε − Eg ) e
= V 2πge
2
h
Eg
Eg
where the origin of the energy scale is chosen at the top of the valence
band, and the one-particle energies of electrons in the conduction band are
2 2
ε = Eg + ~2mk . Calculation yields
e
3
2me π 2 3 −β(Eg −µ)
Ne = V g e
T2e
.
h2
On the other hand, in the MB limit the number of holes is
Nh = V 2πgh
2mh
h2
23 Z0
−∞
√
dε −ε e
−β(−ε+µ)
= V gh
2mh π
h2
32
3
T 2 e−βµ .
In an intrinsic semiconductor at T = 0 the conduction band is empty and
the valence band completely filled, therefore Ne = Nh . Moreover, in the
9.2. MAGNETISM OF DEGENERATE ELECTRON GAS
135
product of the expressions for Ne and Nh the chemical potential is cancelled,
so that
√
2 me mh π 3 3 −βEg
ne nh = n2 = ge gh
T e
h2
and
ne = nh =
√
3
√
2 me mh π 2 3 −βEg /2
ge gh
.
T2e
h2
The chemical potential is readily found from the ratio Ne /Nh as
3 !
gh mh2
1
1
µ = Eg + T ln
,
3
2
2
g me2
e
which corroborates the MB approximation by producing small occupation
numbers both for electrons in the conduction band and holes in the valence
band.
At T = 0 the chemical potential assumes the constant value µ(0) = 21 Eg .
This is sometimes also called the Fermi energy, but it should be borne in
mind that other authors reserve the name Fermi energy to the energy of
the highest occupied electron state at T = 0. In the present model this
is at the top of the valence band, while the zero-temperature value of the
chemical potential does not correspond to any electron state at all.
9.2 Magnetism of degenerate electron gas
Pauli paramagnetism. The magnetic moment of the electron is µe =
γe s ≈ 2γ0 s, where γ0 = −e/(2m) is the classic gyromagnetic ratio. Introducing Bohr’s magneton
µB =
e~
eV
= 5.66 × 10−5
2m
T
we obtain µz = −µB σz , where σz a Pauli spin matrix. In an external magnetic field the energy of the electron is
εpσ =
p2
− µz B = εp + µB Bσ .
2m
(9.23)
(σ = σz = ±1)
The spin degeneracy is lifted, but otherwise the effect is a shift of the chemical potential, so that
Ω = −T
=
X
k
( "
ln 1 + e
−β
#
ℏ2 k2 +µ B−µ
B
2m
"
+ ln 1 + e
−β
#)
ℏ2 k2 +µ B−µ
B
2m
2
1
1
1
2 ∂ Ω0
Ω0 (µ − µB B) + Ω0 (µ − µB B) = Ω0 (µ) + (µB B)
+ ... ,
2
2
2
∂µ2
(9.24)
136
9. FERMIONIC SYSTEMS
where Ω0 is the grand potential of the electron gas, when B = 0. Taking
into account that ∂Ω0 = −N we see that
∂µ
∂N
1
2
+ ... ,
Ω = Ω0 (µ) − (µB B)
2
∂µ T,V
so that the magnetization is
1
M =−
V
∂Ω
∂B
T,V
=
µ2B B
T,V
∂n
∂µ
∂n
∂µ
,
T,V
and the susceptibility
χ=
∂M
∂H
= µ0 µ2B
B=0
.
T,V
In the degenerate electron gas, in which the average occupation number is
nearly a step function
Zµ
ω1 (ε) dε
N =V
0
so that
∂n
∂µ
= ω1 (µ) = D(µ)
T,V
is the density of states (per unit volume) at the Fermi surface (in this limit
µ = εF ) so that
M = µB 2 D(µ)B
and Pauli’s paramagnetic susceptibility assumes the form
χpara = µ0 µB 2 D(µ) ,
(9.25)
which remains valid also in the case, when the dispersion law is different
from the usual isotropic parabolic form (in solids the effective dispersion
law is, as a rule, anisotropic, which leads to a more complex density of
states).
Landau levels. The magnetic field affects the electron energy also
through the orbital motion. This is also a pure quantum effect – in classical statistics there is no diamagnetism of the electron gas. The effect on
the orbital motion is described by the minimal coupling, in which the kinetic momentum p is replaced by the difference p − qA in the Hamilton
function (A is the vector potential and q the charge of the particle). In case
of the electron q = −e and the Hamilton function of a free electron (without
the account of spin) in the magnetic field is
2
H=
(p + eA)
.
2m
9.2. MAGNETISM OF DEGENERATE ELECTRON GAS
137
For the uniform magnetic field B = Bk choose the vector potential in the
Landau gauge: Ax = −By, Ay = Az = 0. The choice of the gauge does
not affect the energy eigenvalues, but it does affect the functional form of
the eigenfunctions, and the Landau gauge turns out to be the simplest to
solve. It should also be borne in mind that the choice of the vector potential
reflects the physical boundary conditions of the problem.
The Hamilton operator corresponding to the Landau gauge is
ie~
−~2
e2 B 2 2
2
b = 1 (b
H
p + eA) =
∂x 2 + ∂ y 2 + ∂ z 2 +
By∂x +
y . (9.26)
2m
2m
m
2m
b commute, solutions of the Schrödinger
Since the operators pbx2 , pbz2 and H
equation with the Hamiltonian (9.26) may be sought in the form
ψ(x, y, z) = eikx+ikz χ(y).
For the function χ the differential equation follows:
ℏkx
p2
~2 ′′ e2 B 2
2
yk =
χ +
(y − yk ) χ = ε − z χ ;
,
−
2m
2m
2m
eB
(9.27)
where ε is the energy eigenvalue of the electron. The equation (9.27) is of
the same form as that of the quantum harmonic oscillator
with the spring
p
constant K = e2 B 2 /m and angular frequency ω = K/m = eB/m = ωc .
The energy eigenvalues are thus grouped into Landau levels with
1
p2
p2z
= z + µB B (2n + 1) .
+ ~ωc n +
(9.28)
εn =
2m
2
2m
The wave functions are localized in the y directions with the width parameter
r
~
.
b0 =
eB
In the x direction they are extended (plane waves) (see Fig. 9–4b).
Each Landau level is heavily degenerate, because the energy eigenvalues (9.28) are independent of the quantum number kx labeling the energy
eigenfunctions. Consider boundary conditions for a rectangular slab with
dimensions Lx × Ly × Lz . Impose periodic boundary conditions in the x
direction, then
kx = kj =
2π
j,
Lx
(j = 0, ±1, ±2, . . .)
⇒ ∆kx =
2π
.
Lx
(9.29)
Wave functions corresponding to consecutive values of j are – according to
(9.27) – shifted with respect to each other in the y direction by the distance
∆y =
~
h
∆k =
.
eB
eBLx
138
9. FERMIONIC SYSTEMS
In the area Lx × Ly there is room for
Ly
eB
=
Lx Ly
∆y
h
eigenfunctions. In case of a thin slab it may be said that each Landau level
has the surface density of states
nL =
eB
.
h
(9.30)
εn
0
x
Figure 9–3: Landau levels and the degeneracy of the electron states,
schematic illustration corresponding to the symmetric gauge.
Landau diamagnetism. Thus, the grand potential of the electron gas
in the uniform magnetic field may be written as
ℏ2 k2
)
(
∞
−β 2mz +µB B(2n+1)−µ
gLx Ly eBT X X
.
ln 1 + e
Ω=−
h
n=0
(9.31)
kz
In the z direction – for a slab of macroscopic thickness – the usual trick
applies:
Z∞
X
Lz
−→
dkz .
2π
kz
−∞
ℏ2 k 2
After the change of variable 2mTz = x and integration by parts we arrive at
the representation
√
∞
gV eB 2πm T 3/2 X
β[µ−µB B(2n+1)]
3
Li
Ω=
−e
.
2
h2
n=0
(9.32)
The desired limit B → 0 cannot be taken directly, since the sum over Landau levels diverges in this limit and the ambiguity of the multiplication of
B and the divergent series must be resolved. To this end, expression (9.32)
9.2. MAGNETISM OF DEGENERATE ELECTRON GAS
139
is traditionally approximated with the aid of the Euler-Maclaurin formula
Zb
n
X
1
f [a + (k − 1)h] =
h
∞
X
Li 32 −eβ[µ−µB B(2n+1)] =
k=1
f (x)dx
a
h
1
[f ′ (b) − f ′ (a)] + O h2 , (9.33)
− [f (b) − f (a)] +
2
12
where b = a + (n − 1)h. Substituting f (x) → Li 23 −eβ(µ−x) , h → 2µB B,
a → µB B and b → ∞ we arrive at the asymptotic relation
n=0
+
1
2µB B
Z∞
µB B
Li 32 −eβ(µ−x) dx
µB Bβ β(µ−µB B) ′ β(µ−µB B) 1
Li 23 −eβ(µ−µB B) −
e
Li 3 −e
+ O B2 .
2
2
6
Expanding the right-hand side to the linear order in B, we obtain
∞
X
n=0
Li 32 −eβ[µ−µB B(2n+1)]
1
=
2µB B
Z∞
µB Bβ βµ ′
Li 23 −eβ(µ−x) dx +
e Li 3 −eβµ + O B 2 ,
2
12
(9.34)
0
in which the limit B → 0 is straightforward. It is clear that the first term
on the right-hand side of (9.34) gives rise to the grand potential without the
magnetic field Ω0 , but this is readily seen by direct calculation as well:
n
Z∞ X
Z∞
∞ −eβ(µ−x)
β(µ−x)
dx = dx
Li 32 −e
n3/2
n=1
0
0
=
∞
X
n=1
n Z∞
n
∞
X
−eβµ
−eβµ
−βx
= T Li 52 −eβµ ,
dx
e
=
T
3/2
5/2
n
n
n=1
(9.35)
0
which gives rise to the expression (9.6) for the grand potential. To identify
the second term on the right-hand side of (9.34), use the important property
of the polylogarithm
∞
∞
d X zn
1 X zn
1
d
Liν (z) =
=
= Liν−1 (z) ,
dz
dz n=1 nν
z n=1 nν−1
z
which allows to estimate the grand potential of the electron gas in the mag-
140
9. FERMIONIC SYSTEMS
netic field (9.32) as
Ω≈
gV (2πm)3/2 T 5/2
Li 25 −eβµ −
3
h
2
(µB B) ∂ 2 gV (2πm)3/2 T 5/2
βµ
Li 52 −e
6
∂µ2
h3
= Ω0 −
2
(µB B) ∂ 2 Ω0
. (9.36)
6
∂µ2
Since N = − ∂Ω0 , we immediately see that in the diamagnetic case
∂µ
2
(µB B)
Ω = Ω0 (µ) +
6
∂N
∂µ
+ ... ,
T,V
so that Landau’s diamagnetic susceptibility may be written as
1
1
χdia = − µ0 µB 2 D(µ) = − χpara ,
3
3
(9.37)
and the total susceptibility of the electron gas is
χ=
2
µ0 µB 2 D(µ) .
3
(9.38)
Quantum Hall effect. In semiconductor interfaces by electric means
conditions may be created such that the conduction electrons are bound to
the interface but free to move along it. This gives rise to an effectively
two-dimensional electron gas with arbitrarily adjustable surface density.
In a current-carrying conductor located in a transversal magnetic field
the Hall effect may be observed, which means induction of a voltage between the faces of the conductor corresponding to an electric field perpendicular to both the current density and the magnetic field induction. The
basic setup is depicted in Fig. 9–4a. To maintain a steady current in the
x direction, the Lorentz force acting on the conduction electrons (or holes
which may also carry current in a semicponductor) −e [v × B]y = evx B
must be compensated by the electric field Ey = vx B. The ratio
ρxy ≡
vx B
B
Ey
=
=
jx
−envx
−en
(9.39)
is the Hall resistivity. In relation (9.39) n is the number density of the
electrons and jx the current density.
In the two-dimensional electron gas the Hall resistance may be correspondingly defined as
Vy
B
Rxy ≡
=
,
(9.40)
Ix
−en2
9.2. MAGNETISM OF DEGENERATE ELECTRON GAS
y
y
z
B
141
b0
Ey
I
Vy
∆y
yj +1
yj
x
x
Figure 9–4: (a) Motion of electrons on a two-dimensional surface and the
Hall effect, (b) transversal wave functions in the Landau gauge.
where n2 is the surface density of electrons (the subscript 2 reminds of the
spatial dimension). The resistance in this case is the same quantity as the
resistivity.
The Hall resistance (9.40) is a linear function of the magnetic field. If
the experimental setup is such that the surface density of electrons n2 is
independent of the magnetic field, the classic argument yields the dashed
straight line in Fig. 9–5. Experimentally it has been observed (Klaus von
Klitzing 1980), however, that at low temperature in the limit of strong magnetic field and small electron density the dependence is not linear, but with
the growth of B the Hall resistance exhibits also constant plateau values
h 1
· ,
(9.41)
e2 ν
where ν = 1, 2, . . .. This is the integer quantum Hall effect. The experimental accuracy
3
is high: 10−5 . . . 10−7 . Later plateaus at fractional values of ν were observed as well. The
4
integer effect may be explained by and large
5
6
with the aid of Landau levels in the effective
two-dimensional electron gas and is thus a
B
one-particle problem. In the fractional effect correlations due to the Coulomb interacFigure 9–5: Hall resistance tion are important and thus this is a genuine
in the quantum Hall effect. many-particle problem. Here, only the integer quantum Hall effect will be considered.
If exactly ν lowest Landau levels are filled, the surface density of electrons is n2 = νnL . Substitution of this value in relation (9.40) yields exactly
the observed plateau value (9.41). These are pointwise values, however,
therefore they does no explain, where the plateaus come from. It is also
not clear that all electron states contribute to the current (in three dimensional electron gas of solids only states in the conduction band do). The
long-distance phase coherence turns out to be important.
R xy
n=2
Rxy =
142
9. FERMIONIC SYSTEMS
The locking to the plateau values may be explained as follows. In the
real matter the Landau levels are broadened to energy bands, with a large
number density at the Landau energy, but very small between the bands. In
solids smallest impurities are able to bring about localized electron states
in the region of low density of states. Electrons in such states do not carry
current, thus forming mobility gaps.
When the electron states are filled up to the Fermi energy εF , the latter may occur in a mobility gap. In this case, the conduction bands are
either completely filled or empty. This brings about a plateau in the Hall
resistance, because the number of extended states in the conduction bands
– and, correspondingly, the number density of current carriers – changes
proportionally to the magnetic induction thus rendering the Hall resistivity intact. With the further increase (decrease) of the magnetic induction
a conduction band adjacent to the mobility gap moves on the Fermi energy causing a decrease (increase) of the occupation of the electron states
of this conduction band. This, however, is compensated by the increase (decrease) of the number of the extended states in all conduction bands, so that
the number density of current-carrying electrons remains constant and the
Hall resistivity increases (decreases) with the magnetic induction.
9.3 Problems
Problem 9.1. It has been observed that the radius R of a large nucleus
depends on its mass number A according to the relation
R = 1.1A1/3 fm,
(1 fm = 10−15 m) .
Using this, calculate the density of saturated nuclear matter, its Fermi
momentum and Fermi energy. Note that the degeneracy factor gs = 4
accounting for both the isospin (proton, neutron) and spin degrees of
freedom (the quantum number of both of them is 1/2). How high is the
degeneracy pressure? Is the matter relativistic? What is its energy
density (including the rest energy of the nucleons mp ≈ mn ≈ 940 MeV
/c2 ).
Problem 9.2. Calculate the compressibility of a highly relativistic (ε =
cp) degenerate electron gas at the accuracy of T 2 .
Problem 9.3. Calculate the total magnetic susceptibility of the free
electron gas in a homogeneous magnetic field B in the Landau gauge
without the decomposition to paramagnetic and diamagnetic susceptibilities, i.e. start with the Hamilton operator
b =
H
pb2y
(pbx − eyB)2
pb2
e
b
+
+ z +
s·B.
2m
2m
2m
m
Problem 9.4. In a semiconductor there are n0 bound electron states
per unit volume (donor states) with the energy −ε0 < 0 all filled at the
temperature T = 0 (see Fig 9–6). When the temperature rises, part of
9.3. PROBLEMS
143
the donor electrons may be transferred to the conduction band, in which
the one-particle density of states is
√
ω1 (E)dE = V A E dE
(A is a constant). Show that at the temperature T the number density
of the conduction electrons is
n = n0
e−(ε0 +µ)/T
,
+1
e−(ε0 +µ)/T
where the chemical potential µ at low temperatures is
1
2n0
1
µ ≈ − ε0 + T ln √ 3/2 .
2
2
A πT
Assume the valence band located so low that its hole excitations need
not to be taken into account.
E
johtavuusvyö
0
-ε0
donoritilat
Figure 9–6: Energy level structure of a simple model of a doped n type
semiconductor.
10. Phase transitions
10.1 Description of phase transitions
Classification of phase transitions. The basic thermodynamic potential in the description of phase transitions is the Gibbs potential G = µN ,
because one of the equilibrium conditions – in addition to that the temperature and the pressure are uniform – is that the chemical potential coincides
in all coexisting phases.
Phase transitions are traditionally classified by the singular behaviour
of the derivatives of the Gibbs potential. If some of these derivatives are
discontinuous, the phase transition is of first order. If all the first derivatives are continuous, but discontinuities (or worse singularities) appear in
the second order derivatives, then we are dealing with a second order phase
transition. These names suggest generalization according to the behaviour
of higher derivatives, but this line of classification could not be consistently
extended.
Order parameter. Another popular classification scheme emphasizes
the behaviour of the central quantity in the modern theory of phase transition, viz. the order parameter. This is a quantity, which by definition is zero
in one phase and assumes finite values in the other. In spite of the loose
definition, in most cases there is a natural choice for the order parameter.
In the gas-liquid transition, for instance, such a natural choice is the difference between the densities of liquid and gas, whereas in the paradigmatic
ferromagnetic transition the order parameter is the magnetization. If the
change of the order parameter in the phase transition is finite, then we
are dealing with a discontinuous phase transition. If the order parameter
tends to the zero value or departs from it in a continuous fashion, then a
continuous phase transition takes place.
It is typical of the continuous phase transitions that the order parameter, various response functions and correlation functions exhibit nonanalytic behaviour as functions of thermodynamic variables in the vicinity
of the critical point, i.e. the point of the continuous phase transition in
the space spanned by the thermodynamic variables. This non-analytic behaviour is usually a powerlike dependence in deviations from the criticalpoint values of variables such as the temperature and some ”external field”
like the pressure in the gas-liquid transition or the magnetic field strength
in the ferromagnetic transition. The exponents appearing in such asymptotic relations are the critical exponents (or critical indices) of the transition.
144
10.1. DESCRIPTION OF PHASE TRANSITIONS
145
The most common critical exponents will be defined and calculated below
within the Landau theory of phase transitions.
In many cases phase transitions involve changes in symmetries of the
system, which are called symmetry breaking. In the ferromagnetic transition, for instance, in the paramagnetic phase the material is often macroscopically isotropic and thus possesses the three-dimensional rotational
symmetry. In the ferromagnetic phase the direction of the macroscopic
magnetization establishes a preferred direction and the rotational symmetry remains at most in the plane perpendicular to the direction of the
magnetization.
Symmetry breaking and order parameter. It is typical of a second
order (or continuous) phase transition that some symmetry of the (usually)
high-temperature phase is spontaneously broken in the low-temperature
phase. The degree of symmetry breaking may be described by an order
parameter vanishing in the symmetric (usually) high-temperature phase,
but finite in the ordered phase with the broken symmetry. Examples:
• Structural transformation of a crystal lattice. In barium titanate
(Fig.) electric polarization is brought about, this is ferroelectricity.
The polarization P is the natural order parameter.
P
Ba,
Ti,
O
Figure 10–1: Structural transformation of BaTiO3 . The polarization P is
the order parameter.
• Ferromagnetic phase transition. The broken symmetry is the spin
rotation symmetry. Below the critical temperature magnetization
M 6= 0 appears; this is the order parameter of the system.
• Superconducting transition of electron system and superfluidity transition of 4 He. Here, a gauge symmetry related with the particle number conservation is broken.
• Symmetry breaking of the electroweak interaction in particle physics.
A gauge symmetry is broken here as well.
146
10. PHASE TRANSITIONS
Figure 10–2: Magnetic ordering.
Universality. It is a remarkable feature of the continuous phase transitions that they are largely insensitive to material properties of the system
apart from such global features like rotational or other symmetries. Physically different phase transitions may be described in a unified fashion as
soon as important global properties such as the number of components and
the tensor character of the order parameter, symmetries of the system and
the space dimension are found to be the same. This is the universality of
continuous phase transitions, and identification of the universality class of
the particular phase transition is one of the important tasks of its analysis.
Most importantly, the values of critical exponents turn out to coincide fairly
accurately in physically different systems belonging to the same universality class.
Singularities in the thermodynamic limit. Mathematical description of continuous phase transitions is much more difficult than that of the
discontinuous transitions. In the latter case different phases have chemical
potentials of their own and at the phase transition the coexistence condition
µ1 = µ2 holds. On the other hand, by definition of the first-order transition,
e.g.,
∂µ1
∂µ2
6=
∂T
∂T
so that rising or lowering the temperature across the transition temperature necessarily involves a reversal in the ordering of the chemical potentials and in equilibrium the phase corresponding to the minimum of the
chemical potential prevails. Chemical potentials of different phases are
smooth functions of the state variables.
In case of continuous phase transition, on the contrary, the number of
state of equilibria changes: a single chemical potential corresponds to the
symmetric phase, whereas several (even infinitely many in case of breaking
of a continuous symmetry) chemical potentials of equal value but different
values of the order parameter are available for the ordered phase. This is a
singularity which is hard to find in the statistical ensembles, whose partib
tion functions, say Z = Tr e−β H are smooth functions of parameters, at least
in a finite system with the physically prevailing effectively short-range interactions (Coulomb force is screened is electrically neutral systems, and
gravity is small).
In the thermodynamic limit, however, singular behaviour may follow
10.2. LANDAU THEORY
147
for suitable values of parameters, as was seen in the case of Bose condensation. Other possible sources of singularity are long-range interactions
and zero-temperature limit, on which, however, we shall not dwell here.
Exact results for physically interesting interacting systems are rare, therefore more or less phenomenological approaches are popular in description
of continuous phase transitions. The most general approach, based on a
variational principle, is that of the Landau theory.
10.2 Landau theory
Effective thermodynamic potential. At the critical point (point of the
continuous phase transition in the parameter space) the order parameter
vanishes in a continuous manner, when the symmetric phase is approached.
In the Landau theory the order parameter ϕ is considered a macroscopic
variable describing an incomplete equilibrium, whose equilibrium value is
found by minimizing the proper thermodynamic potential.
The system is considered to be so close to the critical point that the order
parameter is already small but still so far from the critical point that the
system may be assumed to homogeneous and the thermodynamic potential
a smooth function of the order parameter and state variables. The thermodynamic potential is then expanded in powers of the order parameter and
the leading terms retained. Thus
G(p, T, ϕ) = G0 (p, T ) + αϕ + Aϕ2 + Cϕ3 + Bϕ4 + . . .
(10.1)
The linear term must vanish in case of vector order parameter for a
rotation-invariant Gibbs potential. The third-order term gives rise to a discontinuous transition, so that in case of continuous transition only the second and fourth order terms remain. To guarantee existence of equilibrium,
the coefficient B >, and, although a function of p and T may usually be considered constant. For A > 0 the only minimum of (10.1) is ϕ = 0, whereas
in case A < 0 a twofold degenerate nonvanishing solution for the minimum
exists. The borderline value A = 0 then corresponds to the point of phase
transition. The simplest smooth temperature dependence is A = a(T − Tc ).
Ferromagnetic ordering. Here, the basic idea of Landau theory is
demonstrated in the example of the prototypical ferromagnetic transition.
Denote magnetization by m to emphasize that it is an order parameter
assuming values different from the equilibrium magnetization. In relations
dUsys
dFsys
= T dS + µ0 V h · dm ,
= −SdT + µ0 V h · dm ,
(10.2a)
(10.2b)
the quantity h ≡ h(T, m) is the derivative of the thermodynamic potential
with respect to the order parameter. In equilibrium, however, it must be
equal to the magnetic field strength H. The natural parameters of the
free energy are T and m, i.e. Fsys = Fsys (T, m). If the system is coupled
148
10. PHASE TRANSITIONS
to a magnetic field with the fixed field strength H, then the equilibrium
value of the magnetization m is determined by the condition that the Gibbs
function has a minimum. The magnetic Gibbs potential now becomes an
order-parameter dependent function
Gsys (T, H ; m) = Fsys (T, m) − µ0 V H · m .
(10.3)
dGsys = −SdT − µ0 V M · dH.
(10.4)
1
F (T, m) = F0 (T ) + α2 (T )m2 + α4 (T )m4 + · · · .
2
(10.5)
Choose m to minimize G: δG/δm = δFsys /δm−µ0 V H = µ0 V h−µ0 V H → 0.
In equilibrium we must have h(T, m) = H. In other words, m = m(T, H) ≡
M is the equilibrium magnetization and the familiar result holds
In an isotropic system the free energy F depends on the magnitude of the
order parameter m = |m|. Expand F as power series
Assume simplest possible smooth dependencies to provide stable minima
in the vicinity of Tc :
α2 (T ) = a · (T − Tc ) ;
a>0
(10.6)
α4 (T ) = b = const > 0
F
F
M
T > Tc
M0
M
T < Tc
Figure 10–3: Free energy as a function of the order parameter in Landau
theory.
Let first H = 0. Then the minimum of G is also the minimum of F . From
the condition
∂F
= 2a(T − Tc )m + 2bm3 = 0
∂m
the equilibrium magnetization is found as (cf. Fig. 10–3)
M0 (T ) = 0,
T > Tc
r
a
M0 (T ) = ±
(Tc − T ) .
b
T < Tc
(10.7)
10.2. LANDAU THEORY
149
The latter relation determines the value of the critical exponent β, which
describes the non-analytic behaviour of the order parameter ϕ as a function
of the temperature near the critical point as
β
ϕ(T ) ∼ (Tc − T ) ,
T < Tc ,
(10.8)
H = 0.
Thus, in the Landau theory β = 12 .
The spontaneous magnetization M0 (T ) is depicted in Fig. 10–4. The
minimum value of the zero-field free energy is thus
F (T, M0 ) = F0 (T ) ;
F (T, M0 ) = F0 (T ) −
M0 ( T )
T > Tc
(10.9)
a2
(Tc − T )2 ;
2b
T < Tc
C (T)
Tc
T
Tc
T
Figure 10–4: Magnetization and heat capacity.
Heat capacity. According to the definition
2 ∂S
∂ G
CH = T
= −T
.
∂T H
∂T 2 H
In zero field G = F , more accurately G(T, H = 0) = F (T, M0 (T )), yielding
CH (H = 0) = −T (d2 F/dT 2 ). From relations (10.9) we obtain
d 2 F0
dT 2
T > Tc :
CH (T ) = C0 (T ) ≡ −T
T < Tc :
a2
CH (T ) = C0 (T ) + T
b
(10.10)
The specific heat has a jump
CH (Tc− ) − CH (Tc+ ) =
a2
Tc
b
at the critical temperature (Fig. 10–4).
In general, however, the singular behaviour of the heat capacity in a
continuous transition is characterized by the critical exponents α and α′
150
10. PHASE TRANSITIONS
with the definition
CH (T ) ∼
(
−α
(T − Tc )
, T > Tc ,
−α′
(Tc − T )
, T < Tc ,
H = 0,
(10.11)
where usually both α and α′ are numerically small. The finite discontinuity
in the Landau theory in these terms is described by putting α = α′ = 0.
Susceptibility. If the external field H 6= 0, we arrive at the equilibrium
condition
H
H
M0
M
M
T < Tc
T > Tc
Figure 10–5: Determination of the equilibrium magnetization, relation
(10.13).
∂F
∂G
=0 ⇔
= µ0 V H ,
∂m
∂m
i.e., according to relations (10.5) and (10.6),
2a(T − Tc )M + 2bM 3 = µ0 V H .
(10.12)
(10.13)
In case T > Tc (Fig. 10–5) in the limit of small field H the magnetization
M = χH + O(H 3 ), with the susceptibility χ
χ=
µ0 V
.
2a(T − Tc )
(10.14)
In case T < Tc in the limit of small H obviiously (cf. Fig. 10–5) M = M0 +
δM , where δM ∝ δH is small. The following relation holds,
µ0 V δH
2a(T − Tc )δM + 6bM0 2 δM
a
= 2a(T − Tc )δM + 6b (Tc − T )δM
b
= 4a(Tc − T )δM .
=
This yields for the susceptibility the result
χ=
µ0 V
δM
=
.
δH
4a(Tc − T )
(10.15)
10.3. GINZBURG–LANDAU THEORY
151
From relations (10.14) and (10.15) it is seen that the critical exponents γ
and γ ′ of the susceptibility, defined as
(
−γ
(T − Tc ) , T > Tc ,
χ(T ) ∼
H = 0,
(10.16)
−γ ′
(Tc − T )
, T < Tc ,
in the Landau theory assume the value γ = γ ′ = 1.
M
H <H
1
2
χ = (∂Μ / ∂Η)Η=0
H =0
Tc
Tc
T
T
Figure 10–6: Magnetization and susceptibility.
The susceptibility is depicted in Fig. 10–6. In the same plot dependence
of the magnetization M on the temperature and field strength H has been
sketched. In particular, at the critical temperature T = Tc , from the condition µ0 V H = 2bM 3 it follows
M (Tc , H) = const × H 1/3 .
(10.17)
From this relation it follows that the critical exponent δ, which describes
the non-analytic dependence of the order parameter ϕ of the external field
h at the critical temperature as
ϕ(h) ∼ h1/δ ,
T = Tc ,
(10.18)
in the Landau theory assumes the values δ = 3.
10.3 Ginzburg–Landau theory of superconductivity
In a large system it is possible that the order parameter is not constant
throughout the whole volume, especially, when the broken symmetry is continuous. Deviations from homogeneous order parameter are also necessary
to describe fluctuations near the critical point. If local variability is allowed,
then a field theory is obtained with the order parameter m(r) as a function
of position. Such a generalization was put forward in V. Ginzburg in application to superconductivity and turned out to be very successful explanation of this phenomenon discovered already in 1911 by H. Kamerlingh–
Onnes. The Ginzburg-Landau theory has proved a fruitful starting point
for description of several other ordering phenomena as well.
152
10. PHASE TRANSITIONS
Free energy. The order parameter is assumed to be a complex-valued
function Ψ(r) called macroscopic wave function in this context. Physically,
this is quantity describing the correlated electron pairs (Cooper pairs),
whose motion gives rise to the phenomenon of superconductivity in metals. The ”genuine” wave function of such a pair of electrons depends on
variables of both electrons, of course. The macroscopic wave function here
is related to the motion of the center of mass of the Cooper pair, which is
thus considered a pointlike object in the Ginzburg-Landau theory. The typical length scale up to which the electrons of the pair remain correlated, the
coherence length ξ0 (usually ξ0 >
∼ 1000 Å), must therefore be much less than
the typical spatial scale of the macroscopic wave function for the GinzburgLandau theory to be consistent.
The free energy is written as local functional functional of the macroscopic wave function, where the spatial dependence is taken into account
by the leading term of the gradient expansion:
Z
~2
b
2
2
4
Fsys (T, [Ψ]) = d3 r f0 +
. (10.19)
|∇Ψ|
+
a(T
−
T
)
|Ψ|
+
|Ψ|
c
2m∗
2
Here, m∗ is a parameter of dimension of mass. In a system with charged
particles in an magnetic field gauge invariance is imposed by replacing
the gradient in the canonical momentum operator −i~∇ by the covariant
derivative to obtain −i~∇ − e∗ A(r), where A is the vector potential and e∗
the charge. With this substitution the Ginzburg-Landau free energy (10.19)
remains invariant under the transformation Ψ(r) → Ψ(r)eiα(r) even in case
of position-dependent α(r), when it is accompanied by the proper change of
the vector potential. When the energy of the magnetic field is added, the
total Ginzburg-Landau free energy is obtained in the form
F =
Z
2
ie∗
b
B2
~2 2
4
∇−
A Ψ + aτ |Ψ| + |Ψ| +
,
d r f0 +
2m∗ ~
2
2µ0
3
(10.20)
where for brevity the notation τ = T − Tc has been introduced. Thus, we
are dealing with a 4-parameter (m∗ , e∗ , a, b) phenomenological theory.
Variational conditions. It is convenient to write down the stationarity
equations for the functional (10.20) with the aid of the functional derivative,
whose definition for an arbitrary functional F [f ] of the function f (r) is
Z
δF
δF [f ] ≡ d3 r δf (r)
.
δf (r)
In practice, the usual chain rule together with partial integration is sufficient to arrive at expressions containing
δf (r ′ )
= δ(r ′ − r)
δf (r)
which allows to resolve one spatial integral.
10.3. GINZBURG–LANDAU THEORY
153
Since the electromagnetic field interacts with charged matter, the vector
potential A is a variable quantity as well. Consider first variation of the
magnetic field energy with fixed boundary conditions for the fields varied.
Then
Z
Z
Z
B · (∇ × δA)
B 2 = δ (∇ × A) · (∇ × A) = 2
δ
V
V
V
Z
Z
= −2
∇ · (B × δA) + 2
∇ · (B × δAc )
ZV
ZV
= −2
n · (B × δA) + 2
δA · (∇ × B) ,
(10.21)
∂V
V
where the notation δAc means, that the derivatives in the nabla do not act
on this factor. Taking into account the fixed boundary condition, we arrive
at the result
Z
δ
1
1
∇×B =∇×H.
B2 =
δA(r) 2µ0 V
µ0
Thus example illustrates sufficiently the calculation of the functional
derivatives also for the case of the order parameter Ψ. Therefore, we quote
only the final equilibrium conditions.
Superconductivity. From the requirement of stationarity with respect
to variations of Ψ(r)∗ and δA(r) the Ginzburg–Landau equations for superconductivity follow
2
ie∗
~2
2
A Ψ + aτ + b |Ψ| Ψ = 0 ,
(10.22a)
− ∗ ∇−
2m
~
e∗ ~
(e∗ )2
2
∗
∗
J≡
[Ψ
(∇Ψ)
−
(∇Ψ
)Ψ]
−
A |Ψ| = ∇ × H . (10.22b)
2im∗
m∗
The former is a nonlinear Schrödinger equation for superconducting particles with the mass m∗ and charge e∗ . The latter equation, which determines
the supercurrent density J , is Ampère’s law ∇ × H = J for static fields.
The first term of the current density J is the canonical current, which is
not gauge invariant. Only the account of the second term gives rise to a
gauge invariant current density.
Temperature-dependent coherence length. The Ginzburg–Landau
equations are nonlinear, therefore an exact solution is possible only in special cases. If there is no magnetic field, it is consistent to put A = 0 everywhere. Relation (10.22b) is then automatically fulfilled, if Ψ is real.
Consider a superconductor filling the half-space x > 0. Then Ψ is a
function of x only and the equation (10.22a) assumes the form
−
~2 d2 Ψ(x)
+ aτ Ψ(x) + bΨ(x)3 = 0 .
2m∗ dx2
(10.23)
As a boundary condition, impose Ψ(0) = 0. The equation may be solved by
multiplying by the factor Ψ′ (x) and constructing a first integral. The firstorder differential equation obtained is also solvable. It turns out that a
154
10. PHASE TRANSITIONS
meaningful solution may only be found for temperatures τ < 0, i.e. T < Tc .
For the order parameter the expression
Ψ(x) =
√
x
;
ns tanh
2ξ
follows, where
ns = −
and
ξ=p
(x > 0),
aτ
b
~
2am∗ (Tc − T )
(10.24)
(10.25)
.
(10.26)
The constant ns = |Ψ(∞)|2 is the density of superconducting particles.
The quantity ξ, which describes the thickness of the surface layer, is the
temperature-dependent coherence length.. This is the typical length scale
of the Ginzburg-Landau model. It is approximately equal to the coherence
length
√ ξ0 of the correlated electron pairs far from Tc , but since it diverges
as Tc − T near Tc , it is bound to become much larger than ξ0 (which is
independent of the temperature) close enough to the critical point.
Meissner effect. In a weak magnetic field the vector potential A and
the field strength H are small and the wave function may be formally expanded as
Ψ = Ψ0 + Ψ1 + Ψ2 + . . . ,
where Ψ0 is the zero-field solution obtained above, and Ψn ∝ |A|n .
Let the region x > 0 be superconducting, and the magnetic field directed
along the y axis. Then
A = A(x)ez ;
B = ∇ × A = −A′ (x)ey .
Rewrite equation (10.22a) in more detail,
−
~2 2
ie∗
(e∗ )2 2
∇
Ψ
+
A
·
∇Ψ
+
A Ψ + aτ + b|Ψ|2 Ψ = 0.
2m∗
m∗
2m∗
The zeroth order yields the previous equation (10.23). In the first order the
vector potential is absent, because the vectors A and ∇Ψ0 are orthogonal.
Thus, Ψ1 = 0, and the change of the wave function is of second order in A.
Equation (10.22b) then implies that in the first-order accuracy
−
1
1
(e∗ )2
A|Ψ0 |2 = ∇ × H =
∇ × (∇ × A) = − ∇2 A ,
m∗
µ0
µ0
(10.27)
where the last from is a consequence of the relation ∇ · A = 0.
Equation (10.27) is readily solved deep in the superconductor, where
Ψ0 = cons. For the function A(x) with the account of relations (10.24) and
(10.26) the following equation is obtained
µ0 (e∗ )2 aτ
d2 A(x)
=−
A(x) .
2
dx
m∗ b
(10.28)
10.3. GINZBURG–LANDAU THEORY
155
The physically meaningful solution is the exponentially falling off function
x
A(x) −→ const × exp −
,
(10.29)
λ
x→∞
where the parameter λ is the penetration depth
s
bm∗
.
λ=
aµ0 (e∗ )2 (Tc − T )
(10.30)
Usually, λ ≫ 100 Å. Since deep in the superconductor A = 0, the magnetic
field does not penetrate the matter. The superconductor is thus a perfect
diamagnet. This is the Meissner effect.
In the Ginzburg-Landau theory both the temperature-dependent coherence length (10.26)and the penetration depth (10.30) diverge in the same
way, when the critical temperature is approached. Their temperatureindependent ratio
s
κ=
λ
m∗
= ∗
ξ
~e
2b
,
µ0
(10.31)
the Ginzburg-Landau parameter is an important parameter of the theory,
since its value determines the sign of the surface tension between the superconducting and normal phases and thus the character of the phase transition between them in strong magnetic fields.
Critical field. Superconductors thus expel magnetic field. A strong
enough magnetic field, however, destroys the superconducting state. The
borderline value, the critical field Hc may be determined thermodynamically as follows. A superconducting body possesses a magnetic moment
m = V M = −V H. The potential energy of this magnetic moment in the
RH
external field W = −µ0 0 m·dH = −µ0 12 m·H = 21 V µ0 H 2 is the energy in
excess to the free energy of the superconductor and the energy of the magnetic field in the absence of the superconductor (which occupies the volume
of the superconductor as well). The corresponding interaction energy between the magnetic moment of the same body in the normal state and the
external field is negligible due to the small numerical value of the dia- and
paramagnetic susceptibilities. Therefore, the difference between the energies of the body in the magnetic field in the homogeneous superconducting
state and in the normal state is – up to surface effects –
Fs − Fn = −V
a2 (Tc − T )2
1
+ V µ0 H 2 .
2b
2
(10.32)
Just at the the critical field this difference vanishes, and near the critical
temperature the thermodynamic critical field is
Hc =
a(Tc − T )
√
.
µ0 b
(10.33)
156
10. PHASE TRANSITIONS
Surface effects, however, turn out to be of paramount significance in many
practical superconductors. The point is that the transition from, say, the
superconducting to the normal state requires initial nucleation of small
normal state formations, and the appearance of these is hindered by the
energy cost to build up the surface, when the surface tension between the
normal and superconducting states is positive. In such a case of a superconductor of the I type the phase transition takes place in fields larger than the
thermodynamic critical field so that the magnetic field penetrates a large
volume at once (the surface area between the ordinary and superconducting phases is minimized). In a superconductor of the II type the surface
tension of the interface between the superconducting and normal states is
negative, which leads to the nucleation of the normal phase in a superconducting bulk at field strengths less than the thermodynamic critical field
(the borderline value is the lower critical field Hc1 ) and to the nucleation
of superconducting phase at field strength larger than the thermodynamic
critical field (the borderline value is the upper critical field Hc2 ). In these
materials nothing remarkable happens at Hc .
In superconductors of the II type the magnetic flux penetrates the superconducting bulk as thin filaments (vortices) forming a vortex lattice
(Abrikosov lattice). The flux in the filaments is quantized with the flux
quantum h/e∗ . Observations on this phenomenon as well as on quantization of magnetic flux through a superconducting ring have shown that the
charge of the superconducting particle e∗ = −2e, where −e is the electron
charge. The supercurrent is carried by bound pairs of electrons. These
Cooper pairs are loose formations with the diameter of the of the coherence
length ξ0 and thus much larger than the distances between the conduction
electrons.
The quantization of the magnetic flux may be readily demonstrated in
geometries of a superconducting ring in a magnetic field and for a normalstate filament aligned with an external magnetic field in the superconducting bulk. Imagine a closed contour around the filament or along the ring
deep in the bulk superconductor so that on the loop the magnetic induction
vanishes and the modulus of the order parameter is constant. From relation (10.22b) and the representation Ψ = |Ψ|eiφ it then follows that on the
contour
e∗
∇φ =
A.
(10.34)
~
Integrating over the closed contour we obtain, by virtue of the Stokes theorem,
I
Z
I
e∗
e∗
e∗
A · dl =
n · ∇ × A dS =
ΦB ,
(10.35)
∆φ = ∇φ · dl =
~
~
~
where ∆φ is the change of the phase of the wave function after traversing over the contour and ΦB is the magnetic flux through a surface, whose
boundary is the closed contour at hand. The wave function must, however,
be a single-valued function of the position, which imposes the condition
10.4. FLUCTUATIONS IN LANDAU THEORY
157
∆φ = 2πn with an integer n. Thus, the magnetic flux through a superconducting ring or normal-state filament is quantized as
Z
h
ΦB = n · B dS = ∗ n .
(10.36)
e
In particular, this condition imposes restrictions on the appearance of
normal-state vortices in the phase transition to the normal state in a type
II superconductor in magnetic field.
10.4 Fluctuations in Landau theory
Landau theory is based on an effective thermodynamic potential describing
the system in incomplete equilibrium described by the order parameter ϕ.
The probability of such a state may be estimated in a manner similar to that
used in the Einstein theory of fluctuations. Consider the classical canonical
ensemble (for simplicity of notation).
The partition function is the measure of the phase state with the weight
e−βH :
Z ′
dΓ e−βH(p,q) .
(10.37)
Z=
The measure of the part of the phase state corresponding to the incomplete
equilibrium may be written in a similar form by imposing the condition
ϕ = ϕ(p, q), where ϕ(p, q) is the order parameter expressed as function of
the variables of the phase space. Formally this is effected as
Z ′
dΓ δ (ϕ − ϕ(p, q)) e−βH(p,q) .
(10.38)
Z(ϕ) =
Obviously Z =
Z
dϕ Z(ϕ) and the relative frequency at which the incom-
plete equilibrium occurs in the phase space is
Pr(ϕ) =
Z(ϕ)
= eβ[F −F (ϕ)] ,
Z
(10.39)
where the effective free energy is
F (ϕ) = −T ln Z(ϕ) .
(10.40)
Substituting the expression for F (ϕ) in the Landau theory (inhomogeneous
system)
Z
F (ϕ) = F0 + d3 r g(∇ϕ)2 + a(T − Tc )ϕ2 + Bϕ4
(10.41)
we arrive at the probability density for the order parameter in the form
Pr(ϕ) ∝ e−β
R d3 r
[g(∇ϕ)2 +a(T −Tc )ϕ2 +Bϕ4 ] .
(10.42)
158
10. PHASE TRANSITIONS
Calculations with such a weight are only possible in the form of an expansion in B. The leading order is given by the Gaussian distribution corresponding to B = 0. Already in this approximation problems in definition of
the mathematical quantities involved appear. For instance, the correlation
function of the order parameter is
Z Y
R 3
2
2
dϕ(r) ϕ(r)ϕ(r ′ ) e−β d r [g(∇ϕ) +a(T −Tc )ϕ ]
hϕ(r)ϕ(r ′ )i =
r
Z Y
dϕ(r) e−β
R d3 r
[g(∇ϕ)2 +a(T −Tc )ϕ2 ]
.
(10.43)
r
In continuum space both the denominator and the numerator consist of a
formally infinite-fold integral, to which some meaning should be prescribed.
The simplest thing to do is to put the system on a lattice in a finite box,
which resricts the integrals over values of the order parameter at different
positions to a finite number. In case of a Gaussian integral for a correlation
function it is also possible to use the expression (6.51), in which the dimension of the space of integration does not appear explicitly. This means that
hϕ(r)ϕ(r ′ )i =
−1
T T
(r, r ′ ) = G(r − r ′ ) ,
−g∇2 + a(T − Tc )
2
2
(10.44)
where G(r − r ′ ) is the Green function of the operator −g∇2 + a(T − Tc ), i.e.
the solution of the equation
−g∇2 + a(T − Tc ) G(r) = δ(r)
(10.45)
with vanishing boundary condition at infinity.
More constructively calculation of the correlation function is convenient
to carry out in the wave-vector space. Put the system in a, say, cubic box
and define the coefficients of the Fourier series as
Z
1
d3 r e−ik·r ϕ(r) ,
(10.46)
ϕ(k) =
V
and calculate the Fourier coefficients of the correlation function
Z
Z
′ ′
1
3
d
r
d3 r ′ e−ik·r−ik ·r hϕ(r)ϕ(r ′ )i = hϕ(k)ϕ(k′ )i .
2
V
(10.47)
On the other hand, assuming the usual translation invariance we may
write
hϕ(r)ϕ(r ′ )i = C(r − r ′ )
(10.48)
and
1
V2
Z
d3 r
Z
′
′
d3 r ′ e−ik·r−ik ·r C(r − r ′ )
Z
Z
′
1
3 ′ −i(k+k′ )·r ′ 1
d r e
d3 r e−ik·(r−r ) C(r − r ′ ) = δk′ ,−k C(k) .
=
V
V
(10.49)
10.4. FLUCTUATIONS IN LANDAU THEORY
159
Comparison of relations (10.47) and (10.49) yields
(10.50)
hϕ(k)ϕ(k′ )i = δk′ ,−k hϕ(k)ϕ(−k)i = δk′ ,−k h|ϕ(k)|2 i ,
since for a real ϕ(r) from (10.46) it follows that ϕ∗ (k) = ϕ(−k). Express now
the Gaussian weight in terms of ϕ(k). Substitution of the Fourier series of
ϕ(r) yields
Z
d3 r g(∇ϕ)2 + a(T − Tc )ϕ2
Z
XX
′
= d3 r
−gk · k′ + a(T − Tc ) ϕ(k)ϕ(k′ ) eik·r+ik ·r
k
k′
=V
X
gk 2 + a(T − Tc ) |ϕ(k)|2 . (10.51)
k
In view of expressions (10.50) and (10.51) it appears convenient to carry out
the integration over the real and imaginary parts of ϕ(k). These are not all
independent variables, since
(10.52)
ϕ(k) = ϕR (k) + iϕI (k) = ϕ∗ (−k) = ϕR (−k) − iϕI (−k) .
Therefore, it is sufficient to integrate over values of ϕR (k) and ϕI (k) in a
"half space" of wave vectors, chosen, for instance, by the condition k1 ≥ 0. In
calculation of the correlation function h|ϕ(k)|2 i this feature is unimportant,
however, because all integrals over values of ϕR (k′ ) and ϕI (k′ ) with k′ 6= k
cancel in the expression
P
Z Y
2
−βV [gk2 +a(T −Tc )] |ϕ(k)|2
2
k
dϕR (k)dϕI (k) ϕR (k) + ϕI (k) e
h|ϕ(k)|2 i =
k ,k1 ≥0
Z Y
dϕR (k)dϕI (k) e
−βV
P gk2 +a(T −T
k
[
c)
.
] |ϕ(k)|2
k ,k1 ≥0
(10.53)
Thus, we are left with the following ratio of twofold Gaussian integrals
h|ϕ(k)|2 i =
Z∞
dϕR (k)
−∞
Z∞
−∞
Z∞
−∞
2
2
dϕI (k) ϕ2R (k) + ϕ2I (k) e−βV [gk +a(T −Tc )] |ϕ(k)|
dϕR (k)
Z∞
dϕI (k) e−βV [gk
2
+a(T −Tc )] |ϕ(k)|
−∞
.
2
(10.54)
Calculation yields
h|ϕ(k)|2 i =
therefore
hϕ(k)ϕ(k′ )i =
T
1
,
2
V gk + a(T − Tc )
1
T
δk′ ,−k 2
.
V
gk + a(T − Tc )
(10.55)
(10.56)
160
10. PHASE TRANSITIONS
We see that the length scale of the correlation function, the correlation
length, is
r
g
ξ=
,
T > Tc .
(10.57)
a(T − Tc )
Below Tc a similar relation follows:
r
g
ξ=
,
2a(Tc − T )
T < Tc .
(10.58)
Therefore, in Landau theory values of the critical exponents of the correlation length
(
−ν
(T − Tc ) , T > Tc ,
ξ(T ) ∼
H = 0,
(10.59)
−ν ′
(Tc − T )
, T < Tc ,
are equal and ν = ν ′ = 12 .
Expression for the correlation function as a function of the position vector may now be calculated as the Fourier series
′
T X
eik·(r−r )
.
V
gk 2 + a(T − Tc )
k k′
k
(10.60)
Further, it is customary to pass to the thermodynamic limit, which produces
the familiar integral sum and the correlation function may be calculated as
the inverse Fourier transform
′
Z
′
eik·(r−r )
T e−|r−r |/ξ
d3 k
=
.
(10.61)
hϕ(r)ϕ(r ′ )i = T
(2π)3 gk 2 + a(T − Tc )
4πg |r − r ′ |
hϕ(r)ϕ(r ′ )i =
XX
′
′
hϕ(k)ϕ(k′ )i eik·r+ik ·r =
At the critical temperature ξ → ∞ and
hϕ(r)ϕ(r ′ )i =
T
,
4πg|r − r ′ |
T = Tc ,
1
,
|r − r ′ |1+η
T = Tc
(10.62)
which fixes the value of the critical exponent η of the correlation function
hϕ(r)ϕ(r ′ )i ∼
in the Landau theory as η = 0.
(10.63)
10.5 Problems
Problem 10.1. Consider the following expansion in the order parameter φ of the Gibbs free energy
G(p, T, φ) = G0 (p, T ) + a(T − T0 )φ2 − Cφ3 + Bφ4 ,
where a, B and C are positive constants. Find the equilibrium value of
the order parameter, show that there is a first order phase transition in
this system and find the transition temperature.
10.5. PROBLEMS
Problem 10.2. Calculate the latent heat of the phase transition in the
model of the preceding problem.
Problem 10.3. There are systems in which (on the p,T plane) a line of
second-order transitions changes into a line of first-order transitions at
the tricritical point. Near the tricritical point the Landau expansion of
the Gibbs free energy may be written as
G(φ, p, T ) = G0 (p, T ) + Aφ2 + Bφ4 + Dφ6 ,
where D > 0. The line of second-order transitions is determined by the
conditions A(TC (p)) = 0, B > 0. On the line of first-order transitions
B < 0 so that at the tricritical point A = B = 0. Find the value of
the order parameter on the line of first-order transitions in the ordered
phase and establish a connection between A, B and D (which is the
equation of the line of first-order transitions).
161
Index
absorption ratio, 121
additivity, 56
adiabatic approximation, 121
adiabatic change, 77
adiabatic constant, 31
adiabatic demagnetization, 89
critical exponents, 144, 149, 151
critical point, 144
Curie’s law, 5, 89
De Broglie wave length
thermal, 94
Debye model, 124
degeneracy, 86
degeneration temperature, 130
degree of advancement, 36
degree of freedom, 29
degree of reaction, 36
density of states, 63
density operator, 59
diamagnetism
of Landau, 138
of Meissner, 155
differential, 3
differential form, 3
binomial distribution, 86
black body, 115, 120
Bohr’s magneton, 135
Boltzmann distribution, 69
of MB gas, 91
Born–Oppenheimer approximation, 121
Bose condensation, 110, 113
Bose–Einstein statistics, 62, 101
boson, 102
canonical distribution, 69
classical, 72
canonical ensemble, 69
canonical equations of motion, 49
Carnot’s process, 10
chemical potential, 8
of electron gas, 132
of MB gas, 95
chemical reaction, 36
Clausius–Clapeyron equation, 42
coefficient of thermal expansion,
21
coexistence, 42, 44
coherence length, 152
temperature-dependent, 154
compressibility, 5, 22
conduction electrons, 128
continuity equation, 51
convective time derivative, 51
Cooper pairs, 152, 156
correlation length, 160
critical
point, 44
temperature, 111
efficiency, 9, 11
effusion, 93
electron gas
magnetism, 135
endothermic reaction, 38
energy
free, see free energy
free spin system, 87
internal, 6, 15, 30
of black-body radiation, 118,
119
energy band, 142
energy surface, 52
ensemble, 50
canonical, 69
grand canonical, 75
microcanonical, 53, 54, 66
enthalpy, 17, 18, 38, 87
entropy, 9, 12, 24, 30–32, 63, 70
Boltzmann, 56
of BE and FD gases, 104
of black-body radiation, 119
162
INDEX
of free spin system, 88
of MB gas, 95
statistical, 55
equation of state, 4
of black-body radiation, 119
equilibrium conditions, 24
equilibrium constant, 37
equilibrium distributions, 69
equipartition principle, 92
ergodic flow, 53
ergodic theory, 54
exothermic reaction, 38
extensive variables, 2
factorizing, 98
Fermi energy, 129
Fermi gas, 128
degenerate, 128
ideal, 128
Fermi surface, 129
Fermi temperature, 129
Fermi–Dirac statistics, 62, 102
fermion, 102
Fermmi momentum, 129
ferroelectricity, 145
ferromagnetic ordering, 147
flow, 52
fluctuations, 71, 76, 78, 80, 82
flux quantum, 156
Fock space, 62
free energy, 19, 70
Gibbs, 19
Helmholtz, 19
of black-body radiation, 119
of diatomic ideal gas, 99
of MB gas, 95
free expansion, 31
fugacity, 75
functional, 152
functional derivative, 152
gauge invariance, 152
generalized displacement, 7
generalized force, 7
generating function, 79
Gibbs distribution, 69
Gibbs function, 19, 32, 33, 37
163
of free spin system, 88
Gibbs paradox, 33
Gibbs phase rule, 40
Gibbs–Duhem equation, 20
Ginzburg–Landau equations, 153
Ginzburg–Landau theory, 151
Ginzburg-Landau free energy,
152
Ginzburg-Landau parameter, 155
grand canonical distribution, 75
grand canonical ensemble, 75
grand potential, 20, 76
of BE-gas, 103
of FD gas, 103
of MB gas, 96
gyromagnetic ratio, 87
Hall effect, 140
quantized, 140
Hall resistivity, 140
Hamilton function, 49
harmonic approximation, 122
heat capacity, 8, 22, 29
of electron gas, 133
heat capacity ratio, 31
heat of reaction, 37
heat pump, 10
heteropolar, 97
hole, 140
homopolar, 97, 100
Hooke’s law, 7
hysteresis, 1
ideal gas, 5
classic, 29
classical, 5, 91
diatomic, 97
ideal system, 86
integrating factor, 3
intensive variables, 2
internal energy, 6
inversion temperature, 19
irradiance, 121
Joule process, 31
Joule–Thomson process, 18
Lagrange multiplier, 69, 75
INDEX
164
Landau free energy, 152
Landau levels, 136
Landau theory, 147
latent heat, 41
law of mass action, 37
Legendre transform, 3, 19
Legendre transforms, 17
Liouville equation, 52
Liouville operator, 52
Liouville theorem, 51
Lorentz force, 140
macroscopic wave function, 152
macrostate, 79
magnetic moment, 87
magnetization, 5
of free spin system, 88, 89
Maxwell construction, 45
Maxwell distribution, 92
Maxwell relations, 16, 17, 19–21
Maxwell–Boltzmann gas, 65
Meissner effect, 155
metastable, 45, 101
minimal work
reversible, 83
mixed state, 59
mixing entropy, 32
mixing flow, 54
natural variables, 15
Nernst’s law, 12
non-ergodic, 101, 120
non-ergodic flow, 52
normal coordinates, 122
normal distribution, 87
nuclear demagnetization, 89
nuclear spin, 98
number operator, 75
occupation number, 103
order parameter, 144
ortohydrogen, 100
osmosis, 33, 35
osmotic pressure, 35
parahydrogen, 100
paramagnetism, 5
of Curie, 89
Pauli, 135
particle flux, 93
particle flux density, 93
partition function
canonical, 69
diatomic ideal gas, 98
grand canonical, 75, 96, 102
microcanonical, 66
of BE gas, 103
of FD gas, 103
of free spin system, 88
of MB gas, 95, 96
penetration depth, 155
perfect Bose-Einstein gas, 109
phase diagram, 40
phase equilibrium, 39
phase separation, 45
phase space, 49
measure, 50
phase transition, 40
continuous, 41, 144
discontinuous, 144
ferromagnetic, 145
first order, 41, 144
of second order, 147
second order, 41, 144
phase transtion
ferromagnetic, 147
phonon, 122
photon, 117
Planck distribution, 117
Planck’s radiation law, 118
Poisson brackets, 49
probability density, 50
process, 2
Carnot’s, 10
cyclic, 2, 9
irreversible, 2
isenthalpic, 18
isergic, 31
reversible, 2
pure state, 59
quasistatic, 2
radiance, 120
radianssi, 120
INDEX
Radiant excitance, 120
Rayleigh–Jeans law, 118
real gas, 5
reflection ratio, 121
resistivity, 141
response
thermodynamic, 5, 21
rotation, 97, 99
saturated vapour, 41
solution, 33
Sommerfeld expansion, 132
specific heat, 9
specific heat capacity, 29
speed of sound, 22
spin system
free, 86
stability, 24, 25
stability conditions, 24
state variables, 1
statistical sum
microcanonical, 55
Stefan–Boltzmann constant, 119
Stefan–Boltzmann law, 121
stoichiometric coefficient, 36
superconductivity, 145, 151
superconductor, 153
superfluidity, 145
surface density, 138
surface tension, 7
susceptibility, 6
Landau diamagnetic, 138
magnetic, 89
Pauli paramagnetic, 135
SVN-system, 7
symmetry breaking
spontaneous, 145
system, 1
closed, 1
isolated, 1
open, 1
temperature, 6, 70
absolute, 11
negative, 90
thermal efficiency, 10
165
thermal expansion coefficient, 5,
18
thermal wavelength, 73
thermalization, 91
thermodynamic equilibrium, 1
local, 1
thermodynamic limit, 63
thermodynamic potential, 15
transmission ratio, 121
triple point, 40
työ
vapaa, 19
universality class, 146
Van der Waals equation of state,
44
Van’t Hoff equation, 35
vibration, 97, 99
virial coefficient, 5
virial expansion, 5
Wien’s law, 118
work, 6
electromagnetic, 7
free, 16, 17, 20, 21
Young’s modulus, 7
zeroth law, 6