Download Chapter 1 Review of thermodynamics and statistical mechanics

Chapter 1
Review of thermodynamics and
statistical mechanics
This is a short review that puts together the basic concepts and mathematical expressions
of thermodynamics and statistical mechanics, mainly for future reference.
1.1
Thermodynamic variables
For the sake of simplicity, we will refer in this section to a so-called ’simple system’, meaning: a one-component system, without electric charge or electric or magnetic polarisation,
in bulk (i.e. far from any surface), etc. Such a system is characterised from a thermodynamic point of view by means of three variables: N, number of particles; V volume, and
E, internal energy. Normally one explores the thermodynamic limit,
N → ∞,
V → ∞,
N
= ρ < ∞,
V
(1.1)
where ρ = N/V is the mean density. In a magnetic system the role of volume is played
by the magnetisation M. All these parameters are extensive, i.e. they are proportional
to the system size: if the system increases in size by a factor λ, the variables change by
the same factor.
1.2
Laws of Thermodynamics
Thermodynamics is a macroscopic theory that provides relationships between (but not
quantitative values of) the different thermodynamic parameters; no reference is made to
any microscopic variable or coordinate. It is based on three well-known laws:
1. First law of thermodynamics: it focuses on the energy, E, stating that the energy is
an extensive and conserved quantity. Mathematically:
dE = δW + δQ,
(1.2)
where dE is the change in energy involved in an infinitesimal thermodynamic process, δW is the mechanical work done on the system, and δQ is the amount of
heat transferred to the system. δW and δQ are inexact differential quantities (i.e.
1
quantities such as W or Q do not exist, in the sense that their values cannot be defined for the equilibrium states of the system). However, dE is the same regardless
of the type of process involved, and depends only on the initial and final states: it
is a state function, and every equilibrium state of the system has a unique value of E.
In fact both δW and δQ are ‘work’ terms, but the latter contains the contributions
of microscopic interactions in the system, which cannot be evaluated in practical
terms. δQ is then separated from the ‘thermodynamic’ work δW associated with
the macroscopic (or thermodynamic) variables.
There may be various terms contributing to the thermodynamic work. In general:
δW =
X
xi dXi ,
(1.3)
i
where (xi , Xi ) are conjugate variables. xi is an intensive variable, e.g. µ (chemical
potential), p (pressure), H (magnetic field), ... , and Xi the corresponding extensive
variable, e.g. N, −V , −M (minus the magnetisation), ...
2. Second law of thermodynamics: the central quantity here is the entropy, S, which is
postulated to be a monotonically increasing state function of the energy E. The law
states that, in an isolated system, a process from a state A to a state B is such that
SB ≥ SA . The equality (inequality) stands for reversible (irreversible) processes.
Then, entropy always increases (up to a maximum) or remains unchanged in an
abiabatic process (one not involving energy exchanges with the environment). Possible processes involve changes in microscopic variables; this implies that entropy
is a maximum with respect to changes in these variables at fixed thermodynamic
variables (N, V and E).
Most of this course will be hovering over the entropy concept. The reason is that the
second law is a genuine thermodynamic law in the sense that it introduces a new,
non-mechanical quantity, the entropy S = S(N, V, E), which is a thermodynamic
potential. This means that all other macroscopic quantities can be derived from S:
1
=
T
∂S
∂E
!
,
N,V
p
=
T
∂S
∂V
!
µ
∂S
=−
T
∂N
,
N,E
!
,
(1.4)
V,E
where T , p and µ are the temperature, the pressure and the chemical potential,
respectively. These are intensive variables, which do not depend on the system size.
The above scheme is called entropy representation.
Since S increases monotonically with E, it can be inverted to give E = E(N, V, S).
The intensive parameters in this energy representation are:
T =
∂E
∂S
!
,
N,V
∂E
p=−
∂V
!
,
N,E
µ=
∂E
∂N
!
.
(1.5)
V,E
In a process where both N and V remain fixed (i.e. no mechanical work is involved),
dE = δQ = T dS [the latter equality following from the first of Eqns. (1.5)] and we
have dS = δQ/T . This is valid for reversible processes. For irreversible processes
dS ≥ δQ/T .
2
3. Third law of thermodynamics: A system at zero absolute temperature has zero
entropy.
1.3
Processes and classical thermodynamics
Processes are very important in the applications of thermodynamics. There are three
types of processes:
• quasistatic process: it occurs very slowly in time
• reversible process: a quasistatic process that can be reversed without changing the
system or the environment. The system may always be considered to be in an
equilibrium state. A quasistatic process is not necessarily reversible
• irreversible process: it is a fast process such that the system does not pass through
equilibrium states. If the process is cyclic the system returns to the same state but
the environment is changed
Historically the discipline of thermodynamics grew out of the necessity to understand the
operation and efficiency of thermal engines, i.e. systems that perform cycles while interacting with the environment. The central equation used in this context is δQ = T dS,
which assumes that processes occurring in the system are reversible. Thermal engines
perform cycles, where the system departs from a given state and returns to the same
state after the end of the cycle. The cycle is then repeated. During the cycle, the system absorbs heat Q and performs work −W (remember that W is the work done by the
environment on the system). Obviously Q = −W , since the energy does not change in
a cycle, ∆E = 0 (because the system begins and ends in the same state, and E is a
state function). It turns out that, after the cycle is completed, the system necessarily has
released some heat, and this result follows from the second law.
Consider the cycle depicted in Fig. 1.1, called Carnot cycle. The cycle has been represented in the T − S plane, which is quite convenient to discuss cycles. In the Carnot
cycle the engine operates between two temperature reservoirs T1 and T2 , with T1 > T2 .
From a state A, the system follows an isothermal process at temperature T1 to a state
B, after which the change of entropy is ∆SAB = Q1 /T1 , with Q1 the heat absorbed by
the system from the reservoir. From B an adiabatic process is followed down to C, with
∆SBC = 0 and no heat exchange. Then the system follows a reversed path at temperature T2 from C to D along which it releases a heat −Q2 and changes its entropy by
∆SCD = −Q2 /T2 . Finally, the system revisits state A by following another adiabatic path
with ∆SDA = 0. The total entropy change has to be zero, since S is a state function:
∆S = ∆SAB + ∆SBC + ∆SCD + ∆SDA = 0, which means
Q1
Q2
Q1
T1
=
→
=
> 1,
T1
T2
Q2
T2
(1.6)
the last inequality following from the condition T1 > T2 . This means that the system
absorbs and releases heat, with a positive net amount of heat absorbed, Q = Q1 − Q2 > 0.
3
This heat is used to perform work, W = −Q. The amount of heat or work is given by
the area surrounded by the cycle,
Q = −W =
I
T dS = (T1 − T2 )(S2 − S1 ).
(1.7)
On the other hand, the heat absorbed in the first process is Q1 = T1 (S2 − S1 ). The
efficiency of the heat engine is defined as
η=
(T1 − T2 )(S2 − S1 )
−W
T2
work performed by system
=
=
=1− .
heat absorbed by system
Q1
T1 (S2 − S1 )
T1
(1.8)
Obviously, since 0 < T2 < T1 , the efficiency is never 100% (i.e. there is always some heat
released by the engine, Q2 > 0). The Carnot cycle can be used to measure temperature
ratios by calculating the efficiency of a given cycle.
It can be shown that the Carnot cycle is the most efficient engine possible, which is an
alternative statement of the second law. Other alternative statements exist, notably those
due to Kelvin and Clausius. This point of view is useful in connection with engineering
applications of thermodynamics, and therefore we will not discuss it in the course.
T
T1
A
B
Q=-W
T2
D
C
S1
S2
S
Figure 1.1: Carnot cycle in a heat engine between temperatures T1 and T2 . The net
amount of heat absorbed by the system, which is equal to the work performed by the
system, is given by the area enclosed by the four thermodynamic paths.
1.4
Thermodynamic potentials
The parameters (N, V, E) characterising the system are often inconvenient from various
aspects (experimental or otherwise). Generalisations of thermodynamics are necessary
to contemplate different sets of variables. The maximum-entropy principle can be easily
generalised by defining different thermodynamic potentials .
4
• (N, V, S). This set of variables is not widely used but it is anyway useful sometimes,
especially in theoretical arguments. The thermodynamic potential here is the energy,
E = E(N, V, S), which can always be obtained from S = S(N, V, E) by inversion.
The energy satisfies a minimum principle, in constrast with the entropy (which
satisfies a maximum principle).
• (N, V, T ). These are more realistic variables. The thermodynamic potential is the
Helmholtz free energy, F = F (N, V, T ), such that F = E − T S. This function
satisfies a minimum principle, which is basic to most of the developments that will
be made in this course.
• (N, p, T ). The thermodynamic potential is the Gibbs free energy, G = G(N, p, T ),
with G = F + pV , also obeying a minimum principle.
• (µ, V, T ). The characteristic thermodynamic potential is now the grand potential,
Ω = Ω(µ, V, T ), with Ω = F − µN. The grand potential is again minimum for the
equilibrium state of the system.
The fact that the different thermodynamic potentials satisfy an extremum principle can
be summarised in the following differential expressions:
dS = 0,
d2 S < 0, dF = 0, d2 F > 0, dG = 0, d2 G > 0, dΩ = 0, d2 Ω > 0.
(1.9)
Since variations implied by the differentials are due to changes in microscopic variables,
many times in statistical mechanics we will be taking derivatives of the relevant thermodynamic potentials with respect to microscopic variables, equating these derivatives to
zero in order to obtain the equilibrium state of the system as an extremum condition, and
imposing a sign on second variations in order to analyse the stability of the solutions.
1.5
Thermodynamic processes and first derivatives
Finally, let us consider expressions relating the variables during thermodynamic processes;
these come in terms of relations between differential quantities. For the energy:
dE =
∂E
∂N
!
∂E
dN +
∂V
V,S
!
∂E
dV +
∂S
N,S
!
N,V
dS = µdN − pdV + T dS,
(1.10)
which expresses the first law of thermodynamics when there is a change in the number of
particles. For the Helmholtz free energy:
dF = dE − T dS − SdT = µdN − pdV − SdT,
(1.11)
from which
µ=
∂F
∂N
!
,
V,T
∂F
p=−
∂V
!
,
N,T
∂F
S=−
∂T
!
.
(1.12)
N,V
For the Gibbs free energy:
dG = dF + pdV + V dp = µdN − SdT + V dp,
5
(1.13)
so that one obtains
µ=
∂G
∂N
!
,
∂G
∂V
V =
p,T
!
∂G
S=−
∂T
,
N,T
!
.
(1.14)
N,p
Finally, for the grand potential:
dΩ = dF − µdN − Ndµ = −pdV − SdT − Ndµ,
(1.15)
and
∂Ω
N =−
∂µ
1.6
!
∂Ω
p=−
∂V
,
V,T
!
∂Ω
S=−
∂T
,
µ,T
!
.
(1.16)
µ,V
Second derivatives
Second derivatives of the thermodynamic potentials incorporate useful properties of the
system. They are also called response functions, since they are a measure of how the
system reacts to external perturbations. The most important are:
• Thermal expansion coefficient:
1
α≡
V
∂V
∂T
!
N,p
∂2G
∂T ∂p
1
=
V
!
.
(1.17)
N
This coefficient measures the resulting volume change of a material when the temperature is changed. α > 0, since ∆T > 0 necessarily implies ∆V > 0.
• Isothermal compressibility:
1
κT ≡ −
V
!
∂V
∂p
T
∂2G
∂p2
1
=−
V
!
.
(1.18)
N,T
Compressibility measures the change of volume of a material when a pressure is
applied on the material. A very compressible material has a large value of κ: on
applying a slight incremental pressure the volume changes by a large amount. The
opposite occurs when the material is very little compressible. The sign − in the
definition is needed to insure that κT > 0, since ∆p > 0 implies ∆V < 0.
Another compressibility, the adiabatic compressibility, is also defined:
∂V
∂p
1
κS ≡ −
V
!
.
(1.19)
S
• Specific heat (or heat capacity) at constant volume:
CV ≡ T
∂S
∂T
!
=
N,V
∂E
∂T
!
=
N,V
∂Q
∂T
!
N,V
= −T
∂2F
∂T 2
!
.
(1.20)
N,V
The heat capacity per particle, cv = Cv /N is also commonly used. The heat capacity
is a measure of how energy changes when the temperature of a material is changed.
If high, a large variation in energy corresponds to a small change in temperature
(this is the case in water, which explains the inertia of water to change temperature
and its quality as a good thermoregulator). Obviously CV > 0 since, when ∆T > 0,
energy must necessarily increase, ∆E > 0.
6
• Specific heat at constant pressure:
Cp ≡ T
∂S
∂T
!
=
N,p
∂E
∂T
!
=
N,p
∂Q
∂T
!
N,p
= −T
∂2G
∂T 2
!
.
(1.21)
N,p
Likewise the heat capacity per particle, cp = Cp /N is commonly used. This heat
capacity is more easily accessible experimentally. There is a relation between Cv
and Cp involving some of the previously defined coefficients:
Cp = CV +
T V α2
.
NκT
(1.22)
A useful additional result is the Euler theorem. To obtain the theorem, we rst write the
extensivity property of the energy as
E(λN, λV, λS) = λE(N, V, S),
(1.23)
and differentiate with respect to λ:
∂E
∂E
∂E
N+
V +
S = E(N, V, S).
∂λN
∂λV
∂λS
(1.24)
Taking λ = 1 (this is legitimate since λ is arbitrary) and using the definition of the
intensive parameters:
E = µNpV + T S
(Euler theorem).
(1.25)
From this we can obtain the Gibbs-Duhem equation by differentiating:
dE = µdN + Ndµ − pdV − V dp + T dS + SdT,
Ndµ − V dp + SdT = 0
(Gibbs-Duhem equation).
(1.26)
The Gibbs-Duhem equation reflects the fact that the intensive parameters are not all
independent.
When the Euler theorem is applied to the other thermodynamic potentials, the following
relations are obtained:
Ω = −pV,
G = µN,
F = µN − pV.
(1.27)
For example, let us obtain the second relation. We have:
G(λN, p, T ) = λG(N, p, T ).
(1.28)
Differentiating with respect to λ:
∂G
N = G(N, p, T ).
∂λN
Taking λ = 1 and using µ = (∂G/∂N)(p,T ) , the relation G = µN follows.
7
(1.29)
1.7
Equilibrium and stability conditions
Intensive parameters are very important for various reasons. One is that they are easily
controlled in experiments so that one can establish certain conditions for the environment
of the system. Another reason is that they define specific criteria for the equilibrium
conditions of a system.
Let us consider two systems in contact, 1 and 2, so that energy is allowed to flow from
one to the other, but the system as a whole is otherwise isolated. Let E1 and E2 be their
energies, V1 and V2 their volumes, and N1 and N2 their numbers of particles. Volumes
and number of particles are constant, but energies can change subject to the condition
E1 + E2 =const. Since dS = 0 for arbitrary internal changes, with S = S1 + S2 , and in
any process dE1 = dE2 , we have:
0 = dS =
∂S1
∂E1
!
∂S2
dE1 +
∂E2
(N1 ,V1 )
= dE1
!

∂S1
dE2 = dE1 
∂E1
(N2 ,V2 )
!
∂S2
−
∂E2
(N1 ,V1 )
!
1
1
.
−
T1 T2
(N2 ,V2 )


(1.30)
Therefore, at equilibrium, T1 = T2 . Now if we also allow for variations in the volumes V1
and V2 , with the restriction V1 + V2 =const., then in any process
0 = dS =

= dE1 
∂S1
∂E1
!
∂S1
∂E1
!
∂S2
dE1 +
∂E2
(N1 ,V1 )
(N1 ,V1 )
−
∂S2
∂E2
!
!
∂S1
dE2 +
∂V1
(N2 ,V2 )
(N2 ,V2 )


 + dV1 
∂S1
∂V1
!
!
∂S2
dV1 +
∂V2
(N1 ,E1 )
(N1 ,E1 )
−
∂S2
∂V2
!
!
(N2 ,E2 )
dV2
(N2 ,E2 )

,
(1.31)
so that
0 = dE1
1
1
p2
p1
−
+ dV1
−
,
T1 T2
T1 T2
(1.32)
from which T1 = T2 and p1 = p2 . Finally, if transfer of particles is allowed between the
two systems, but with the restriction N1 + N2 = const. and the volumes of the systems
fixed,
0 = dS =
∂S1
∂E1
!
∂S2
dE1 +
∂E2
(N1 ,V1 )
= dE1
!
∂S1
dE2 +
∂N1
(N2 ,V2 )
1
1
µ1 µ2
− dN1
,
−
−
T1 T2
T1 T2
!
∂S2
dN1 +
∂N2
(V1 ,E1 )
!
dN2
(V2 ,E2 )
(1.33)
from which T1 = T2 and µ1 = µ2 . Therefore, for the two systems to be in equilibrium,
thermal, mechanical and chemical equilibria are required at the same time. These conditions will be very important when we discuss phase transitions of the first order in
subsequent chapters. Using the same arguments, one can also say that in a single system
at equilibrium the local temperature, the local pressure and the local chemical potential
must be equal everywhere, i.e. all spatial gradients of the intensive parameters must be
8
zero (of course this requires a suitable definition for local quantity). Any nonzero gradient
gives rise to a current or flow (transport of energy and mass) that restores equilibrium.
Relation between current and gradient is usually given, in linear response theory, by a
proportionaly law of the type of Fouriers equation or similar laws.
The above equilibrium conditions are obtained from the equation dS = 0, which is an
extremum condition, and involve the equality of the intensive parameters. The condition
on second differential, d2 S < 0, can also be exploited, giving rise to the stability conditions
for the system. These involve the response functions. As an example, we consider the
Gibbs free energy G(N, p, T ) which, by virtue of the Euler equation, can be written
G = F + pV = ET S + pV , i.e.
G(N, p, T ) = E(N, V, S)T S + pV.
(1.34)
Suppose we fix (N, p, T ) and consider an internal process in the system. This will change
the values of S and V and, correspondingly, of E. The variation in G to all orders will
be:
∆G = ∆E − T ∆S + p∆V =
1
+
2
"
!
!
∂E
∂E
dV +
dS − T dS + pdV
∂V
∂S
∂2E
∂2E
∂2E
2
dV
dS
+
dV
+
2
dS 2 + · · ·
2
2
∂V
∂V ∂S
∂S
!
!
#
!
(1.35)
Since
∂E
∂V
!
∂E
∂S
= −p,
!
= T,
(1.36)
we get
∂2E
∂2E
∂2E
2
dV
dS
+
dV
+
2
dS 2 + · · ·
∂V 2
∂V ∂S
∂S 2
#
(1.37)
∂2E
∂2E
∂2E
2
dV
dS
+
dV
+
2
dS 2 > 0,
∂V 2
∂V ∂S
∂S 2
(1.38)
1
∆G =
2
"
!
!
!
The condition d2 G > 0 implies
!
!
!
which requires
∂2E
∂V 2
!
> 0,
∂2E
∂S 2
!
> 0,
∂2E
∂V 2
!
∂2E
∂S 2
!
∂2E
−
∂V ∂S
!2
> 0.
(1.39)
Using the denition of the response functions, the following conditions of stability are
obtained:
CV > 0,
κS > 0,
T
>
V κS CV
∂T
∂V
!2
.
(1.40)
S
Following the same procedure but using other thermodynamic potentials one can obtain
other stability conditions, e.g. κT > 0, Cp > 0, ...
9
1.8
Statistical ensembles
While thermodynamics derives relationships between macroscopic variables, statistical
mechanics aims at providing numerical values for such quantities. It also explains relations between thermodynamic quantities as derived by thermodynamics. It is based on
knowledge of the interactions between the microscopic entities (either real or defined)
making up the system.
In principle statistical mechanics takes account of the time evolution of all degrees of
freedom in the system. Thermodynamic mechanical quantities (thermodynamic energy,
pressure, etc.) are identified with appropriate time averages of corresponding quantities
defined for each microscopic state. Since it is not possible to follow the time evolution of
the system in detail, a useful mathematical device, the ensemble, is used by equilibrium
statistical mechanics. The basic postulate is that time averages coincide with ensemble
averages (the ergodic hypothesis). Ensemble theory tremendously facilitates calculations,
but a statistical approach is implicit via probability distributions. The loss of information
inherent to a probabilistic approach is not relevant in practical terms.
An ensemble is an imaginary collection of static replicas of the system, each in one of
the possible microscopic states; possible means compatible with the imposed macroscopic
conditions. Corresponding to different possibilities to define these conditions, different ensembles can be defined. The one associated with the primitive variables (N, V, E) is the
so-called microcanonical ensemble: here we mentally collect all the possible microscopic
states (Ω in number; not to be confused with the grand potential) that are compatible
with the imposed conditions (N, V, E). A basic postulate of statistical mechanics is that,
as the system dynamically explores all these possible states, the time spent on any one
of them is the same. This means that the number (or fraction) of replicas of the system
to be found in the ensemble in any one state is the same, irrespective of the state. If we
consider the states to be numerable (this is actually the case quantum-mechanically) with
an integer ν = 1, 2, ..., Ω, all states are equally probable and assigned the same probability
pν = Ω−1 . Connection with thermodynamics is realised via the famous expression
S = k log Ω,
(1.41)
where k is Boltzmann’s constant, k = 1.3806503 × 10−23 J K−1 . All macroscopic variables
can be obtained from S using Eqns. (1.4). The microcanonical ensemble is difficult to use
since counting the number of microstates (i.e. computing Ω) is often impractical. Also,
in classical mechanics, where states form a continuum and therefore Ω would in principle
be infinite, one has to resort to quantum-mechanical arguments (to demand compatibility
with the classical limit of quantum mechanics) and associate one state to a volume hn in
phase space (with h Planck’s constant and n the number of degrees of freedom). One can
still use Eqn. (1.41), but its interpretation is a bit awkward.
Eqn. (1.41) connects the microscopic and the macroscopic worlds. Also it gives an
intuitive picture of what entropy means: it is a quantity associated with order. In effect,
an ordered system has very few possible configurations Ω and therefore low entropy (consider a solid where atoms are located in the neighbourhood of the sites of a lattice and
cannot wander very far), while a disordered system has a large number of possible states
10
which means high entropy (for instance, a fluid, where atoms can explore the available
volume more or less freely). The history of the development of the entropy concept and its
relation to mechanics is a fascinating one, involving the irreversibility paradox solved by
Boltzmann (and the heated debate associated with it that allegedly led to Boltzmann’s
suicide). The paradox states that the microscopic (either classical or quantum) equations of motion are time-reversible, but macroscopically there is an arrow of time which
processes have to respect (for example, a gas spontaneously expands into the available
volume, but never compresses locally and leaves a void region). We will not go into many
technicalities here, but simply mention an intuitive solution for the problem of why an
ideal gas expands but the reversed process never occurs, Fig. 1.2. Thermodynamic’s
second law states that entropy always increases when an isolated system undergoes an
irreversible process. Eqn. (1.41) then implies that, at the microscopic level, the number
of available microstates should increase. If we take a gas confined to one half of a volume
V , i.e. to V /2, and at some instant of time let the gas expand into the whole volume V ,
we know how to calculate the entropy increase: since the entropy of an N-particle ideal
gas in a volume V is S = S0 + Nk log (V Λ3 /N) [see Eqn. (1.59)], the change of entropy
when the gas expands from a volume V /2 to a volume V is
∆S = Sfinal − Sinitial = Nk log V Λ3 /N − Nk log V Λ3 /2N = Nk log 2 > 0.
(1.42)
Incidentally, this result can also be obtained by direct application of Eqn. (1.41). Since
particles are independent, the number of ways in which the N particles can be distributed
in one half of the volume, with respect to the total number, is given by (1/2)N , since there
are only two choices for each particle (it is either in the left or the right half). Then:
N
Ωinitial
1
∆S = k (log Ωfinal − log Ωinitial ) = −k log
= −k log
Ωfinal
2
= Nk log 2.
(1.43)
According to experience, the reversed process where the gas compresses spontaneously
into the left half never occurs, and indeed thermodynamics prohibits so. Microscopically
it is easy to see that Ω has also increased: a particle has a larger phase-space volume
available when the gas expands than before. Therefore Eqn. (1.41) is reasonable. But
what about the time evolution? When the gas has expanded, since every gas particle
follows a reversible trajectory, there should be phase-space trajectories leading to states
where all particles are again only in half of the volume, but these trajectories are very few
with respect to the total number. In other words, these trajectories are very improbable
or, put it differently, we should wait a disproportionately long time for that state to be
reached. The microscopic explanation of the second law is statistical: entropy generally
increases, but there may be an insignificantly small number of particular processes which
lead to an entropy decrease.
The microcanonical ensemble is very important from a conceptual point of view, but
impractical for doing calculations. A more convenient ensemble is the canonical ensemble,
associated to the variables (N, V, T ). The system is in contact with a thermal reservoir
at temperature T . Here the probability p of the ν-th state depends on the energy of the
state, Eν :
e−βEν
pν =
Q
11
(1.44)
V/2
V/2
V
Figure 1.2: Irreversible expansion of a gas from a volume V /2 to a volume V .
where β = 1/kT and Q is the canonical partition function, which appears as a normalisation of the probability function pν . For a classical conservative system of N identical
particles characterised by a Hamiltonian H(q, p), with (q, p) a set of canonically conjugated variables q = {q1 , q2 , ..., qn }, p = {p1 , p2 , ..., pn },
1 ZZ
Q=
dqdpe−βH(q,p).
n
N!h
(1.45)
For a quantum system
Q=
X
e−βEν .
(1.46)
ν
Connection with thermodynamics is made through the expression
F = −kT log Q,
(1.47)
and all macroscopic variables can be obtained from F using Eqns. (1.12). One obvious
difficulty with this ensemble is that it is not always possible to evaluate the partition function, which involves a complicated sum over states. Very often the activity in statistical
mechanics involves ways to avoid having to sum over states (by doing it in a different
but approximate way), even though this is actually what the theory asks for. For the
mechanical variables (variables that can be linked to each microscopic state) a different
method to have access to macroscopic properties is via ensemble averaging using the probability pν . This implies that thermodynamic quantities are directly identified with these
averages. For example, in the canonical ensemble, where the energy fluctuates, the mean
energy hEi is
hEi =
X
Eν pν ,
pν =
ν
e−βEν
.
Q
(1.48)
In classical mechanics we would have
hEi =
ZZ
dqdpH(q, p)ρ(q, p),
ρ(q, p) =
e−βH(q,p)
.
Q
(1.49)
Other typical mechanical variables for which this method can be applied are pressure and
magnetisation (for a magnetic system). Ensemble averaging finds important applications
12
in computer-simulation methods.
A simple example is a gas of N identical classical noninteracting monoatomic molecules,
with Hamiltonian
H(q, p) =
N
X
p2i
.
i=1 2m
(1.50)
Only kinetic terms are included. The effect of interactions, however, has to be taken into
account implicitely, as it is the only possible mechanism that allows energy to flow among
the degrees of freedom. The partition function of the gas can be factorised into a product
of reduced molecular partition functions:
Q=
1
N!h3N
Z
dp
Z
dqe−βH(q,p) =
VN
N!h3N
Z
∞
−∞
dpe−βp
2 /2m
3N
=
qN
,
N!
(1.51)
with q the ‘reduced’ molecular partition function:
V
q= 3
h
Z
∞
−βp2 /2m
−∞
dpe
3
=
V
,
Λ3
Λ= √
h
.
2πmkT
(1.52)
The factor N! in (1.51) accounts for particle indisguishability. Λ is the so-called thermal
wavelength, which is a measure of the extension of the wavepacket associated to the
molecule (thus q is the fraction of total volume available to the molecule with respect
to the quantum volume). The multiplicative property of the partition function goes over
into the additive property of the free energy, and all other thermodynamic functions can
be derived from it. We have:
F = −kT log Q = −NkT log q − kT log N! = −NkT log
V
− kT log N!.
Λ3
(1.53)
Since N is normally very large, the logarithm of N! is usually approximated using Stirlings
approximation:
log N! = N log N − N + ... ≃ N log N − N,
N ≪ 1,
(1.54)
which finally gives
F
= ρ log ρΛ3 − ρ,
V kT
(1.55)
where ρ = N/V is the mean density. From Eqn. (1.55) the pressure is
p=
∂F
∂V
!
=
N,T
NkT
,
V
(1.56)
which is the ideal-gas equation of state. The energy can be obtained from F = ET S:
E = F + TS = F − T
∂F
∂T
!
N,V
= −T 2
∂F/T
∂T
!
= kT 2
N,V
∂ log Q
∂T
!
,
(1.57)
N,V
which gives
3
E = NkT.
2
13
(1.58)
The entropy of the ideal gas follows from F = E − T S:
S=
5
E−F
= Nk − Nk log ρΛ3 = S0 − Nk log ρΛ3 .
T
2
(1.59)
Note that the fact that the Hamiltonian can be separated into independent terms, in this
case one for each particle, brings about a factorisation of Q into N identical reduced partition functions that can be computed (in this case easily); the free energy is then obtained
as a sum of N identical reduced free energies. This is a clear example of an ideal system.
Frequently the Hamiltonian of a given system is approximated by uncoupling particular
variables or set of variables in order to use this mathematical trick, the so-called ideal
approximation.
For completeness, we mention the statistical-mechanical expressions for the grand canonical ensemble, with variables (µ, V, T ). Here the system is in contact with a reservoir at
temperature T and chemical potential µ, with which the system exchanges energy and
particles. We have:
Ξ=
∞
X
1
nN
N =0 N!h
ZZ
dqdpe−β[HN (q,p)−µN ] ,
(1.60)
where Ξ is the grand partition function and nN is the number of degrees of freedom for
N particles. For a quantum system:
Ξ=
∞ X
X
e−β[Eν (N )−µN ]
(1.61)
N =0 ν
(here Eν (N) is the energy spectrum of a system of N particles). Connection with thermodynamics is made through the expression
Ω = −kT log Ξ,
(1.62)
and macroscopic variables can be obtained from Ω using Eqns. (1.16). This ensemble is
useful when discussing phase transitions. Sometimes it is simpler to use than the canonical ensemble, since summation over N may lead to simplication in the state-counting
operation (an obvious example is the Bose ideal gas).
1.9
Fluctuations
In statistical mechanics averages (first moments) are identified with thermodynamic mechanical quantities. But in different ensembles the fluctuating mechanical quantities are
different. Equivalence of ensembles when N is large holds at the level of averaged mechanical quantities (meaning, for example, that the pressure hpi obtained as an average
for given volume V in the canonical ensemble (N, V, T ) gives rise, in the isobaric ensemble
(N, hpi , T ), to a value of the average volume hV i that coincides with V ). But it should
also hold at the level of second-order moments; this means that fluctuations must remain
small in large systems (and zero in the thermodynamic limit). What about higher-order
averages (i.e. higher than first moments) then? It turns out that these are also linked to
thermodynamic properties of the system. We focus on second moments or ‘fluctuations’.
14
Their values depend on the type of ensemble.
In the canonical ensemble the system is coupled to a thermal reservoir at temperature T ,
with which it interchanges energy. The fluctuation in energy is
rD
σE =
E
(E − hEi)2 =
q
hE 2 i − hEi2 .
(1.63)
A simple derivation using Eqn. (1.48) leads to
D
E
2
σE2 = E 2 − hEi = kT 2
∂ hEi
∂T
!
= kT 2 CV .
(1.64)
(N,T )
Therefore the fluctuation in energy is related to the specific heat at constant volume.
This is an example of the fluctuation-dissipation theorem in statistical physics, which
relates fluctuations of a statistical variable to some thermodynamic quantity. This result
immediately implies (since hEi ∼ NkT ) that
1
σE
∼√ ,
hEi
N
(1.65)
so that the relative fluctuation is very small in real materials. A similar result in the
macrocanonical ensemble is
2
σN
D
= N
2
E
2
− hNi = kT
∂ hNi
∂µ
!
= ρkT κT
(1.66)
(V,T )
or the fluctuation in the number of particles. Also,
σN
1
∼√ .
hNi
N
(1.67)
The law N −1/2 is quite general in equilibrium statistical physics.
1.10
Examples of some ideal systems (reminder)
As a reminder, we list a few classical examples where the ideal approximation can be
successfully applied. In most cases this approximation provides a starting point from
which more sophisticated approximations, incorporating the effect of interactions, can be
implemented.
• Harmonic solid
In a crystalline solid molecules are arranged into a periodic three-dimensional lattice, each molecule being attached to a particular lattice site, about which it moves
in an oscillatory way. The Hamiltonian is
H(p, q) =
N
X
=
N
X
N X
p2i
1X
+
φ (|ri − rj |)
2 i=1 j6=i
i=1 2m
N X
1X
p2i
+ U0 +
ui · Dij · uj + · · · ,
2 i=1 j6=i
i=1 2m
15
(1.68)
where φ(r) is the pair potential between two molecules (we assume it, for the sake
of simplicity, to be isotropic, i.e. to depend on the modulus of the relative vector).
In the last equality the displacement vector of the i-th molecule has been defined,
ui = ri − Ri , with Ri the location of its associated lattice site, and the interaction
part has been approximated using a Taylor expansion in all the vectors ui . D is a
nondiagonal 3N × 3N matrix, which essentially gives the curvatures of the potential
at the equilibrium (lattice) sites. U0 is the (constant) Madelung energy. Now,
restricting the expansion to the quadratic term (this can be done at sufficiently low
temperatures), and diagonalising the D matrix, the Hamiltonian adopts a diagonal
form:
H(ξ, ξ̇) =
3N 1X
ξ˙k2 + ωk2ξk2 ,
2 k=1
(1.69)
which corresponds to 3N decoupled harmonic oscillators of frequencies {ωk }. Again,
as in the ideal gas, the degrees of freedom can be grouped into a set of 3N independent, decoupled coordinates ξk , called normal modes. In this case the original
(natural) Cartesian coordinates were coupled but, within the context of the harmonic approximation, particular linear combinations of these coordinates (the normal modes) are decoupled. From this point of view, the harmonic solid is an ‘ideal
gas’. The statistical mechanics of the problem is now trivial; the partition function
is
Q=
3N
Y
qk ,
(1.70)
k=1
with qk the reduced partition function of the i-th vibrational mode. At the low
temperatures where the harmonic approximation is supposed to be valid, a quantum
treatment is required. The quantum partition function (reduced partition function)
of the k-th harmonic oscillator is:
qk =
∞
X
(k)
−βEn
e
=
∞
X
−β(n+1/2)h̄ωk
e
n=0
n=0
e−βh̄ωk /2
=
.
1 − e−βh̄ωk
(1.71)
Here En(k) = (n + 1/2)h̄ωk is the quantized energy of a single quantum harmonic
oscillator. The complete partition function is the product of the reduced partition
functions:
Q=
3N
Y
qk =
k=1
3N
Y
e−βh̄ωk /2
,
1 − e−βh̄ωk
k=1
(1.72)
and the free energy is
F = −kT log Q = U0 + kT
3N
X
k=1
(
h̄ωk
+ log 1 − e−βh̄ωk
2kT
)
(1.73)
(the Madelung energy has been added). Since there are so many frequencies distributed along a finite interval in frequencies, it is a good approximation to introduce
16
a frequency distribution g(ω), such that g(ω)dω gives the number of modes with
frequencies in the interval [ω, ω + dω]. Obviously:
Z
∞
0
dωg(ω) = 3N.
(1.74)
Going from the sum over modes k to an integral over ω involves the transformation
3N
X
Z
→
i=1
∞
0
dωg(ω).
(1.75)
The free energy can then be written as
F = U0 + kT
Z
∞
0
(
)
h̄ω
dωg(ω)
+ log 1 − e−βh̄ω ,
2kT
(1.76)
and the energy is
E = U0 + h̄
Z
∞
0
e−βh̄ω
1
+
dωg(ω)ω
2 1 − e−βh̄ω
(
)
.
(1.77)
A standard approximation is the Debye theory, which assumes
g(ω) =





αω 2, 0 < ω < ωmax ,
(1.78)
0,
ωmax < ω < ∞.
3
From the normalisation of g(ω) one gets α = 9N/ωmax
. Then the energy can be
written as
E = U0 +
9Nh̄ωmax 9NkT
+
8
u3
Z
0
u
dx
x3
,
ex − 1
(1.79)
where u ≡ βh̄ωmax . From here the specific heat is
"
∂E
12
CV =
= 3Nk 3
∂T
u
Z
0
u
3u
x3
− u
dx x
e −1 e −1
#
(1.80)
It is easy to analyse the low- and high-temperature limits of CV : for T → 0,
CV ∝ T 3 , while for T → ∞, CV → 3Nk. Both limits are correct, despite the
crudeness of the Debye theory (for metals, an additional contribution to CV due
to free electrons changes this scenario a bit; the above would be the contribution
coming from ionic degrees of freedom).
• Paramagnetism
A paramagnetic substance exhibits a finite magnetisation when subject to an external magnetic field, whereas it possesses no net magnetisation at zero field. Langevin
paramagnetism is due to the interaction of the intrinsic magnetic moments of the
molecules of a substance with the magnetic field, and it can be studied classically
(this is in contrast with Pauli paramagnetism, which is due to free electrons in a
metal and has to be analysed using quantum statistics). Thermodynamically, the
17
magnetisation M and the magnetic field H are conjugate variables. The following
derivatives can be defined:
∂G
M =−
∂H
!
,
!
∂M
∂H
χT =
T
,
CH = −T
T,H→0
∂2G
∂T 2
!
.
(1.81)
H
[here we are using the Gibbs free energy as the relevant thermodynamic potential,
G = G(H, T ), since the relevant variables are taken as H –which plays the role of a
pressure– and temperature; one can also use M and T as variables, in which case the
relevant potential is the Helmholtz potential F = F (M, T )]. These are completely
equivalent to those defined for a simple substance, noting that M is to be associated
with the volume V , and H with the pressure; thus, χT (magnetic susceptibility at
zero field) is equivalent to the compressibility, and CH is the specific heat at constant
magnetic field (similar to Cp ). κT and CH are response functions. In particular, κT
will play an important role in future developments.
A very simple model of paramagnetism consists of assigning a magnetic moment
(spin) to each atom, disregarding any interaction between the spins. We assume
that spins are independent and that there is an applied magnetic field H. Each spin
has a magnitude S = 1/2, so that only two values of the spin along the direction
of the magnetic field are possible, s = ±1. The spins will have an associated
magnetic moment µB (the Bohr magneton), so that µ = µB s. The Hamiltonian of
the paramagnetic solid is then
H=−
N
X
i=1
µi H = −µB H
N
X
si .
(1.82)
i=1
Let us calculate the magnetisation M. There are two ways: one is to calculate the
average value by thermal averaging:
hMi =
X
X
s1 =±1 s2 =±1
X
···
s =±1
XN
s1 =±1 s2 =±1
= µB
N
X
i=1

= µB



X
s1 =±1

eβµB Hs1  


X
s1 =±1
X
s=±1
= NµB X
X
s1 =±1

eβµB Hs1  
X
s2 =±1

s1 =±1
seβµB Hs
eβµB Hs
X
···

i=1
e−βH
sN =±1
X



X
s2 =±1
si =±1
N
X
i=1
18




X
si =±1


si eβµB Hsi  · · · 

X
sN =±1

sN =±1

eβµB HsN 

X
X
sN =±1

si eβµB Hsi  · · · 
eβµB Hs2  · · · 
eβµB H − e−βµB H
eβµB H + e−βµB H
s=±1
X
eβµB Hs2  · · · 
eβµB Hs2  · · ·
eβµB Hs1  
= NµB
µB si e−βH
eβµB Hs2  · · · 
s2 =±1
s2 =±1

!
X

eβµB Hs1  
X
N
X
X
X
sN =±1

eβµB HsN 

eβµB HsN 

eβµB HsN 
= NµB tanh (βµB Hs).
(1.83)
Note that hMi = NµB hsi, where hsi is the thermal average of a single spin. This
is of course a consequence of the spins being independent.
The other route is via the free energy. The partition function is:
Q = qN ,
q=
X
eβµB Hs = eβµB H + e−βµB H
(1.84)
s=±1
(we consider the spins to be distinguishable since they are arranged on the sites of
a crystalline lattice). Then:
G = −NkT log q = −NkT log eβµB H + e−βµB H .
Using the thermodynamic derivative:
∂G
M =−
∂H
!
= NµB
T
(1.85)
eβµB H − e−βµB H
= NµB tanh (βµB H),
eβµB H + e−βµB H
(1.86)
which coincides with the previous expression. The magnetic susceptibility at zero
field is:
µ2B M
.
kT
(1.87)
= log [2 cosh (βµB H)] − βµB H tanh (βµB H).
(1.88)
χT =
∂M
∂H
!
=
T,H→0
The entropy is
1
S
=−
Nk
Nk
∂G
∂T
!
H
It can be demonstrated that the limits of M and S for small H are, respectively:
Nµ2B H
M →
,
kT
S
1 µB H
→ log 2 −
Nk
2 kT
2
.
(1.89)
The first equation is the famous Curie’s law, M = cH/T , which is the equation of
state of an ideal paramagnet. It is only valid for small M (M should be small compared with the saturation magnetisation M0 , which is the maximum magnetisation
that a particular magnet can exhibit, in this case NµB ). Note that M → χT H, as
it corresponds to χT being a response function at zero field. The first term in the
entropy is the entropy of the system at infinite temperature (or zero magnetic field),
where all spins are independent and S = k log 2N .
Other typical examples where the ideal approximation can be applied are: diatomic
molecules (where rotational, translational and vibrational degrees of freedom are approximately decoupled and treated as independent), chemical reactions, mixtures of ideal
gases, and ideal quantum systems (giving rise to quantum statistics and their typical
applications: Bose-Einstein condensation, black-body radiation, non-interacting fermion
systems such as electrons in metals, etc.)
19
Figure 1.3: Equations of state of (a) a fluid, and (b) a magnet, for different temperatures.
The discontinuous line is the ideal approximation.
1.11
Phenomena that cannot be explained by ideal
approximation
The ideal approximation enjoys a number of successes. We have reviewed just three:
derivation of ideal-gas law relating pressure, temperature and density for a dilute gas,
Eqn. (1.56), correct low- and high-temperature limits of the heat capacity of solids (ionic
contribution in metals), and explanation of Curies law. The most severe limitation of the
ideal approximation is its failure to explain a very important class of phenomena, namely
the occurrence of phase transitions in Nature (the only exception is the Bose-Einstein condensation, which occurs in a quantum ideal system of bosons and is therefore a quantum
phase transition). Phase transitions are collective phenomena due to interactions; any
approximation that neglects interactions will not be able to reproduce any phase transition. Since phase transitions are ubiquitous, the ideal approximation must be corrected.
But many other features of materials require consideration of interactions. For example, most properties of liquids are not even qualitatively reproduced by adopting an ideal
approximation. The experimental behaviour of the equation of state of a fluid can be
used to illustrate these features. Experimentally, the equation of state can be written as
p
= ρ + B2 (T )ρ2 + B3 (T )ρ3 + · · ·
kT
(1.90)
where Bn (T ) are the virial coefficients, smooth functions of the temperature. The first
term in the right-hand side accounts for the ideal-gas approximation. Fig. 1.3(a) depicts
qualitatively the equation of state p = p(ρ, T ) of a typical fluid at two temperatures.
Several gross discrepancies with the ideal-gas approximation are worth-mentioning:
• As ρ increases from zero the pressure departs from linear behaviour. This effect is
due to interactions in the fluid.
• At low temperature there is a region where the pressure is constant; this reflects the
20
presence of a phase transition: the low-density gas phase passes discontinuously to
the high-density liquid phase.
• At high temperatures, T > Tc , the phase transition disappears. At Tc (the critical
temperature), the system exhibits very peculiar properties; for example, the first
derivative of the pressure with respect to density vanishes, implying an infinite
compressibility.
In the case of a paramagnet, the equation of state H = H(M, T ) is represented schematically in Fig. 1.3(b). In this case the ideal approximation works relatively well for low
values of H. However, in a ferromagnetic material the phenomenon of spontaneous magnetisation takes place at a temperature T = Tc : below this temperature, a non-zero
magnetisation arises in the material, even at H = 0. The origin of this is the interaction
between spins, which reinforces the effect of any external magnetic field and may create
an internal magnetic field even when an external magnetic field is absent.
21