Download Statistical Mechanics: An overview

Chapter 1
Statistical Mechanics: An overview
1.1
Introduction
Thermodynamics deals with the thermal properties of macroscopic system by determining the relationship between different thermodynamic parameters of the system.
In order to determine different relationships between the system parameters, the internal structure of matter is completely ignored in thermodynamics. The motion of
atoms, molecules or ions of a system, their interactions with each other or with the
external field are not considered in this subject. Thermodynamics is purely based
on principles formulated by generalizing experimental observations. On the other
hand, statistical mechanics describes the thermodynamic behaviour of macroscopic
systems from the laws which govern the behaviour of the constituent elements at the
microscopic level. In the formalism of statistical mechanics, a macroscopic property
of a system is obtained by taking a “statistical average” (or “ensemble average”) of
the property over all possible “microstates” of the system at thermodynamic equilibrium. A microstate of a system is defined by specifying the states of all of its
constituent elements. Below, a brief statistical mechanical description of fluid and
magnetic systems will be given.
A macroscopic therodynamic system, generally, is composed of a large number (of
the order of Avogadro number NA ≈ 6.022 × 1023 per mole) of microscopic elements
such as atoms, molecules, dipole moments or magnetic moments, etc. Each element
may have a large number of internal degrees of freedom associated with different
types of motion such as translation, rotation, vibration etc. The constituent elements
usually interact among each other via certain complex interactions. They may as
well interact with the external fields applied to the system. In the thermodynamic
limit, the macroscopic properties of a thermodynamic system is thus determined
by the properties of the constituent molecules, their internal interactions as well as
interaction with external fields.
The thermodynamic limit of a macroscopic system of density ρ with volume V and
1
Chapter 1. Statistical Mechanics: An overview
number of elements N is defined as
lim N → ∞
lim V → ∞ but ρ =
N
= finite.
V
(1.1)
In this limit, the extensive parameters (such as volume, entropy, etc.) of the system
become directly proportional to the size of the system (N or V ), while the intensive
parameters (such as pressure, teperature, external field, etc.) become independent
of the size of the system.
1.2
Specification of microstates
The macroscopic state of a thermodynamic system at equilibrium is specified by
the values of a set of measurable thermodynamic parameters. For example, the
macrostate of a fluid system can be specified by pressure P , temperature T and
volume V . For a magnetic system it can be described by external magnetic field
B, magnetization M and temperature T . A microstate of a system, on the other
hand, is obtained by specifying the states of all of its constituent elements. However,
it depends on the nature of the constituent elements (or particles) of the system.
Specification of microstates are made differently for classical and quantum particles.
We will describe both the situations below.
Microstates of classical particles: In order to specify the microstates of a system
of classical particles, one needs to specify the position q and the conjugate momentum p of each and every constituent particle of the system. In a classical system,
the time evolution of q and p is governed by the classical Hamiltonian H(p, q) and
Hamilton’s equation of motion
q̇i =
∂H (p, q)
∂pi
and
ṗi = −
∂H (p, q)
,
∂qi
i = 1, 2, 3, · · · , 3N
(1.2)
for a system of N particles in 3-dimensions. The state of a single particle at any time
is then given by the pair of conjugate variables (qi , pi ), a point in the phase space.
Each single particle then constitutes a 6-dimensional phase space (3-coordinate and
3-momentum). For N particles, the state of the system is then completely and
uniquely defined by 3N canonical coordinates q1 , q2 , · · · , q3N and 3N canonical moment p1 , p, · · · , p3N . These 6N variables constitute a 6N-dimensional Γ-space or
phase space of the system and each point of the phase space represents a microstate of the system. The locus of all the points in Γ-space satisfying the condition
H(p, q) = E, total energy of the system, defines the energy surface.
Example: Consider a free particle of mass m inside a one dimensional box of length
L, such that 0 ≤ q ≤ L, with energy between E and E + δE. The macroscopic state
of the system is defined by (E, N, L) with N = 1. The microstates are specified in
certain region of phase space.
Since the energy of the particle is E = p2 /2m, the
√
momentum will be p = ± 2mE and the position q is within 0 and L. However, there
2
1.2 Specification of microstates
(a)
(b)
Figure 1.1: (a) Accessible region of phase space for a free particle of mass m and energy
E in a one dimensional box of length L. (b) Region of phase space for a one dimensional
harmonic oscillator with energy E, mass m and spring constant k.
is a small
p width in energy δE, so the particles are confined in small strips of width
δp = m/2EδE as shown in Fig.1.1(a). Note that if δE = 0, the accessible region of
phase space representing the system would be one dimensional in a two dimensional
phase space. In order to avoid this artifact a small width in E is considered which
does not affect the final results in the thermodynamic limit. In Fig.1.1(b), the phase
space region of a one dimensional harmonic oscillator with mass m, spring constant
k and energy between E and E + δE is shown. The Hamiltonian of the particle is:
H = p2 /2m + kq 2 /2 and for a given energy E, the accessible region is an ellipse:
p2 /(2mE) + q 2/(2E/k) = 1. With thepenergy between E and E + δE, the accessible
region is an elliptical shell of area 2π m/kδE.
Microstates of quantum particles: For a quantum particle, the state is characterized by the wave function ψ (q1 , q2 , q3 , · · · ). Generally, the wave function is
written in terms of a complete orthonormal basis of eigenfunctions of the Hamiltonian operator of the system. Thus, the wave function may be written as
X
Ψ=
cn φn , Ĥφn = En φn
(1.3)
n
where En is the eigenvalue corresponding to the state φn . The eigenstates φn ,
characterized by a set of quantum numbers n provides a way to count the microscopic
states of the system.
Example: Consider a localized magnetic ion of spin 1/2 and magnetic moment µ
~.
The particle has two eigenstates, (1, 0) and (0, 1) associated with spin up (↑) and
~ the
down spin (↓) respectively. In the presence of an external magnetic field H,
energy is given by
~ = +µH for spin ↓
E = −~µ.H
−µH for spin ↑
Thus, the system with macrostate (N, H, T ) with N = 1 has two microstates with
3
Chapter 1. Statistical Mechanics: An overview
~ and down spin
energy −µH and +µH corresponding to up spin (parallel to H)
~ If there are two such magnetic ions in the system, it will have
(antiparallel to H).
four microstates: ↑↑ with energy −2µH, ↑↓ & ↓↑ with zero energy and ↓↓ with
energy +2µH. For a system of N spins of spin-1/2, there are total 2N microstates
and specification of the spin-states of all the N spins will give one possible microstate
of the system.
1.3
Statistical ensembles
An ensemble is a collection of a large number of replicas (or mental copies) of
the microstates of the system under the same macroscopic condition or having the
same macrostate. However, the microstates of the members of an ensemble can be
arbitrarily different. Thus, for a given macroscopic condition, a (classical) system
of an ensemble is represented by a point in the phase space. The ensemble of
a macroscopic system of given macrostate then corresponds to a large number of
points in the phase space. During time evolution of a macroscopic system in a fixed
macrostate, the microstate is supposed to pass through all these phase points.
Depending on the interaction of a system with the surroundings (or universe), a
thermodynamic system is classified as isolated, closed or open system. Similarly,
statistical ensembles are also classified into three different types. The classification
of ensembles again depends on the type of interaction of the system with the surroundings which can either be by exchange of energy only or exchange of both energy
and matter (particles or mass). In an isolated system, neither energy nor matter
is exchanged and the corresponding ensemble is known as microcanonical ensemble.
A closed system exchanging only energy (not matter) with its surroundings is described by canonical ensemble. Both energy and matter are exchanged between the
system and the surroundings in an open system and the corresponding ensemble is
called a grand canonical ensemble.
1.4
Statistical equilibrium
Consider an isolated system with the macrostate (E, N, V ). A point in the phase
space corresponds to a microstate of such a system and its internal dynamics is
described by the corresponding phase trajectory. The density of phase points ρ(p, q)
is the number of microstates per unit volume of the phase space and it is the probability to find a state around a phase point (p, q). At any time t, the number of
representative points in the volume element d3N qd3N p around the point (p, q) of the
phase space is then given by
ρ(p, q)d3N qd3N p.
(1.4)
By Liouville’s theorem, in the absence of any source and sink in the phase space,
the total time derivative in the time evolution of the phase point density ρ(p, q) is
4
1.5 Postulates of statistical mechanics
given by
where
dρ
∂ρ
=
+ {ρ, H} = 0
dt
∂t
(1.5)
3N X
∂ρ ∂H
∂ρ ∂H
−
{ρ, H} =
∂q
∂p
∂pi ∂qi
i
i
i=1
(1.6)
is the Poisson bracket of the density function ρ and the Hamiltonian H of the system.
Thus, the cloud of phase points moves in the phase space like an incompressible fluid.
The ensemble is considered to be in statistical equilibrium if ρ(p, q) has no explicit
dependence on time at all points in the phase space, i.e.; ∂ρ
= 0.
∂t
Under the condition of equilibrium, therefore,
3N X
∂ρ ∂H
∂ρ ∂H
−
=0
{ρ, H} =
∂qi ∂pi
∂pi ∂qi
i=1
(1.7)
ρ(p, q) = constant.
(1.8)
and it will be satisfied if ρ is an explicit function of the Hamiltonian H(q, p) or ρ is
a constant independent of p and q. That is
The condition of statistical equilibrium then requires no explicit time dependence
of the phase point density ρ(p, q) as well as uniform distribution of ρ(p, q) over the
relevant region of phase space. The value of ρ(p, q) will, of course, be zero outside
the relevant region of phase space. Physically the choice corresponds to an ensemble
of systems which at all times are uniformly distributed over all possible microstates
and the resulting ensemble is referred to as the microcanonical ensemble. However,
in canonical ensemble it can be shown that ρ(q, p) ∝ exp[−H(q, p)/kB T ].
1.5
Postulates of statistical mechanics
The principles of statistical mechanics and their applications are based on the following two postulates.
Equal a priori probability: For a given macrostate (E, N, V ), specified by the
number of particles N in the system of volume V and at energy E, there is usually
a large number of possible microstates of the system. In case of classical noninteracting system, the total energy E can be distributed among the N particles in
a large number of different ways and each of these different ways corresponds to a
microstate. In the fixed energy ensemble, the density ρ(q, p) of the representative
points in the phase space corresponding to these microstates is constant or the phase
points are uniformly distributed. Thus, any member of the ensemble is equally
likely to be in any of the various possible microstates. In case of a quantum system,
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Chapter 1. Statistical Mechanics: An overview
the various different microstates are identified as the independent solutions of the
Schrödinger equation of the system, corresponding to an eigenvalue E. At any time
t, the system is equally likely to be in any one of these microstates. This is
generally referred as the postulate of equal a priori probability for all microstates of
a given macrostate of the system.
Principle of ergodicity: The microstates of a macroscopic system are specified
by a set of points in the 6N-dimensional phase space. At any time t, the system is
equally likely to be in any one of the large number of microstates corresponding to
a given macrostate, say (E, N, V ) as for an isolated system. With time, the system
passes from one microstate to another. After a sufficiently long time, the system
passes through all its possible microstates. In the language of statistical mechanics,
the system is considered to be in equilibrium if it samples all the microstates with
equal a priori probability. The equilibrium value of the observable X can be obtained
by the statistical or ensemble average
RR
X(p, q)ρ(p, q)d3N qd3N p
R
hXi =
.
(1.9)
ρ(p, q)d3N qd3N p
On the other hand, the mean value of an observable (or a property) is given by its
time-averaged value:
Z
1 T
X̄ = lim
X(t)dt.
(1.10)
T →∞ T 0
The ergodicity principle suggests that statistical average hXi and the mean value
X̄ are equivalent: X̄ ≡ hXi.
1.6
1.6.1
Thermodynamics in different ensembles
Microcanonical ensemble (E,N,V)
In this ensemble, the macrostate is defined by the total energy E, the number of
particles N and the volume V . However, for calculation purpose, a small range of
energy E to E + δE (with δE → 0) is considered instead of a sharply defined energy
value E. The systems of the ensemble may be in any one of a large number of
microstates between E and E + δE. In the phase space, the representative points
will lie within a hypershell defined by the condition E ≤ H(p, q) ≤ E + δE.
At statistical equilibrium, all representative points are uniformly distributed and
the phase point density ρ(q, p) is constant between E and E + δE otherwise zero.
As per equal a priori probability, any accessible state is equally probable. Therefore,
the probability Px to find a system in a state x corresponding to energy Ex between
6
1.6 Thermodynamics in different ensembles
E and E + δE is given by
Px = R E+δE
E
ρ(q, p)
(1.11)
ρ(q, p)d3N qd3N p
if E = Ex = E + δE, otherwise zero. One may note that
R E+δE
E
Px d3N qd3N p = 1.
The number of accessible microstates Ω is proportional to the phase space volume
enclosed within the hypershell and it is given by
1
Ω(E, N, V ) = 3N
h
Z
E+δE
d3N qd3N p
(1.12)
E
for a system of N particles and of total energy E. If the particles are indistinguishable, the number of microstates Ω should be divided by N! as the Gibb’s correction.
The factor of 1/h3N suggests that a tiny volume element of h3N in the phase space
represents the microstates of the system. This means that a small displacement
within the volume element h3N around the phase point does not correspond to any
measurable change in the macrostate of the system. By Heisenberg’s uncertainty
principle (∆q∆p∼h) for quantum particles, it can be shown that h is the Planck’s
constant. However, if the energy states are discrete, the particles are distributed
among the different energy levels as, ni particles in the energy level εi and satisfies
the following conditions
X
X
N=
ni and E =
ni εi ,
(1.13)
i
i
The total number of possible distributions or microstates of N such particles is then
given by
N!
Ω=
.
(1.14)
n1 !n2 ! · · ·
Hence, the micorcanonical phase-space density is given by

1

forE ≤ H(p, q) ≤ E + δE
ρ(q, p) =
Ω(q, p)
 0
otherwise
(1.15)
The expectation value (ensemble average) of a thermodynamic quantity X(q, p)
would be given by
Z Z
1
hX(q, p)i = 3N
X(q, p)ρ(q, p)d3N qd3N p.
(1.16)
h
The thermodynamic properties can be obtained by associating entropy S of the
system to the number of accessible microstates Ω. The statistical definition of
7
Chapter 1. Statistical Mechanics: An overview
entropy by Boltzmann is given by
S(E, N, V ) = kB ln Ω
(1.17)
where kB is the Boltzmann constant, 1.38 × 10−23 JK−1 . In a natural process the
equilibrium corresponds to maximum Ω or equivalently maximum entropy S as is
stated in the second law of thermodynamics. It is to be noted that, as T → 0,
the system is going to be in its ground state and the value of Ω is going to be 1.
Consequently, the entropy S → 0 which is the third law of thermodynamics.
It is important to note that the entropy defined in Eq. 2.17, can be obtained as
ensemble avergae of −kB ln ρ(q, p). Following the definition of ensemble average in
microcanonical ensemble one has
Z Z
1
S(E, N, V ) = 3N
ρ(q, p) {−kB ln ρ(q, p)} d3N qd3N p
h
Z E+δE
1
1
1
3N
3N
−kB ln
= 3N
d qd p
h
Ω
Ω
(1.18)
E
Ω
1
=
−kB ln
Ω
Ω
= kB ln Ω
Therefore, the definition of entropy in microcanonical ensemble can also be taken
as S = h−kB ln ρi.
Since the thermodynamic potential such as entropy S is known in terms of the
number of the microstates, the thermodynamic properties of the system can be
obtained by taking suitable derivative of S with respect to the relevant parameters.
Let us start from the differential form of the first law of thermodynamics given by
dE = T dS − P dV + µdN
or dS =
1
P
µ
dE + dV − dN
T
T
T
(1.19)
where µ is the chemical potential. The thermodynamic parameters are then obtained
as
kB ∂Ω
∂ ln Ω
∂S
1
= kB
=
(1.20)
=
T
∂E V,N
∂E V,N
Ω ∂E V,N
P
∂ ln Ω
kB ∂Ω
∂S
= kB
=
(1.21)
=
T
∂V E,N
∂V
Ω ∂V E,N
E,N
kB ∂Ω
∂ ln Ω
∂S
µ
= kB
=
(1.22)
− =
T
∂N E,V
∂N E,V
Ω ∂N E,V
8
1.6 Thermodynamics in different ensembles
1.6.2
Canonical ensemble for fluid system
In the micro-canonical ensemble, a microstate was defined by a fixed number of
particles N, a fixed volume V and a fixed energy E. However, the total energy E
of a system is generally not measured. Furthermore, it is difficult to keep the total
energy fixed. Instead of energy E, temperature T is a better alternate parameter
of the system which is directly measurable and controllable. the corresponding
ensemble is known as canonical ensemble. In the canonical ensemble, the energy
E can vary from zero to infinity. In addition to fixed temperature T , the volume
V and the number of particles N can also be kept fixed and such an ensemble is
called constant volume canonical ensemble. Alternately, one may keep the pressure
P and the number of particles N be fixed along with the fixed temperature T . Such
an ensemble is called constant pressure canonical ensemble. Below we will describe
statistical thermodynamics for both the situations.
1.6.2.1
Constant volume canonical ensemble (N,V,T):
Let us consider an ensemble whose microstate is defined by N, V and T . The set
of microstates can be continuous as in most classical systems or it can be discrete
like the eigenstates of a quantum mechanical Hamiltonian. Each microstate s is
characterised by the energy Es of that state. If the system is in thermal equilibrium
with a heat-bath at temperature T , then the probability ps that the system to be in
the microstate s is ∝ e−Es /kB T , the Boltzmann factor. Since
the system has to be
P
in a certain state, the sum of all ps has to be unity, i.e.; s ps = 1. The normalized
probability
exp(−Es /kB T )
1
ps = P
(1.23)
= e−Es /kB T
Z
s exp(−Es /kB T )
is the Gibbs probability and the normalization factor
Z (N, V, T ) =
X
s
− kEsT
e
B
=
X
e−βEs , where β =
s
1
kB T
(1.24)
is the constant volume canonical partition function or simply canonical partition function.
The expectation (or average) value of a macroscopic quantity X is given by
P
Xs exp(−βEs )
1X
Xs e−Es /kB T
(1.25)
hXi = P s
=
Z
exp(−E
/k
T
)
s
B
s
s
where Xs is the property X measured in the microstate s.
Since the energy of the system fluctuates from zero to infinity, the average energy
9
Chapter 1. Statistical Mechanics: An overview
is then given by
1X
Es e−βEs
Z s
(
!)
X
∂
ln
e−βEs
=−
∂β
s
hEi =
=−
(1.26)
∂
ln Z(N, V, T ).
∂β
Immediately, one could calculate a thermal response function, the specific heat at
constant volume CV as
∂E
CV =
(1.27)
∂T V
where E ≡ hEi.
If the consecutive energy levels are very close and can be considered continuous
as in a classical system, the summation in Eqs.2.24, 2.25 should be replaced by
integration. The Hamiltonian that describe the system of N number of particles of
fixed volume V is given by
N
X
H=
Hi (qi , pi )
(1.28)
i
where Hi is the Hamiltonian for the ith particle.
In this limit, the canonical partition function can be written as
Z Z
1
Z(N, V, T ) = 3N
exp {−βH(p, q)} d3N qd3N p
h N!
where the volume element d
3N
qd
3N
p=
N
Y
(1.29)
d3 qi d3 pi and N! is for indistinguishable
i
particles only. Note that for non-interacting system of N particles, the partition
function Z can be written as Z = N1 ! Z1N where Z1 is the partition function for a
single particle. Consequently one obtains ideal gas behaviour for N → ∞.
The expectation value of X would be given by
Z Z
1
X(q, p) exp {−βH(p, q)} d3N qd3N p.
hXi =
Z(N, V, T )
(1.30)
The Helmholtz free energy F (N, V, T ) = E − T S, where E is the internal energy
and S is the entropy, is the appropriate potential or free energy to describe a thermodynamic system of fixed volume V and fixed number of particles N is in thermal
equilibrium with a heat-bath at temperature T . The statistical definition of entropy
in the canonical ensemble can be obtained from the ensemble average of −kB ln ps ,
ps is the probability to find the system in the state s. The entropy S in terms of ps
10
1.6 Thermodynamics in different ensembles
then can be obtained as
S = h−kB ln ps i = −kB
= −kB
X
ps ln ps
X e−βEs
1X
ln ps e−βEs = −kB
ln ps
Z s
Z
s
(1.31)
s
Following the definition of ps given in Eq. 2.23, one has
S = −kB
X e−βEs
s
Z
1
(−βEs − ln Z) =
T
E
= + kB ln Z
T
1X
Es e−βEs
Z s
!
+ kB ln Z
(1.32)
where E ≡ hEi. Therefore, the Helmholtz free energy F (N, V, T ) of the system is
given by
F (N, V, T ) = E − T S = −kB T ln Z(N, V, T ).
(1.33)
The thermal equilibrium of the system corresponds to the minimum free energy or
maximum entropy at finite temperature. All equilibrium thermodynamic properties
can be calculated by taking appropriate derivatives of the free energy F (N, V, T )
with respect to an apropriate parameter. Since the differential form of the first law
of thermodynamics given by
dE = T dS − P dV + µdN
(1.34)
where µ is the chemical potential, the differential form of the Helmholtz free energy
F (N, V, T ) is given by
dF = dE − SdT − T dS
(1.35)
= −SdT − P dV + µdN
Then, the thermodynamic parameters of the system can be obtained as
∂F
∂F
∂F
S=−
, P =−
, µ=
∂T V,N
∂V T,N
∂N T,V
(1.36)
Since we will be using these relations frequently, we present here different derivatives
of the Helmholtz free energy F (N, V, T ) with respect to its parameters for a fluid
system at constant volume in the following flowchart.
The thermodynamic response functions now can be obtained by taking second
derivatives of the Helmholtz free energy F (N, V, T ). The specific heat at constant
volume can be obtained as
2 ∂ F
∂S
= −T
(1.37)
CV = T
∂T V,N
∂T 2 V,N
11
Chapter 1. Statistical Mechanics: An overview
and the isothermal compressibility κT is given by
2 1
∂P
∂ F
= −V
=V
.
κT
∂V T,N
∂V 2 T,N
Z(N, V, T ) =
1
h3N N !
R
(1.38)
e−βH d3N qd3N p
F (N, V, T ) = −kB T ln Z(N, V, T )
F = E − TS
dF = −SdT − P dV + µdN
Pressure
∂F
P = − ∂V
T,N
Entropy
S = − ∂F
∂T V,N
Chemical Potential
∂F
µ = ∂N
T,V
Specific heat as fluctuation in energy: The fluctuation in energy is defined a
h(∆E)2 i = h(E − hEi)2i = hE 2 i − hEi2 .
(1.39)
By calculating hE 2 i, it can be shown that
h(∆E)2 i = −
1.6.2.2
∂hEi
= kB T 2 CV
∂β
or
CV =
1
(hE 2 i − hEi2 ).
kB T 2
(1.40)
Constant pressure canonical ensemble (N,P,T):
The NPT ensemble is also called the isothermal-isobaric ensemble. It describes
systems in contact with a thermostat at temperature T and a bariostat at pressure
P . The system not only exchanges heat with the thermostat, it also exchanges
volume (and work) with the bariostat. The total number of particles N remains
fixed. But the total energy E and volume V fluctuate at thermal equilibrium.
In the NPT canonical ensemble, the energy E as well as the volume V can vary
from zero to infinity. Each microstate s is now characterised by the energy Es of
that state and the volume of the system V . The probability ps that the system to
be in the microstate s is propotional to e−(Es +P V )/kB T . P
Since the system has to be
in a certain state, the sum of all ps has to be unity, i.e.; s ps = 1. The normalized
12
1.6 Thermodynamics in different ensembles
probability
exp{−(Es + P V )/kB T }
P
dV s exp{−(Es + P V )/kB T }
0
1
e−(Es +P V )/kB T
=
Z(N, P, T )
ps = R ∞
is the Gibbs probability and the normalization factor
Z ∞
X − (Es +P V )
Z (N, P, T ) =
dV
e kB T
0
=
Z
s
∞
dV
0
X
(1.41)
(1.42)
−β(Es +P V )
e
s
where β = 1/(kB T ) and Z (N, P, T ) is the constant pressure canonical partition
function.
It can be noted here that the canonical partition function Z(N, V, T ) under constant
volume is related to the canonical partition function Z(N, P, T ) under constant
pressure by the foillowing Laplace transform.
Z ∞
Z ∞
X
X
−β(Es +P V )
Z (N, P, T ) =
e
=
dV
e−βEs
e−βP V dV
0
0
s
s
(1.43)
Z ∞
−βP V
Z (N, V, T ) e
dV
=
0
The expectation (or average) value of a macroscopic quantity X is given by
R∞
P
dV s Xs exp{−β(Es + P V )}
0
P
hXi = R ∞
dV s exp{−(Es + P V )/kB T }
0
Z ∞
(1.44)
X
1
−(Es +P V )/kB T
dV
Xs e
=
Z(N, P, T ) 0
s
where Xs is the property X measured in the microstate s when system volume is V .
Under this macroscopic condition, both the enthalpy H = E + P V and the volume
V of the system fluctuate. The average enthalpy hHi of tye system is given by
hHi = hEi + P hV i
Z ∞
X
1
=
dV
(Es + P V ) e−(Es +P V )/kB T
Z(N, P, T ) 0
s
(
!)
Z ∞
X
∂
ln
dV
e−β(Es +P V )
=−
∂β
0
s
=−
(1.45)
∂
ln Z(N, P, T )
∂β
Immediately, one could calculate a thermal response function, the specific heat at
13
Chapter 1. Statistical Mechanics: An overview
constant pressure CP as
CP =
∂H
∂T
(1.46)
P
where H ≡ hHi.
In the continuum limit of the energy levels, the summations in the above expressions should be replaced by integrals. In this limit, the constant pressure canonical
partition function can be written as
Z ∞
Z Z
1
Z(N, P, T ) = 3N
dV
exp {−β [H(p, q) + P V ]} d3N qd3N p
(1.47)
h N! 0
where H =
N
X
i
Hi (qi , pi ) and the volume element d
3N
qd
3N
p=
N
Y
d3 qi d3 pi . N! is for
i
indistinguishable particles only.
The expectation value of X is given by
Z ∞
Z Z
1
dV
X(q, p) exp {−β [H(p, q) + P V ]} d3N qd3N p (1.48)
hXi =
Z(N, P, T ) 0
The Gibbs free energy G(N, P, T ) = E − T S + P V is the appropriate potential
or free energy to describe a thermodynamic system of fixed number of particles in
thermal equilibrium with a heat-bath at temperature T as well as in mechanical
equilibrium at a constant pressure P . The statistical definition of entropy S under
this macroscopic condition The statistical definition of entropy under this conditions
can be obtained from the ensemble average of −kB ln ps , ps is the probability to find
the system in the state s. The entropy S in terms of ps then can be obtained as
S = h−kB ln ps i
Z
X
1
= −kB
dV
e−β(Es +P V ) ln ps
Z
s
Z
X 1
−β(Es +P V )
= −kB dV
ln ps
e
Z
s
Z
X
= −kB dV
ps ln ps
(1.49)
s
Following the definition of ps given in Eq. 2.41, one has
Z
X e−β(Es +P V )
S = −kB dV
{−β(Es + P V ) − ln Z}
Z
s
)
( Z
X
1 1
dV
(Es + P V )e−β(Es +P V ) + kB ln Z
=
T Z
s
=
1
(E + P V ) + kB ln Z
T
14
(1.50)
1.6 Thermodynamics in different ensembles
where E ≡ hEi and V ≡ hV i. Therefore, the Gibbs free energy G(N, P, T ) of the
system is given by
G(N, P, T ) = E − T S + P V = −kB T ln Z(N, P, T ).
(1.51)
The thermodynamic equilibrium here corresponds to the minimum of Gibbs free energy. All equilibrium, thermodynamic properties can be calculated by taking appropriate derivatives of the Gibbs free energy G(N, P, T ) with respect to an appropriate
parameter of it. Since the differential form of the first law of thermodynamics given
by
dE = T dS − P dV + µdN
(1.52)
where µ is the chemical potential, the differential form of the Gibbs free energy
G(N, P, T ) = E − T S + P V is given by
dG = dE − SdT − T dS + P dV + V dP
= −SdT + V dP + µdN
Then, the thermodynamic parameters of the system can be obtained as
∂G
∂G
∂G
S=−
, V =−
, µ=
∂T P,N
∂P T,N
∂N T,P
(1.53)
(1.54)
Since we will be using these relations frequently, we present here different derivatives
of the Gibbs free energy G(N, P, T ) with respect to its parameters for a fluid system
at constant pressure in the following flowchart.
Z(N, P, T ) =
1
h3N N !
R
dV
R
e−β(H+P V ) d3N qd3N p
G(N, P, T ) = −kB T ln Z(N, P, T )
G = E − TS + PV
dG = −SdT + V dP + µdN
Volume
∂G
V = ∂P
T,N
Entropy
S = − ∂G
∂T P,N
Chemical Potential
∂G
µ = ∂N
T,P
The thermodynamic response functions now can be calculated by taking second
derivatives of the Gibbs free energy G(N, P, T ). The specific heat at constant pres15
Chapter 1. Statistical Mechanics: An overview
sure can be obtained as
CP = T
∂S
∂T
P,N
= −T
∂2G
∂T 2
,
the isothermal compressibility κT can be obtained as
1 ∂2G
1 ∂V
=−
,
κT = −
V ∂P T,N
V ∂P 2 T,N
and the volume expansion coefficient αP is given by
2 1 ∂V
1
∂ G
αP =
=
.
V ∂T P,N
V ∂T ∂P N
1.6.3
(1.55)
P,N
(1.56)
(1.57)
Canonical ensemble for magnetic system
We will be using the second form of the first law dE = T dS + BdM and define
other state functions and thermodynamic potentials such as enthalpy H(N, S, B),
the Helmholtz free energy F (N, M, T ) and the Gibbs free energy G(N, B, T ). The
definitions of these thermodynamics state functions and differential change in a
reversible change of state are given by
H(N, S, B) = E − MB
F (N, M, T ) = E − T S
G(N, B, T ) = E − T S − MB
and
and
and
dH = T dS − MdB
dF = −SdT + BdM
dG = −SdT − MdB
(1.58)
where explicit N dependence is also avoided. If one wants to take into account of
number of particles ther must me another term µdN in all differential forms of the
state functions. It can be noticed that the thermodynamic relations of a magnetic
system can be obtained from those in fluid system if V is replaced by −M and P is
replaced by B.
Note that if the other form of the first law dE = T dS −MdB is used, the differential
change in Helmholtz free energy would be given by dF = −SdT − MdB same
as the differential change in Gibbs free energy given in Eq.2.58 considering dE =
−T dS + BdM. The Helmholtz free energy becomes F (B, T ) function of B and T
instead of F (M, T ) function of M and T . In many text books, F (B, T ) is used
as free energy and the reader must take a note that in this situation a different
definition of magnetic work and energy is used.
Now one can obtain all the thermodynamic parameters from the state functions by
16
1.6 Thermodynamics in different ensembles
taking appropriate derivatives as given
∂E
T =
∂S M
∂F
S=−
∂T M
∂E
B=
∂M S
∂H
M =−
∂B S
below
∂H
T =
,
∂S B
∂G
S=−
,
∂T B
∂F
B=
,
∂M
T
∂G
M =−
.
∂B T
or
or
or
or
(1.59)
Now, one needs to fix the parameters in order to define the canonical ensemble for
a magnetic system. Let us first take (N, M, T ), the number of magnetic moments
N, the total magnetization M and the temperature T as our system parameters.
Under such macroscopic condition, Helmholtz free energy F (N, M, T ) describes the
equilibrium condition of the system. It is similar to the constant volume canonical
ensemble for a fluid system. If a microstate s of the (N, M, T ) canonical ensemble
is characterised by an energy Es and the system is in thermal equilibrium with
a heat-bath at temperature T , then the probability ps that the system to be in
the microstate s must be ∝ e−Es /kB T , the Boltzmann factor. The constant field
canonical partition function Z(N, M, T ) would be given by
Z (N, M, T ) =
X
− kEsT
e
B
=
s
X
e−βEs
(1.60)
s
where β = 1/(kB T ) and the Gibbs free energy is then given by
F (N, B, T ) = −kB T ln Z (N, M, T ) .
(1.61)
Different derivatives of the Helmholtz free energy F (N, M, T ) with respect to its
parameters for a magnetic system at constant external field are given in the following
flowchart.
However, the natural parameters for a magnetic system are the number of magnetic
moments N, the external magnetic induction B and the temperature T . Under
this macroscopic condition, the Gibbs free energy G(N, B, T ) of the system defibnes
the thermodynamics equilibrium. It is similar to the constant pressure canonical
ensemble for a fluid system. One needs to calculate the constant field canonical
partition function Z(N, B, T ) and which would be given by
Z (N, B, T ) =
X
− kEsT
e
s
B
=
X
e−βEs
(1.62)
s
where β = 1/(kB T ) and the Gibbs free energy is then given by
G(N, B, T ) = −kB T ln Z (N, B, T ) .
17
(1.63)
Chapter 1. Statistical Mechanics: An overview
Different derivatives of the Gibbs free energy G(N, B, T ) with respect to its parameters for a magnetic system at constant external field are given in the following
flowchart.
Z(N, M, T ) =
P
s
e−βEs
F (N, M, T ) = −kB T ln Z(N, M, T )
F
= E − TS
dF = −SdT + BdM
Entropy
S = − ∂F
∂T M
Magnetic field
∂F
B = ∂M
T
Z(N, B, T ) =
P
s
e−βEs
G(N, B, T ) = −kB T ln Z(N, B, T )
G = E − T S + BM
dG = −SdT − MdB
Entropy
S = − ∂G
∂T B
Magnetization
∂G
M = − ∂B
T
The magnetic response functions now can be calculated by taking second derivatives
of the Gibbs free energy G(N, B, T ). The specific heat at constant magnetization
18
1.6 Thermodynamics in different ensembles
and constant external field B can be obtained as
2 ∂S
∂ F
CM = T
= −T
,
∂T M,N
∂T 2 M,N
and
CB = T
∂S
∂T
B,N
= −T
∂2G
∂T 2
(1.64)
(1.65)
B,N
respectively. The isothermal susceptibility χT can be obtained as
2 ∂M
∂ G
χT =
=−
,
∂B T,N
∂B 2 T,N
and the coefficient αB is given by
2 ∂ G
∂M
=
.
αB =
∂T B,N
∂T ∂B N
(1.66)
(1.67)
Susceptibility as fluctuation in magnetization: It can be shown that the
isothermal susceptibility is proportional to the fluctuation in magnetization
χT =
1.6.4
kB T
(hM 2 i − hMi2 ).
N
(1.68)
Grand canonical ensemble (µ,V,T)
Consider a system in contact with an energy reservoir as well as a particle reservoir and the system could exchange energy as well as particles (mass) with the
reservoirs. Canonical ensemble theory has limitations in dealing these systems and
needs generalization. It comes from the realization that not only the energy E but
also the number of particles N of a physical system is difficult to measure directly.
However, their average values hEi and hNi are measurable quantities. The system
interacting with both energy and particle reservoirs comes to an equilibrium when
a common temperature T and a common chemical potential µ with the reservoir is
established. In this ensemble, each microstate (r, s) corresponds to energy Es and
number of particles Nr in that state. If the system is in thermodynamic equilibrium
at temperature T and chemical potential µ, the probability pr,s is given by
pr,s = C exp(−αNr − βEs ),
(1.69)
where α = −µ/kB T and β = 1/kB T . After normalizing,
exp(−αNr − βEs )
pr,s = P
,
r,s exp(−αNr − βEs )
19
since
X
r,s
pr,s = 1
(1.70)
Chapter 1. Statistical Mechanics: An overview
where the sum is over all possible states of the system. The numerator exp(−αNr −
βEs ) is the Boltzmann factor here and the denominator
X
Q=
exp(−αNr − βEs )
(1.71)
rs
is called the grand canonical partition function. The grand canonical partition
function then can be written as
X
X
µNr
Es
Q=
exp
=
zNr e−Es /kB T
(1.72)
−
kB T
kB T
r,s
r,s
where z = eµ/kB T is the fugacity of the system. In case of a system of continuous
energy levels, the grand partition function can be written as
∞
X
1
Q=
3N
h N!
N =1
Z Z
µN
exp −βH(p, q) +
kB T
d3N qd3N p.
(1.73)
Note that division by N! is only for indistinguishable particles.
The mean energy hEi and the mean number of particle hNi of the system are then
given by
P
X
∂
r,s Es exp(−αNr − βEs )
ln{
e−αNr −βEs }
=−
hEi = P
∂β
exp(−αN
−
βE
)
r
s
r,s
r,s
(1.74)
∂ ln Q
1 ∂Q
=−
=−
∂β
Q ∂β
and
P
X
∂
r,s Nr exp(−αNr − βEs )
ln{
e−αNr −βEs }
=−
hNi = P
∂α
exp(−αN
−
βE
)
r
s
r,s
r,s
∂ ln Q
1 ∂Q
=−
=−
∂α
Q ∂α
(1.75)
where hEi ≡ E and hNi ≡ N are the internal energy and number of particles of the
system respectively.
The grand potential Φ(T, V, µ) = E − T S − µN is the appropriate potential or
free energy to describe the thermodynamic system of fixed volume in equilibrium
at constant temperature T and chemical potential µ. The statistical definition of
entropy under this conditions is given by
X
S = −kB
pr,s ln pr,s
(1.76)
r,s
where pr,s is the probability to find the system in the state r, s. Following the
20
1.6 Thermodynamics in different ensembles
definition of pr,s given in Eq. 2.70, one has
S = −kB
1
=
T
=
X e−αN −βEs
r,s
Z
{(−αN − βEs ) − ln Q}
1 X
(Es − µNr ) e(αNr −βEs )
Q r,s
!
+ kB ln Q
(1.77)
1
(E − µN) + kB ln Q
T
where E ≡ hEi and N ≡ hNi. Therefore, the grand potential Φ(T, V, µ) of the
system is given by
Φ(µ, V, T ) = E − µN − T S = −kB T ln Q.
(1.78)
The thermodynamic equilibrium corresponds to minimum of the grand potential
Φ(µ, V, T ) under these macroscopic conditions. All equilibrium thermodynamic
properties can now be calculated by taking appropriate derivatives of the grand
potential Φ(T, V, µ) with respect to its parameters. Since dE = T dS − P dV + µdN,
the differential change of the grand potential can be expressed as
dΦ = dE − T dS − SdT − µdN − Ndµ = −SdT − P dV − Ndµ
(1.79)
and the thermodynamic variables in terms of Φ(V, T, µ) are then obtained as
∂Φ
∂Φ
∂Φ
,
−P =
,
−N =
.
(1.80)
−S =
∂T V,µ
∂V T,µ
∂µ T,V
Response functions can be obtained by taking higher oder derivatives of the parameters calculated.
Compressibility as fluctuation in number density: The fluctuation in number
of particles N is defined as
∂hNi
hNi2 kB T
h(∆N) i = h(N − hNi) i = hN i − hNi = kB T
=
κT
∂µ
V
2
2
2
2
(1.81)
where κT is the isothermal compressibility. The isothermal compressibility is then
proportional to density fluctuation. If κ0T = V /(hNi KB T ), one has
(N − hNi)2
κT
.
(1.82)
=
κ0T
hNi
21
Chapter 1. Statistical Mechanics: An overview
1.7
Quantum statistics
The ensemble theory developed so far is applicable to classical system or to quantum
mechanical systems composed of distinguishable entities. The complete description
of the state of a system in quantum mechanics is given by the wave function |ψ(r, t)i,
the solution of the Schrödinger equation
i~
d|ψ(r, t)i
= Ĥ|ψ(r, t)i
dt
(1.83)
where Ĥ is the Hamilton operator of the system in a suitable Hilbert space. The
wave function |ψ(r, t)i is thus the quantum-mechanical analogue of the point in phase
space of classical statistical mechanics. Any dynamical state or the wave function
|ψ(r, t)i of the system can be expressed as a linear combination of the stationary
eigenstates |ni as
X
|ψ(r, t)i =
cn |nie−iEn t/~
(1.84)
n
where the coefficients cn (t) define a point in the (Hilbert) space of stationary eigenstates |ni. The stationary eigenstates |ni are the solutions of the time independent
Schrödinger equation and is given by
Ĥ|ni = En |ni
(1.85)
where En is the energy of the system corresponding to the eigenstate state |ni.
The expectation value of a physical observable  in quantum mechanics is given by
X
hÂi = hψ(r, t)|Â|ψ(r, t)i =
c∗n cm hn|Â|mi
(1.86)
m,n
Note that averaging here is performed over a very large number of measurements,
each performed on a system whose state is precisely specified by |ψ(r, t)i. The
collection of such states in quantum mechanics is called a pure ensemble.
However, we are interested in statistical ensemble average of this quantum expectation value. Quantum ensemble average is performed by constructing density matrix.
1.7.1
The density matrix
As the number of particles in the system increases, the separation between the energy
levels decreases rapidly. In the thermodynamic limit, they become extremely close.
On the other hand, the energy levels are never completely sharp. They are broadened
for many reasons, including the uncertainty principle. In a macroscopic system (with
extremely dense energy levels), the levels will thus always overlap. Consequently, a
macroscopic system will never be in a strictly stationary quantum state (pure state),
it will always be in a dynamical state, in a time-dependent mixture of stationary
states (mixed state). Let us assume that the system is ergodic and the time average
22
1.7 Quantum statistics
can be replaced by an ensemble average over many quantum systems at the same
instance. The dynamical state is no longer a unique quantum state but is a statistical
mixture of quantum states.
Say, the system is in a compound state given by a wave function |ψs i corresponding
to an energy Es . The compound state |ψs i is of course a linear combination of many
eigenstates |ni of the Hamiltonian (say) and can be written as
X
|ψs i =
csn |ni.
(1.87)
n
Then the pure ensemble average or the quantum expectation values of an observable
 is given by
X
hÂi = hψs |Â|ψs i =
csn ∗ csm hn|Â|mi
(1.88)
m,n
Note that the last expression holds for any complete set of states |ni even if they
are not the eigenstates of the Hamiltonian.
Now, the system could have many such possible microstates |ψs i corresponding to
the given macroscopic conditions. If the probability of finding the system in one of
the microstate is ps , the quantum ensemble average of an observable  is given by
X
hhÂii =
ps hψs |Â|ψs i
s
=
X X
m,n
ps csn ∗ csm
s
!
(1.89)
hn|Â|mi
Let us define the density matrix ρ̂, a statistical operator, as
X
ρ̂ =
|miρmn hn|
(1.90)
m,n
where ρmn = hm|ρ̂|ni =
observable  is given by
P
hhÂii =
=
s
ps csn ∗ csm . Then, the quantum ensemble average of an
X
m,n
X
m
hm|ρ̂|nihn|Â|mi
hm|ρ̂Â|mi = Tr ρ̂Â = Tr Âρ̂ .
(1.91)
The density matrix, a statistical operator, has the following important properties:
1. The density matrix completely defines an ensemble of quantum systems and it
carries all the information that is available for a quantum statistical ensemble.
2. Its diagonal elements ρnn tell the probability that the system is in the state
|ni whereas its off-diagonal elements ρmn tell the probability of a transition
23
Chapter 1. Statistical Mechanics: An overview
from the state |ni to the state |mi. For stationary states, thus, ρ̂ has to be
diagonal, it commutes with the Hamiltonian.
3. The density matrix is normalized,
Tr(ρ̂) = 1.
(1.92)
The trace runs over any complete set of states |ni and is independent of the
choice of the basis set.
4. The dynamics of the system is completely described by
i
∂ ρ̂ h
= Ĥ, ρ̂ .
i~
∂t
(1.93)
This is the Schrödinger equation of the density matrix.
1.7.2
Quantum mechanical ensembles
Now, we shall specify the density matrix and its relation to thermodynamic quantities.
1.7.2.1
Quantum microcanonical ensemble
The energy of systems belonging to a microcanonical ensemble is fixed. Therefore,
it is convenient to use the energy eigenstates as a basis and in this basis the density
matrix is diagonal. According to the postulate of equal a priori probability, all the
states |ni with the energy En between E and E + δE are equally probable and let
there be Ω such states. The probability for the system being in one of these states
is ρ = 1/Ω and the entropy is equal to:
S = h−kB ln ρi.
(1.94)
However, one must use the density matrix ρ̂ instead of simple probabilities ρ and
the entropy of a quantum system is:
S = h−kB ln ρ̂i = −kB Tr (ρ̂ ln ρ̂) .
(1.95)
Again, the average means ensemble and quantum averages, both averages cannot
be separated.
1.7.2.2
Quantum canonical ensemble
In the energy representation ρ̂ is diagonal and one can use the same arguments as
in the case of classical canonical distributions. The matrix elements are equal to:
ρmm =
1 −βEm
e
Z
with Z =
X
m
24
e−βEm
(1.96)
1.7 Quantum statistics
However, in an arbitrary basis, the density matrix can be written as
1
ρ̂ = e−β Ĥ with Z = Tr e−β Ĥ
Z
(1.97)
where ρ̂ and Ĥ are operators and represented by matrices. Thus, for a canonical
ensemble, the average (thermal and quantum mechanical) of an operator  is
1 A = hhÂii = Tr ρ = Tr e−β Ĥ Â
Z
(1.98)
In particular, the internal energy E is
E = hhĤii =
∂
1 −β Ĥ Tr Ĥe
=−
ln Tr e−β Ĥ
Z
∂β
∂
ln Z.
=−
∂β
(1.99)
The free energy is
F = −kB T ln Z.
(1.100)
All the thermodynamic relations are the same as before,
theonly difference is that
the partition function has to be calculated as Z = Tr e−β Ĥ .
1.7.2.3
Quantum grand-canonical ensemble
In the grand canonical ensemble the density operator ρ̂ operates on a generalized
Hilbert space which is the direct sum of all Hilbert spaces with fixed number of
particles. The density matrix is given by
ρ̂ =
1 −β(Ĥ−µN̂ )
e
Q
(1.101)
where the grand partition function is
Q = Tr e−β(Ĥ−µN̂ ) .
(1.102)
Notice that now N̂ is an operator in the generalized Hilbert space.
The ensemble average of an operator  in the grand canonical ensemble is given by
1 (1.103)
A = hhÂii = Tr Âe−β(Ĥ−µN̂ ) .
Q
1.7.3
Wave function and statistics
The ensemble theory developed in the language of operators and wave functions
reveals new physical concepts. The behaviour of even a non-interacting ideal gas
25
Chapter 1. Statistical Mechanics: An overview
departs considerably from the results obtained in the classical ensemble theory. In
the presence of interactions, the behaviour becomes even more complicated. However, in the limit of high temperatures and low densities, the behaviour of all physical
systems tends asymptotically to what is expected in classical treatments.
Consider a gas of N non-interacting particles described by the Hamiltonian
Ĥ(q, p) =
N
X
i=1
Ĥi (qi , pi )
(1.104)
where (qi , pi ) are the coordinate and momentum of the ith particle, Ĥi is the Hamiltonian operator. A stationary system of N particles in a volume V then can be in
any one of the quantum states determined by the solutions of the time independent
Schödinger equation
ĤψE (q) = EψE (q)
(1.105)
where E is the eigenvalue of the Hamiltonian and ψE is the corresponding eigenfunction.
If there are ni particles inX
an eigenstate εi , then
X the distribution should satisfy
ni = N and
ni εi = E
(1.106)
i
i
and the wave functions for N particles {a1 , a2 , · · · aN } with ai particle in the ni th
state can be written as
(i)
ψ (a1 , a2 , · · · aN ) =
√1
N!
(ii)
ψ (a1 , a2 , · · · aN ) =
√1
N!
(iii) ψ (a1 , a2 , · · · aN ) =
√1
N!
QN
i=1
φanii ,
P QN
i=1





φan11
φan12
..
.
Product
φan11 ,
φan21
φan22
..
.
φan1N φan2N
Symmetric
· · · φanN1
· · · φanN2
..
..
.
.
aN
· · · φ n1



,

Anti − symmetric
where φi is the eigenfunction of the single particle Hamiltonian Ĥi with eigenvalue
εi . Each single particle wave function φiPis always a linear combination of a set of
orthonormal basis functions {ϕj }, φi = j cij ϕj . The particles described by these
three wave functions obey different statistics. (i) The particles described by the
product function correspond to different microstate by interchanging particles between states. These are then distinguishable particles and obey Maxwell-Boltzmann
statistics. (ii) In the case of symmetric wave functions, interchanging of particles
does not generate a new microstate. Thus, the particles are indistinguishable. Also,
all the particles in a single state corresponds to a non-vanishing wave function. That
means accumulation of all the particles in a single state is possible. These particles
obey Bose-Einstein statistics and are called bosons. (iii) For the anti-symmetric
wave function, if the two particles are exchanged, the two columns of the deter26
1.7 Quantum statistics
minant are exchanged and leads to the same wave function with a different sign.
Thus, the particles are again indistinguishable. However, if any two particles are
in one state then the corresponding rows of the determinant are the same and the
wave function vanishes. This means that a state cannot be occupied by more than
one particle. This is known as Pauli principle. These particles obey Fermi-Dirac
statistics and they are called fermions.
1.7.4
Distribution functions
Consider an ideal gas of N identical particles. Let s represents the single particle
state and S denotes the state of the whole system. At the state S, the total energy
ES and the number of particlesX
N are given by
X
ns εs and N =
ns .
ES =
s
s
The distribution functions can be calculated by obtaining the appropriate partition
function.
MB Statistics: In this case the particles are distinguishable. The logarithm of
canonical partition function is given by
!
X
−βεs
ln Z = N ln
e
.
(1.107)
s
The mean number of particles in state s is then given by
hns iM B = −
1 ∂ ln Z
Ne−βεs
= P −βεs .
β ∂εs
se
(1.108)
This is the Maxwell-Boltzmann (MB) distribution as already obtained in classical
statistical mechanics.
BE Statistics: The grand canonical partition function Q of N indistinguishable
bosons is given by
X
ln Q = −
ln 1 − e−β(εs −µ)
(1.109)
s
The number of particles in an grand canonical ensemble is given by
(
)
X
1 ∂ X
1 ∂ ln Q
=−
ln 1 − e−β(εs −µ)
=
hns i
N=
β ∂µ
β ∂µ
s
s
(1.110)
Thus the average number of molecules in the s level is
hns iBE = −
1 ∂ e−β(εs −µ)
1
ln 1 − e−β(εs −µ) =
= β(εs−µ)
.
−β(ε
−µ)
s
β ∂µ
1−e
e
−1
27
(1.111)
Chapter 1. Statistical Mechanics: An overview
Figure 1.2: Plot of mean occupation number hns i of a single-particle energy state εs in
a system of non-interacting particles: curve 1 is for fermions, curve 2 for bosons and curve
3 for the particles obeying Maxwell-Boltzmann statistics.
This is the Bose-Einstein (BE) distribution where always µ < εs , otherwise hns i
could be negative.
FD Statistics: The fermions have only two states, ns = 0 or 1. Thus, the grand
canonical partition function for N indistinguishable fermions is given by
X
ln 1 + e−β(εs −µ)
(1.112)
ln Q =
s
The number of particles is given by
X
1 ∂ ln Q X e−β(εs −µ)
N=
=
=
hns i
β ∂µ
1 + e−β(εs −µ)
s
s
(1.113)
Thus the average number of molecules in the s level is
hns iF D =
1
eβ(εs −µ) + 1
.
(1.114)
This is the Fermi-Dirac (FD) Distribution. However, in the classical limit T → ∞
and ρ → 0,BE and FD statistics lead to the classical MB statistics.
28