Statistical Mechanics
April 1, 2026
Contents
1
The Motivation for Statistical Mechanics
3
2
Microscopic Description of a thermodynamic system
2.1 Why Not Use Classical or Quantum Mechanics Directly? . . . . . . . . . . . . .
4
5
3
Ensemble
5
4
Microstates of a Gas of Particles: Phase Space
7
5
Postulates of Statistical Mechanics
5.1 Postulate of Equal A Priori Probabilities . . . . . . . . . . . . . . . . . . . . . .
5.2 Ergodic Hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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6
Liouville’s Theorem
6.1 Principle of Conservation of Extension . . . . . . . . . . . . . . . . . . . . . . .
6.2 Statistical Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Statistical ensembles
7.1 Microcanonical Ensemble . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.1.1 Example: Ideal Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2 Thermodynamic Quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2.1 What are physical meaning of S and T ? . . . . . . . . . . . . . . . . . .
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Derivation of Thermodynamics
8.1 First Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.2 Second Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.3 Statistical Interpretation of Entropy and the Second Law . . . . . . . . . . . . .
8.3.1 Reversible and Irreversible Processes . . . . . . . . . . . . . . . . . . .
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Classical Ideal Gas Again
9.1 Entropy and more . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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10 Canonical Ensemble
10.1 Equivalence of Canonical and Microcanonical Ensembles . . . . . . . . . . . . .
10.2 Free Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.3 Classical ideal gas in the canonical ensemble . . . . . . . . . . . . . . . . . . .
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1
11 Grand Canonical Ensemble
11.1 Equivalence Between Canonical and Grand Canonical Ensembles . . . . . . . .
11.2 Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.3 Ideal Gas in the Grand Canonical Ensemble . . . . . . . . . . . . . . . . . . . .
11.3.1 Average Number of Particles . . . . . . . . . . . . . . . . . . . . . . . .
11.3.2 Pressure and Equation of State . . . . . . . . . . . . . . . . . . . . . . .
11.3.3 Average Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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12 Equipartition theorem
12.1 Statement of the Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.2 Quadratic Degrees of Freedom . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.3 Derivation in the Canonical Ensemble . . . . . . . . . . . . . . . . . . . . . . .
12.4 Generalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.5.1 Ideal Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.5.2 One-Dimensional Harmonic Oscillator . . . . . . . . . . . . . . . . . .
12.5.3 Diatomic Molecule (High-Temperature Limit) . . . . . . . . . . . . . .
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13 Quantum Statistical Mechanics
13.1 Pure and Mixed Ensembles . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.2 Density Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.2.1 Properties of the Density Operator . . . . . . . . . . . . . . . . . . . . .
13.2.2 Time Evolution of the Density Operator . . . . . . . . . . . . . . . . . .
13.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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14 Ensembles in Quantum Statistical Mechanics
14.1 Microcanonical Ensemble . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14.2 Canonical Ensemble . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14.2.1 3D harmonic oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . .
14.2.2 Maxwell-Boltzmann gas . . . . . . . . . . . . . . . . . . . . . . . . . .
14.3 Grand Canonical Ensemble . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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15 Ideal Bose gas
15.1 Grand partition function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15.2 Thermodynamic limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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16 Ideal Fermi gas
16.1 High temperature limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16.2 Low temperature limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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17 Magnetism
17.1 Paramagnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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18 Blackbody radiation
18.1 Partition function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18.2 Average occupancy and equation of state . . . . . . . . . . . . . . . . . . . . . .
18.3 Planck’s distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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2
Statistical mechanics is the discipline that links microscopic physics to macroscopic phenomena. In this course, we learn how the collective behaviour of an enormous number of particles
gives rise to familiar thermodynamic concepts such as temperature, pressure, and entropy. By
replacing detailed microscopic information with statistical reasoning, we uncover how universal
laws emerge from underlying dynamics and gain a deeper understanding of matter in equilibrium.
1
The Motivation for Statistical Mechanics
Let us begin by asking a simple but fundamental question:
Why do we need a separate subject called Statistical Mechanics?
To answer this, we must first recall what thermodynamics provides. Thermodynamics offers a
phenomenological description of macroscopic systems. It summarizes and organizes experimental
facts into mathematical relations without addressing the microscopic mechanisms responsible for
them.
In thermodynamics, we characterize a system using macroscopic variables such as pressure
P , volume V , temperature T , and external fields (electric E, magnetic B, etc.). These parameters
are experimentally measurable, and a thermodynamic state is specified by giving their values. A
system is said to be in thermodynamic equilibrium when these parameters no longer change with
time.
The equilibrium properties are encoded in relations of the form
f (T, V, P ) = 0,
(1.1)
which are determined empirically. For example, an ideal gas satisfies
P V = nRT.
(1.2)
This equation of state is based entirely on observation—we do not ask why it holds, nor do we
examine the microscopic behavior of molecules that leads to this relation. The same is true for the
laws of thermodynamics: they are given to us without providing their microscopic origin.
Goal of Statistical Mechanics: Statistical mechanics is the framework that allows us to understand macroscopic thermodynamic behaviour from the underlying microscopic physics. It answers
questions such as:
• What is the microscopic origin of pressure and temperature?
• Why does the ideal gas law hold?
• What is entropy in microscopic terms?
Thus, the central motivation for statistical mechanics is our desire to go beyond phenomenology and understand the microscopic origin of macroscopic relations.
In this course we will focus on equilibrium statistical mechanics. This subject does not address
how systems relax to equilibrium or whether they equilibrate at all—that is part of non-equilibrium
statistical mechanics. Here we determine the properties of systems once they are already in equilibrium.
3
2
Microscopic Description of a thermodynamic system
In statistical mechanics, the distinction between microscopic and macroscopic (thermodynamic)
variables is fundamental. They describe the same physical system but at different levels of description.
Microscopic variables: A microscopic description of a system specifies the exact state of all its
constituent degrees of freedom. The number of such variables is enormous—typically of the order
of Avogadro’s number for macroscopic systems. The precise choice of microscopic variables
depends on both the nature of the system and the theoretical framework used to describe it.
For example, in a classical gas consisting of N particles, the microscopic variables are the
positions and momenta of all particles: {⃗r1 , · · · , ⃗rN } and {⃗p1 , · · · , p⃗N }. Together, these variables
specify a point in the 6N -dimensional phase space and uniquely determine the microstate of the
system.
In contrast, for a quantum many-body system, the microscopic description is given by the
many-body quantum state, typically represented by a wavefunction |Ψ⟩ or, more generally, a density operator. Equivalently, one may describe the microstate in terms of occupation numbers of
energy eigenstates (or other suitable single-particle bases), depending on the problem at hand.
Macroscopic variables: Macroscopic (thermodynamic) variables describe a system in terms of
bulk, averaged properties rather than individual microscopic details. Examples include energy,
volume, temperature, pressure, and entropy.
The number of macroscopic variables is very small, and they are directly measurable in experiments. A thermodynamic system in equilibrium is completely specified by an appropriate set of
such macroscopic variables.
Main comparison between microscopic and macroscopic variables:
• In equilibrium, macroscopic variables are time-independent, apart from small thermodynamic fluctuations. In contrast, the microscopic variables—such as the positions and momenta of individual particles—change rapidly with time. For example, while the positions
and momenta of molecules in a gas evolve continuously and chaotically, macroscopic quantities such as the pressure and total energy remain essentially constant.
• Macroscopic variables of a system can be directly controlled and adjusted through external
constraints or experimental conditions. In contrast, microscopic variables—such as the positions and momenta of individual particles—cannot be controlled or specified in practice,
as they depend on uncontrollable and rapidly fluctuating details of the system.
Microscopic description of a system: In statistical mechanics, a microscopic description of a
system means a description that specifies the complete physical state of the system at the most
fundamental level allowed by the theory (classical or quantum), without any coarse-graining or
averaging.
A microscopic description answers the question: What exactly is the state of every constituent
of the system right now? It uniquely determines the system’s future evolution given the underlying
dynamical laws.
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2.1
Why Not Use Classical or Quantum Mechanics Directly?
A natural question arises: Why do we need a new theoretical framework when classical and quantum mechanics already provide the fundamental laws of physics? The answer is not a limitation of
these laws themselves, but rather the enormous number of degrees of freedom involved in macroscopic systems.
Consider a gas confined in a box containing N ∼ 1023 molecules. Even if a purely classical
description is sufficient, the complete microscopic description requires solving
• 3N ∼ 3 × 1023 coupled second-order differential equations,
• together with 6N initial or boundary conditions specifying all particle positions and momenta.
Such a task is not merely difficult—it is fundamentally impractical. One cannot even specify, let
alone solve, equations of motion for so many degrees of freedom.
Quantum mechanics faces an analogous problem. In principle, the system is governed by a
many-body Schrödinger equation, but solving it for 1023 interacting particles is completely infeasible.
The difficulty, therefore, is not conceptual but practical: macroscopic systems are far too complex to be treated by tracking their microscopic dynamics directly. This overwhelming complexity necessitates a new framework— statistical mechanics—which replaces exact microscopic
descriptions with a probabilistic treatment and allows macroscopic behavior to be derived from
microscopic laws.
Turning a Difficulty into an Advantage
The central idea of statistical mechanics is that although we cannot track each particle individually,
the enormous number of particles becomes an advantage.
In everyday life we recognize this principle:
• When performing an experiment, we take many readings and compute an average. A few
readings are unreliable, but with many data points the average becomes stable.
• To estimate the performance of a class, we need marks from many students.
Likewise, in a gas with 1023 molecules, fluctuations around average values are negligible.
Average properties become extremely well-defined.
This is the essence of statistical mechanics:
Instead of tracking every microscopic detail, we compute statistical averages, which accurately
describe macroscopic behaviour.
Statistical mechanics is therefore not merely a convenience—it is a necessity for understanding
systems with an enormous number of degrees of freedom, and it provides the bridge between
microscopic physics and macroscopic thermodynamics.
3
Ensemble
One of the central questions of statistical mechanics is: How do we extract the average value of a
macroscopic parameter for a system with an enormous number of degrees of freedom? Statistical
mechanics provides the answer by introducing the idea of a statistical ensemble. To motivate this,
let us begin with a simple example.
5
Definition 3.1. In statistical mechanics, an ensemble is a large collection of virtual copies of a
system, all prepared under the same macroscopic conditions, but possibly in different microscopic
states. The ensemble does not represent many systems existing physically at the same time; rather,
it is a conceptual device that allows us to assign probabilities to different microstates and compute
macroscopic averages.
Coin Toss: A Motivating Example
Imagine an experimentalist repeatedly tossing a coin and recording each outcome as Head (H) or
Tail (T). To determine the average occurrence of H, the experiment should ideally be repeated a
very large number of times—formally, infinitely many times—so that a well-defined average can
be obtained.
Rather than imagining infinitely many tosses performed sequentially by a single person, consider an equivalent picture: an infinite number of experimentalists, each performing one coin toss
under identical macroscopic conditions. All coins are identical, and the experiments are carried
out in the same room, on the same floor, under identical lighting and ambient conditions. The
outcomes from all experimentalists are then collected to compute the average occurrence of H.
At first sight, one might object that since the coin toss is governed by classical mechanics,
identical initial conditions should lead to identical outcomes. This reasoning, however, is misleading. While the macroscopic conditions are the same, the microscopic initial conditions (that can
not be controlled from outside) are inevitably different in each trial. Tiny variations in the release
angle, initial velocity, finger motion, air currents, or surface interactions cannot be controlled or
even measured. These microscopic differences, though negligible at the macroscopic level, lead
to different outcomes—sometimes H and sometimes T.
The collection of all such experimental realizations forms an ensemble. Each experimentalist
contributes one member of the ensemble. The defining property of an ensemble is:
All ensemble members are identical at the macroscopic level, but differ in their microscopic
states.
Once the ensemble is defined, the average value of any observable is obtained as an ensemble
average. Conceptually, we examine each member of the ensemble and evaluate the observable. In
the coin-toss example, we assign
(
1, if the outcome is Head,
H=
0, if the outcome is Tail.
Summing this quantity over all ensemble members and dividing by the total number of members
yields the average occurrence of H.
Experimentally, this average is known to be 1/2. Hence, in the ensemble description, half of
the members correspond to H and the other half to T.
This construction relies on a fundamental assumption of statistical mechanics, known as the
postulate of a priori probability:
All accessible microstates are equally probable.
In general, an ensemble consists of a very large collection of virtual copies of a system, each
constructed to satisfy the same macroscopic constraints (energy, volume, number of particles,
external fields, etc.) but differing in their microscopic details. Each member of the ensemble is a
complete replica of the actual system and is, for all practical purposes, an acceptable representation
of the real physical system. Constructing a suitable ensemble for a given system is the first step in
statistical mechanics.
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Another illustration- Counting Microstates: This is not a very good thermodynamic example.
But this example helps us to have some idea of microstates for a given macroscopic variable.
Consider another simple system: five indistinguishable particles and five energy levels labelled
0, 1, 2, 3, 4. Suppose the total energy of the system is 4. How many microscopic arrangements
(microstates) are possible? The allowed configurations are:
• One particle in level 4 and the rest in level 0.
• One particle in level 3, one in level 1, and the rest in level 0.
• Two particles in level 2 and the rest in level 0.
• One particle in level 2, two in level 1, and the rest in level 0.
• Four particles in level 1 and one in level 0.
There are five different microstates. Although each microstate differs microscopically, they all
correspond to the same macroscopic condition: the total energy is 4. These five microstates form
the ensemble corresponding to this macroscopic constraint.1
If the macroscopic condition changes, the ensemble changes as well. For example, if the total
energy were 20, then all particles must occupy level 4, and there is only one possible microstate.
Thus, statistical mechanics proceeds by identifying all possible microscopic configurations
consistent with the imposed macroscopic conditions and assigning appropriate probabilities to
them. Macroscopic quantities are then computed as statistical averages over the ensemble.
4
Microstates of a Gas of Particles: Phase Space
Let us now return to a concrete physical system—a gas confined in a container—and address the
question: How do we construct a statistical ensemble for a gas of particles?
We begin with the simplest case: a single free particle moving in one dimension. Treating
the system classically, the state of the particle is completely specified by its position q and momentum p. Geometrically, the state of the system can therefore be represented as a point in a
two-dimensional space with Cartesian axes labeled q and p. This space is called the phase space,
denoted by Γ.
At any given time t, specifying the values of q(t) and p(t) is equivalent to specifying a point
in phase space. As the particle evolves in time, its representative point traces out a curve in phase
space.
This construction generalizes naturally to a system of N particles moving in three dimensions.
In this case, the phase space has dimension 6N , with coordinates
{q1 , q2 , . . . , q3N ; p1 , p2 , . . . , p3N }.
These 6N variables are collectively denoted by (q, p).
The dynamics of the system is governed by the Hamiltonian H, from which the equations of
motion follow as
∂H
,
∂pi
∂H
ṗi = −
.
∂qi
q̇i =
1
(4.1)
(4.2)
Do we need infinitely many copies of a system to study it statistically? Not necessarily. For many problems, it is
sufficient to know the number of distinct microstates and their probabilities.
7
Thus, each point in phase space corresponds to a unique microscopic state of the system, and vice
versa.
If the total energy of the system is fixed at H = E, then the set of all points in phase space
satisfying this condition defines a constant-energy surface (or energy hypersurface). Since the
total energy is conserved, the phase-space trajectory of the system always remains confined to this
hypersurface.
In principle, each point (q, p) in phase space represents a distinct microstate. For macroscopic
systems, however, it is neither possible to determine nor to control this point with arbitrary precision. Moreover, states that differ only within an infinitesimally small region of phase space are
physically indistinguishable. From a quantum-mechanical perspective, this limitation arises from
the uncertainty principle, which prevents the simultaneous precise specification of position and
momentum.
Instead of attempting to follow the exact microscopic trajectory of the system, we adopt a
statistical description. We then introduce the phase space density ρ(q, p, t), defined such that
ρ(q, p, t) d3N p d3N q
gives the number of microstates contained in the phase-space volume element d3N p d3N q around
the point (q, p) at time t. The total number of microstates is therefore
Z
Ω = ρ(q, p, t) d3N p d3N q.
(4.3)
The phase-space density can also be interpreted probabilistically. A larger value of ρ(q, p, t) implies that the system spends more time in that region of phase space, and hence the probability
of finding the system there is higher. The probability of finding the system within the volume
element d3N p d3N q around (q, p) at time t is given by
ρ(q, p, t) d3N p d3N q
,
P (q, p, t) d3N p d3N q = R
ρ(q, p, t) d3N p d3N q
(4.4)
where P (q, p, t) is the normalized probability density in phase space.
Thus, specifying a statistical ensemble for a given system amounts to specifying the phasespace density function ρ(q, p, t). The central task of statistical mechanics is therefore to determine
the appropriate form of ρ(q, p, t) under given physical conditions.
Thus, the introduction of the phase space density replaces the notion of a single microscopic
trajectory by a statistical distribution over phase space. Specifying a statistical ensemble for a
given system is therefore equivalent to specifying the phase space density ρ(q, p, t).
5
Postulates of Statistical Mechanics
In order to determine the phase-space density function ρ(q, p, t), statistical mechanics relies on a
small number of fundamental postulates. These postulates are not derived from first principles but
are motivated by empirical success and physical reasoning.
5.1
Postulate of Equal A Priori Probabilities
The central postulate of equilibrium statistical mechanics states that:
All accessible microstates of an isolated system in equilibrium are equally probable.
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This postulate forms the foundation of the statistical description of systems in thermal equilibrium. It asserts that, in the absence of any additional information, there is no reason to favor
one microscopic state over another, provided all such states are consistent with the macroscopic
constraints (such as fixed energy, volume, and particle number).
For example, in the case of a gas confined to a container with fixed energy, all points on
the constant-energy surface in phase space are assigned equal probability. Thus, the system is
equally likely to be found in any accessible region of phase space consistent with the macroscopic
constraints. Therefore, phase space density ρ(q, p) is constant.
5.2
Ergodic Hypothesis
In statistical mechanics, we are primarily interested in the average or mean value of physical
observables. One natural way to define such an average is through a time average, obtained by
observing the system over a very long period of time.
Let F (t) be a physical quantity measured along the trajectory of the system. Its time average
is defined as
Z
1 T
F (t) dt,
(5.1)
⟨F ⟩t = lim
T →∞ T 0
where T is the total observation time. Since F depends on the microscopic variables q(t) and p(t),
this may be written as
Z
1 T
F q(t), p(t) dt.
(5.2)
⟨F ⟩t = lim
T →∞ T 0
Geometrically, this means that as the system evolves in time, its representative point moves
through phase space, and the time average is obtained by evaluating F along this trajectory and
averaging over time.
To make this more transparent, we discretize the time interval [0, T ]. Suppose the phase space
is divided into small cells labeled by j = 1, 2, . . . , M . During the interval T , the system visits the
Figure 1: Ergodic hypothesis
j th cell mj times and spends a total time
Tj = mj ∆tj
in that cell. Clearly,
M
X
Tj = T.
j=1
9
The time average can then be written as
M
1X
⟨F ⟩t = lim
F (qj , pj ) Tj
T →∞ T
j=1
=
M
X
F (qj , pj )
j=1
Tj
.
T
(5.3)
(5.4)
The quantity
Tj
T
may be interpreted as the probability of finding the system in the j th phase-space cell during a
long-time observation. Passing to the continuum limit, the time average takes the form
Z
⟨F ⟩t = d3N q d3N p P̃ (q, p) F (q, p).
(5.5)
P̃j =
Earlier, we introduced the probability density P (q, p) of finding the system at the phase-space
point (q, p) from the ensemble description. The crucial question is whether the probability P̃ (q, p),
defined from long-time dynamics, is the same as the ensemble probability P (q, p).
The ergodic hypothesis asserts that these two probabilities are identical:
P̃ (q, p) = P (q, p).
As a consequence, the time average of any observable is equal to its ensemble average:
R
F (q, p) ρ(q, p) d3N q d3N p
R
≡ ⟨F ⟩e .
⟨F ⟩t =
ρ(q, p) d3N q d3N p
(5.6)
Thus, the ergodic hypothesis states that time averages are equal to ensemble averages. This is
one of the fundamental assumptions of statistical mechanics. Since both averages are assumed to
be identical, we shall henceforth omit the subscripts t and e.
In statistical mechanics, the expectation value of any physical observable defined over an ensemble is meaningful only if its fluctuations are negligible compared to its mean value. Quantitatively, this requires
⟨F 2 ⟩ − ⟨F ⟩2
≪ 1.
(5.7)
⟨F ⟩2
If this condition is not satisfied, the ensemble average cannot be identified with the experimentally observed value, and the validity of the statistical description must be questioned.
For macroscopic physical systems, one finds that the mean square fluctuation of any extensive
observable typically scales as
⟨F 2 ⟩ − ⟨F ⟩2 ∼ O(N ),
(5.8)
while the square of its mean scales as
⟨F ⟩2 ∼ O(N 2 ),
(5.9)
where N is the number of particles (or degrees of freedom) in the system. Consequently,
⟨F 2 ⟩ − ⟨F ⟩2
1
∼ .
2
⟨F ⟩
N
(5.10)
Thus, in the thermodynamic limit N → ∞, relative fluctuations vanish, and ensemble averages
become sharply defined and physically reliable.
We therefore conclude that statistical mechanics provides a valid and self-consistent description of nature only when the number of degrees of freedom of the system is sufficiently large.
10
6
Liouville’s Theorem
In order to determine the phase-space density ρ(q, p, t), we must understand its fundamental properties. A key question is: How does the phase-space density evolve with time?
As discussed earlier, at any instant a classical system is represented by a single point in phase
space. As the system evolves in time, this point moves along a trajectory determined by the
equations of motion. Equivalently, we may think of phase space as being populated by an infinite collection of representative points, each corresponding to a possible realization of the same
macroscopic system. As time evolves, all these points flow through phase space according to
Hamiltonian dynamics.
Now consider a small volume element V in phase space. The representative points within this
volume are not fixed: there is a continuous inflow and outflow of points across the boundary of V .
Figure 2: Liouville theorem
The number of microstates (representative points) inside the volume V at time t is
Z
∆N =
ρ(q, p, t) dV.
(6.1)
V
The rate of change of ∆N is therefore
Z
∂ρ
dV.
V ∂t
(6.2)
Since representative points in phase space can neither be created nor destroyed, any change in
∆N must be due to the net flux of points across the boundary ∂V . Hence,
Z
Z
∂ρ
⃗
dV = −
ρ ⃗v · dS,
(6.3)
V ∂t
∂V
⃗ is the outward-directed surface element and ⃗v is the phase-space velocity field,
where dS
⃗v = (q̇i , ṗi ) .
Using the divergence theorem, this equation becomes
Z ∂ρ
+ ∇ · (ρ⃗v ) dV = 0.
∂t
V
11
(6.4)
Since the volume V is arbitrary, the integrand must vanish everywhere, yielding the continuity
equation in phase space:
∂ρ
+ ∇ · (ρ⃗v ) = 0.
(6.5)
∂t
Using Hamilton’s equations,
q̇i =
∂H
,
∂pi
ṗi = −
∂H
,
∂qi
we may write
∂
∂H
∂
dpi
∇ · (ρ⃗v ) =
ρ
+
ρ
∂qi
∂pi
∂pi
dt
∂ρ ∂H
∂ρ ∂H
=
−
∂qi ∂pi
∂pi ∂qi
= {ρ, H}PB .
(6.6)
Thus, the evolution equation for the phase-space density becomes
∂ρ
+ {ρ, H}PB = 0,
∂t
(6.7)
or equivalently,
dρ
= 0.
(6.8)
dt
This result is known as Liouville’s theorem: the phase-space density is conserved along the
trajectories of the system.
6.1
Principle of Conservation of Extension
Consider a fixed number of representative points δN occupying a small volume δV in phase space:
δN = ρ δV.
(6.9)
As time evolves, each point traces out a trajectory in phase space. At a later time t1 , the original
volume δV is mapped into a new region δV ′ . Although the shape of the region may be distorted,
the number of representative points it contains remains the same.
Taking the time derivative of δN ,
dρ
d
d
(δN ) =
δV + ρ (δV ) = 0.
dt
dt
dt
(6.10)
Since Liouville’s theorem implies dρ/dt = 0, we obtain
d
(δV ) = 0.
dt
(6.11)
Thus, phase-space volume is conserved under Hamiltonian evolution. A group of representative points occupying a volume δV at one time will occupy another region of the same volume at
any later time, although its shape may change. This result is known as the principle of conservation of extension.
12
6.2
Statistical Equilibrium
A statistical ensemble is said to be in equilibrium if the phase-space density has no explicit time
dependence:
∂ρ(q, p, t)
= 0.
(6.12)
∂t
From Liouville’s equation, this implies
{ρ, H}PB = 0.
(6.13)
Hence, in equilibrium, the phase-space density ρ(q, p) is a constant of motion and depends
only on conserved quantities such as the energy.
7
Statistical ensembles
In statistical mechanics, there are three standard ensembles that are commonly used to model real
physical systems under different experimental conditions. These are:
• Microcanonical ensemble,
• Canonical ensemble,
• Grand canonical ensemble.
The classification of ensembles is based on how a system interacts with its surroundings. In
particular, it depends on whether the system can exchange energy, particles (matter), or neither
with its environment. Each ensemble is characterized by a different form of the phase-space
density ρ.
A physical system may interact with its surroundings in the following ways:
• exchange energy,
• exchange particles,
• exchange both energy and particles,
• or exchange neither.
If the system can exchange energy with its surroundings but not particles, the appropriate
description is the canonical ensemble. If the system can exchange both energy and particles, it is
described by the grand canonical ensemble. Finally, if the system exchanges neither energy nor
particles with its surroundings, it is described by the microcanonical ensemble.
Statistical mechanics provides systematic methods to determine the appropriate distribution
function ρ for each ensemble, and hence to compute the equilibrium thermodynamic properties of
macroscopic systems. Although the microscopic distributions differ from one ensemble to another,
all three ensembles yield the same thermodynamic results in the thermodynamic limit.
The choice of ensemble is therefore guided by the physical situation being modeled and by
experimental constraints, rather than by any fundamental difference in the underlying physics.
13
7.1
Microcanonical Ensemble
The microcanonical ensemble describes an isolated system that does not exchange either energy
or particles with its surroundings. The macroscopic state of such a system is completely specified
by the fixed values of the total energy E, the volume V , and the number of particles N .
Since the system is isolated, all microstates that are consistent with these macroscopic constraints are accessible. According to the postulate of equal a priori probabilities, each such microstate is equally likely. Consequently, the phase-space density ρ(q, p) is uniform over the region
of phase space compatible with
H(q, p) = E,
and vanishes elsewhere.
More precisely, the microcanonical ensemble is defined by the distribution
(
constant, E ≤ H(q, p) ≤ E + ∆E,
ρ(q, p) =
0,
otherwise,
(7.1)
where ∆E is a small energy interval chosen to account for the finite resolution of macroscopic
1
measurements. Since ρ(q, p) has dimension (angular momentum)3N we write the constant as 3N .
h0
The normalization of ρ(q, p) determines the total number of accessible microstates,
Z
1
d3N q d3N p,
(7.2)
Ω(E, V, N ) = 3N
h0 E≤H(q,p)≤E+∆E
which plays a central role in equilibrium thermodynamics.
Since the phase-space density ρ(q, p) is defined such that ρ d3N q d3N p is dimensionless, and
since the phase-space volume element d3N q d3N p has dimensions (angular momentum)3N , the
density ρ(q, p) must have dimensions (angular momentum)−3N . We therefore write the constant
phase-space density as
1
ρ(q, p) = 3N ,
h0
where h0 is a constant with the dimensions of angular momentum.
The introduction of the constant h0 effectively partitions phase space into small elementary
cells of volume h3N
0 . All microstates lying within such a minimum phase-space volume are treated
as indistinguishable, since no physical measurement can resolve differences on scales smaller than
this cell.
Thus, each elementary cell represents a single microscopic state in the statistical description.
In classical statistical mechanics, the value of h0 is not fixed by the theory and may be chosen arbitrarily small. However, when quantum mechanics is taken into account, the uncertainty principle
imposes a natural lower bound on the phase-space volume associated with a single microstate. In
this case, the appropriate choice is
h0 ∼ ℏ,
reflecting the fact that phase space is effectively quantized into cells of volume ℏ3N . This identification ensures consistency between classical and quantum statistical mechanics.
The microcanonical ensemble provides the most fundamental statistical description of equilibrium, from which other ensembles can be obtained as appropriate limits.
Z
ΩMC (E) =
d3N q d3N p.
(7.3)
E≤H(q,p)≤E+∆
14
The integration over the configuration variables q depends on the size (volume) V of the system. Consequently, the phase–space volume ΩMC (E) depends on V , the energy width ∆, and the
number of particles N .
We define the integrated phase–space volume
Z
Σ(E) =
d3N q d3N p,
(7.4)
H(q,p)≤E
which represents the volume of the region in phase space bounded by the energy surface H(q, p) =
E.
Then,
ΩMC (E) = Σ(E + ∆) − Σ(E).
(7.5)
For sufficiently small ∆, we may write
ΩMC (E) = ω(E) ∆,
where
ω(E) =
∂Σ(E)
∂E
is called the density of states at energy E.
Using the identity
d
Θ(E − H) = δ(E − H),
dE
we obtain
Z
ω(E) = d3N q d3N p δ H(q, p) − E .
(7.6)
(7.7)
(7.8)
Therefore, the microcanonical phase–space volume can be written as
Z
ΩMC (E) = ∆
7.1.1
d3N q d3N p δ H(q, p) − E .
(7.9)
Example: Ideal Gas
Let us consider an ideal gas enclosed in a box of volume V :
• Collisions of gas molecules with the walls are elastic; hence the total energy is conserved.
• Interactions among the molecules are neglected.
The Hamiltonian of the system is given by
H(p) =
3N
X
p2
i
2m
i=1
= E.
The microcanonical phase–space volume is
Z E+∆ Z
ΩMC (E, V ) = C
de d3N q d3N p δ H(p) − e
E
!
Z
Z
3N
2
X
p
i
= C de d3N q d3N p δ
−e .
2m
i=1
15
(7.10)
(7.11)
Since the Hamiltonian is independent of the coordinates qi , the configuration integral gives
Z
d3N q = V N .
(7.12)
Introducing dimensionless variables
pi
ui = √
,
2me
we obtain
ΩMC (E, V ) = C V N
Z
de (2me)3N/2
!
Z
d3N u δ e
X
u2i − 1
i
=CVN
Z
3N/2 Z
(2me)
de
e
!
d3N u δ
X
u2i − 1 .
(7.13)
i
Defining
Z
f (N ) ≡
!
d3N u δ
X
u2i − 1 ,
i
and performing the e–integration over the narrow interval [E, E + ∆], we finally obtain
ΩMC (E, V ) = C V N
(2mE)3N/2
∆ f (N ).
E
(7.14)
Thus, for an ideal gas,
3N
Ω(E, V ) ∝ V N E 2 −1 .
(7.15)
Since N is very large, Ω(E, V ) is a rapidly increasing function of both E and V . Although this
result has been derived explicitly for an ideal gas, the qualitative behavior is generic for macroscopic statistical systems.
7.2
Thermodynamic Quantities
We define the following thermodynamic quantities in the microcanonical ensemble:
S(E, V ) = k ln ΩMC (E, V ),
∂S
−1
T =
,
∂E qi , other external parameters fixed
∂S
µi =
.
∂qi E, other external parameters fixed
7.2.1
(7.16)
(7.17)
(7.18)
What are physical meaning of S and T ?
Consider a macroscopic system with total volume V and particle number N , divided into two
subsystems:
• Subsystem 1: volume V1 , particle number N1
• Subsystem 2: volume V2 , particle number N2
16
Figure 3: Subsystems
with V1 + V2 = V and N1 + N2 = N .
Both N1 and N2 are assumed to be very large so that statistical mechanics applies. The two
subsystems are separated by a membrane that allows energy exchange but no particle exchange.
The interaction energy between the subsystems is negligible compared to their internal energies.
The total Hamiltonian of the combined system is
H(q, p) = H1 (q1 , p1 ) + H2 (q2 , p2 ),
(7.19)
where (q1 , p1 ) and (q2 , p2 ) denote the phase space variables of subsystems 1 and 2 respectively.
We treat the combined system in the microcanonical ensemble with total energy E. The phase
space volume is then
Z
ΩM C (E, V ) = ∆
dp dqδ (H(q, p) − E) .
Writing the phase space integrals explicitly,
Z
ΩM C (E, V ) = ∆ d3N1 q1 d3N1 p1 d3N2 q2 d3N2 p2 δ (H1 + H2 − E) .
(7.20)
(7.21)
Introduce two auxiliary integrations over energies E1 and E2 :
Z
Z
ΩM C (E, V ) = ∆ dE1 dE2 dΓ1 dΓ2 δ(H1 − E1 ) δ(H2 − E2 ) δ(E1 + E2 − E)
Z
= ∆ dE1 dE2 Ω1 (E1 ) Ω2 (E2 ) δ(E1 + E2 − E)
Z
= ∆ dE1 Ω1 (E1 ) Ω2 (E − E1 ).
(7.22)
Here Ωi (Ei ) denotes the microcanonical phase space volume of subsystem i.
Additivity and Extensivity of Entropy For macroscopic systems, Ω(E) is a rapidly increasing
function of E. Consequently:
• Ω1 (E1 ) increases rapidly with E1 ,
• Ω2 (E − E1 ) decreases rapidly with E1 .
Their product is therefore sharply peaked at some value E1 = Ē.
Thus the dominant contribution to ΩM C comes from a narrow region around Ē:
ΩM C (E, V ) ≃ ∆ Ω1 (Ē) Ω2 (E − Ē).
(7.23)
k ln ΩM C (E, V ) = k ln Ω1 (Ē, V1 ) + k ln Ω2 (E − Ē, V2 ) − (· · · ) ln ∆.
(7.24)
Taking the logarithm,
17
Figure 4: Sharpness
Defining entropy as
S(E, V ) = k ln Ω(E, V ),
(7.25)
and noting that ln Ωi ∼ Ni while ln ∆ is finite, we obtain in the thermodynamic limit:
S(E, V ) = S1 (Ē, V1 ) + S2 (E − Ē, V2 ).
(7.26)
This property is exactly the extensive property of the entropy of a system which says, if a
system is composed of two subsystems whose entropies are, respectively, S1 and S2 , the entropy
of the total system is S1 + S2 , when the subsystems are sufficiently large. We identify S to be the
entropy of the system. Still we need to check if S satisfy second law of thermodynamics.
Equilibrium Condition and Temperature The next question is, what is that maximum value
of ΩM C and what is the condition for that. The condition is given by,
∂
(Ω1 (E1 )Ω2 (E − E1 )) = 0,
∂E1
(7.27)
which implies,
1 ∂Ω1 (E1 )
1 ∂Ω2 (E2 )
=
Ω (E ) ∂E1
Ω (E ) ∂E2
1 1
2 2
∂S1 (E1 )
∂S2 (E2 )
=
∂E1
∂E2
E1 =Ē
E2 =E−Ē
=⇒
T1 (Ē) = T2 (Ē).
(7.28)
Therefore when T1 ̸= T2 two subsystems keep exchanging energy in the form of heat and they
come to equilibrium when T1 becomes equal to T2 i.e. exchange of heat stops. Therefore, we
identify T to be the temperature of the system.
In calculating ΩM C (E, V ) the dominant contribution comes from a region of phase space
where,
H1 (q 1 , p1 ) = Ē1
H2 (q 2 , p2 ) = Ē2
(7.29)
where, Ē1 and Ē2 are determined from,
T1 (Ē1 ) = T2 (Ē2 ).
18
(7.30)
Sharpness of the Energy Distribution We have shown the extensive properties of S and equilibrium condition for the system assuming that the function Ω1 (E1 )Ω2 (E2 ) is a very sharply
peaked function of energy. Now the question is how sharp the peak is. Because, whole statical description depends on that answer.
Define
1
S1 (E1 ) + S2 (E − E1 ) .
(7.31)
I(E1 ) = Ω1 (E1 )Ω2 (E − E1 ) = exp
k
Expanding the entropy around the equilibrium point Ē,
"
#
c
1
I(E1 ) = exp
S1 (Ē) + S2 (E − Ē) − (E1 − Ē)2 + · · · ,
k
2
where
1 ∂ 2 S1 ∂ 2 S2
+
.
c=
k ∂E12
∂E22 eq
(7.32)
(7.33)
Hence the energy distribution is Gaussian:
h c
i
I(E1 ) ∝ exp − (E1 − Ē)2 .
2
(7.34)
Thus we see that if (E1 − Ē)2 >> 1c then the distribution goes to zero. Therefore I is peaked
about E1 = Ē with a small spread. We want to compute the spread of this distribution.
The fluctuation about the point E1 = Ē is given by,
p
⟨(E1 − Ē)2 ⟩
.
(7.35)
Ē
Calculate
R
q
Ω1 (E1 )Ω2 (E − E1 )(E1 − Ē)2 dE1
R
⟨(E1 − Ē)2 ⟩ =
Ω1 (E1 )Ω2 (E − E1 )dE1
R
c
2
dE1 e− 2 (E1 −Ē) (E1 − Ē)2
=
R
c
2
dE1 e− 2 (E1 −Ē)
1
×C
=
c
(7.36)
Therefore,
p
⟨(E1 − Ē)2 ⟩
C
=v (
u
Ē
u
2
tĒ 2 ∂ 2 S2
+ ∂∂ES22
∂E
1
Ē1
2
1
)≡P
(7.37)
Ē2
We want to study the N dependence of fluctuation. For large N , S ∼ N and E ∼ N . Hence,
1
∂ 2S
=
2
∂E
N
2
∂
S
=⇒ E 2 2 = N
∂E
Hence
1
1
∼√
P
N
19
(7.38)
. Therefore for large N the fluctuation E1 − Ē is small compared to Ē.
The quantity P 2 goes N for large N . Other terms are suppressed with respect to the leading
term. But in P 2 , if the coefficient of order N term vanishes then we have to look at the sub leading
terms ∼< N . In that case fluctuations are not small.
2
1
2
2∂ S
2 ∂
P =E
= E
2
∂E
∂E T
1
2 ∂
where e = E/N, energy per dof
= Ne
∂e T
e2 ∂T
e2 −1
= −N 2
= −N 2 CV .
(7.39)
T ∂e
T
This quantity is expected to be negative for positive specific heat. Statistical mechanics works
when CV > 0. From this expression we see that the coefficient of N vanishes for CV → ∞. Then
the fluctuation becomes large: happens at phase transition.
Pressure Now consider two subsystems that can exchange both energy and volume, with fixed
total E and V :
Z
Ω(E, V ) = dE1 dV1 Ω1 (E1 , V1 ) Ω2 (E − E1 , V − V1 ).
(7.40)
Maximization yields the equilibrium conditions:
∂S2
∂S1
=
.
∂V1
∂V2
T1 = T2 ,
(7.41)
Defining
∂S
,
(7.42)
∂V
we identify p as the statistical pressure. Equilibrium requires equality of temperature and pressure across subsystems.
Later, we shall show that this statistical definition coincides with mechanical pressure.
p≡T
Exercise
• Show that volume fluctuations satisfy
p
1
⟨(V1 − V̄ )2 ⟩
∼√ .
V̄
N
• Show that thermodynamic stability requires
∂p
< 0.
∂V
8
Derivation of Thermodynamics
8.1
First Law
We now ask: what is the analog of a quasistatic thermodynamic transformation? A quasistatic
thermodynamic transformation corresponds to a slow variation of the macroscopic variables E
and V , induced by coupling the system to external agents.
20
In the language of statistical mechanics, during a quasistatic transformation of a system the
representative points in phase space are uniformly distributed over a slowly changing region of
phase space. The transformation is sufficiently slow that, at every instant of time, the system can
be described by a microcanonical ensemble.
As the accessible region of phase space changes, the entropy of the system also changes. The
differential change in entropy is given by
∂S
∂S
dE +
dV
dS(E, V ) =
∂E V
∂V E
1
p
=
dE + dV.
(8.1)
T
T
Multiplying both sides by T , we obtain
T dS = dE + p dV.
(8.2)
The dynamical significance of this relation becomes clear once we establish that p coincides
with the mechanical pressure. This expression is precisely the first law of thermodynamics. We
therefore conclude that p represents the mechanical pressure of the system, defined as force per
unit area.
8.2
Second Law
(0)
(0)
Consider a system with energy E1 and entropy S1 (E1 , . . .). Consider another system with
(0)
(0)
energy E2 and entropy S2 (E2 , . . .). Bring these two systems into thermal contact so that they
can exchange energy.
The total phase-space volume of the combined system is
Z
(0)
(0)
Ω0 =
dE1 Ω1 (E1 ) Ω2 (E (0) − E1 ),
E (0) = E1 + E2
Z
1
(0)
S1 (E1 ) + S2 (E − E1 )
=
dE1 exp
k
Z
h c
i
1
2
S1 (Ē1 ) + S2 (Ē2 )
dE1 exp − (E1 − Ē1 ) ,
= exp
k
k
where Ē1 is the energy of the first system at equilibrium.
(8.3)
Evaluating the Gaussian integral, we obtain
1
C
Ω0 = exp
S1 (Ē1 ) + S2 (Ē2 ) × √ .
(8.4)
k
c
Therefore, the entropy of the combined system is
1
Sf = k ln Ω0 = S1 (Ē1 ) + S2 (Ē2 ) − ln c + const.
(8.5)
2
Since S1 ∼ N1 , S2 ∼ N2 , and c ∼ N , the last two terms are negligible in the large-N (and
large N1 , N2 ) limit compared to the leading terms. Hence,
Sf = S1 (Ē1 ) + S2 (Ē2 ).
(8.6)
Because
the phase-space volume is maximized at E1 = Ē1 , the quantity ln Ω1 (E1 )Ω2 (E (0) −
E1 ) also attains its maximum at this value. Therefore,
(0)
(0)
Sf = S1 (Ē1 ) + S2 (Ē2 ) ≥ S1 (E1 ) + S2 (E2 ).
(8.7)
Thus, the final entropy of the system is always greater than or equal to the initial entropy. This
establishes the law of increase of entropy.
21
8.3
Statistical Interpretation of Entropy and the Second Law
From the point of view of statistical mechanics, entropy counts the number of accessible microstates of a system. If this number is large, the system has more freedom; if it is small, the
system has less freedom. In this sense, entropy provides a measure of the randomness or disorder
of a given system.
Since entropy is given by the logarithm of the phase-space volume, the second law of thermodynamics implies that, after any change in the system, the total accessible phase-space volume
never decreases. That is,
Ωf ≥ Ωi .
To make this idea more precise, consider an isolated system. Suppose the system is constrained
by specifying certain parameters y1 , y2 , . . . , yn . These parameters are not necessarily external;
rather, they characterize the system at a macroscopic level. Consequently, the number of accessible
microstates of the system depends on the values of these parameters. We write
Ω = Ω(y1 , . . . , yn ),
(8.8)
where Ω denotes the number of accessible states of the system when each parameter yi lies in the
interval [yi , yi + dyi ].
Suppose the system is initially in equilibrium, with the parameters restricted to some given
small ranges. In this situation, all microstates counted by Ωi are equally probable. Now consider a
process in which some of these constraints are removed. All states that were previously accessible
to the system remain accessible, but in addition many new states become accessible.
Therefore, the number of accessible microstates can only increase, implying
Ωf ≥ Ωi .
8.3.1
Reversible and Irreversible Processes
Suppose that the final equilibrium state has been reached, so that the systems in the ensemble are
uniformly distributed over Ωf accessible microstates. If we now restore the original constraints,
the systems in the ensemble will still occupy Ωf states with equal probability. Thus, if Ωf > Ωi ,
simply restoring the constraints does not recover the initial situation.
Once the system is randomly distributed over Ωf states, reimposing the constraints cannot
cause the system to move spontaneously out of some of its accessible states in order to occupy a
more restricted set of states.
Now consider an isolated system undergoing a process that takes it from an initial state to a
final state. If the final situation is such that neither the imposition nor the removal of constraints
can restore the initial state of the isolated system, then the process is called irreversible. On the
other hand, if the imposition or removal of constraints can restore the initial state, the process is
called reversible.
Therefore, for a reversible process,
Ωf = Ωi ,
(8.9)
and hence,
dS = 0.
22
(8.10)
9
Classical Ideal Gas Again
We have seen that the phase-space volume of a classical ideal gas is given by
Z
∆
ΩMC (E, V ) = 3N
d3N q d3N p δ H(p) − E
h0
!
Z
3N
2
X
p
∆ N
i
d3N p δ
−E .
= 3N V
2m
h0
i=1
Define
pi =
(9.1)
√
2mE ui .
Then,
Z
3N
d
pδ
!
3N
X
p2
i
i=1
−E
2m
= (2mE)
3N/2
Z
d
3N
uδ E
3N
X
!!
u2i − 1
.
i=1
Thus we have,
∆
(2mE)3N/2
ΩM C = 3N V N
h
E
!
Z
d3N u δ
X
u2i − 1 .
(9.2)
i
The E dependence is explicit. The integration is independent of E, it only depends on N . Let us
find that explicitly.
!
Z
Z
n
n
X
X
n
2
In =
d uδ
ui − 1 =
dn u δ(u2 − 1),
u2 ≡
u2i .
i=1
i=1
Introduce n-dimensional spherical coordinates,
r ≥ 0,
u = r Ω̂,
for which the measure becomes
dn u = r n−1 dr dΩn−1 ,
where dΩn−1 is the solid-angle element on the unit (n − 1)-sphere.
The integral then factorizes as
Z
Z ∞
In = dΩn−1
r n−1 δ(r2 − 1) dr.
0
Using the identity
1 δ(r − 1) + δ(r + 1) ,
δ(r2 − 1) = δ (r − 1)(r + 1) =
2|r|
and noting that r ≥ 0, only the root r = 1 contributes. Hence
δ(r2 − 1) =
Therefore,
Z ∞
r
0
Thus we obtain
n−1
1
δ(r − 1) dr =
2
2
1
In =
2
1
δ(r − 1).
2r
Z ∞
0
1
r n−2 δ(r − 1) dr = .
2
Z
dΩn−1 =
23
1
Ωn−1 .
2
The surface area of the unit (n − 1)-sphere is related to the volume Cn of the unit n-ball by
Ωn−1 = n Cn ,
Cn =
π n/2
.
Γ n2 + 1
Hence,
Z
dn u δ
n
X
!
u2i − 1
=
i=1
n
Cn
2
and in particular, for n = 3N ,
Z
d3N u δ
3N
X
!
u2i − 1
=
i=1
3N
C3N .
2
Therefore ΩM C is given by
∆ N (2mE)3N/2 3N
C3N .
ΩM C = 3N V
h
E
2
(9.3)
S(E, V, N ) = k ln ΩM C (E, V, N )
(9.4)
Entropy is given by
We recall that
Cn =
π n/2
.
Γ n2 + 1
Using Stirling’s approximation,
1
1
ln Γ(z + 1) = z ln z − z + ln(2πz) + O
,
2
z
z → ∞,
and setting z = n2 , we obtain
n
n n n 1
1
ln Γ
+ 1 = ln − + ln(πn) + O
.
2
2 2
2 2
n
Therefore,
n
n
ln π − ln Γ
+1
2
2
n
n n n 1
1
= ln π − ln + − ln(πn) + O
.
2
2 2
2 2
n
ln Cn =
Keeping only the leading extensive terms, we have
ln Cn =
n
n n n
ln π − ln + + O(ln n)
2
2 2
2
or equivalently,
i
nh
2π
ln Cn =
1 + ln
+ O(ln n).
2
n
24
Putting everything together we find,
" 3/2 #
3
4πmE
+ Nk .
S(E, V, N ) = k ln ΩM C = N k ln V
2
3N h
2
(9.5)
The average temperature is given by
T̄ =
∂S
∂E
−1
2 E
.
3 Nk
=
(9.6)
Therefore we have
3
E = N k T̄
2
(9.7)
The pressure is given by
p=T
∂S
∂V
=
N kT
.
V
(9.8)
Hence we have,
pV = N k T̄ .
Using the expression for temperature, we can write
3
S(T, V ) = N k ln V + ln T + σ .
2
where
9.1
2πmk
3
3
+ .
σ = ln
2
2
h
2
(9.9)
(9.10)
(9.11)
Entropy and more
There are a few important conceptual issues associated with the classical expression for entropy,
which we discuss below.
• Third law of thermodynamics: From the classical expression for entropy we find that, as
T → 0, the entropy behaves as
S → −∞,
which is in clear contradiction with the third law of thermodynamics. The third law states
that at absolute zero temperature the entropy of any system approaches a universal constant,
which is conventionally taken to be zero for a perfect crystal.
This apparent contradiction arises because the third law is fundamentally a quantum mechanical statement, whereas the derivation of the entropy used here is purely classical. At
very low temperatures the energy of the system approaches its quantum mechanical ground
state, and a classical description is no longer valid. Therefore, the classical expression for
entropy should not be trusted in the low-temperature regime.
• Extensive property: Entropy is an extensive thermodynamic quantity. This means that if
we scale the size of the system by a factor α, all extensive quantities should scale by the
same factor. In particular, if we scale the number of particles and the volume as
N → αN,
25
V → αV,
the entropy should also scale as S → αS.
However, the classical expression for entropy obtained earlier does not satisfy this scaling
property. This signals an inconsistency in the classical treatment and indicates that the
expression for entropy must be modified.
• Gibbs paradox: From the expression for the entropy of an ideal gas, (9.10), we see that
the entropy is proportional to the total number of particles N . This dependence leads to
an apparent violation of the second law of thermodynamics, as illustrated by the following
example.
Consider a container of volume V containing a gas of N particles at temperature T . Now
insert a partition that divides the container into two parts of volumes V1 and V2 , with V1 +
V2 = V . Suppose that N1 particles are in volume V1 and N2 particles are in volume V2 , with
N1 + N2 = N .
Since inserting the partition is a reversible process, the total entropy should remain unchanged. Therefore,
S = S1 + S2 ,
where S is the entropy before inserting the partition, and S1 and S2 are the entropies of the
two subsystems after the partition is inserted.
However, using (9.10), we find
S − (S1 + S2 ) = N ln V − N1 ln V1 − N2 ln V2 ̸= 0.
Equivalently,
∆S = (S1 + S2 ) − S < 0,
which violates the second law of thermodynamics.
What went wrong?
This inconsistency is known as the Gibbs paradox. The problem does not lie in the subadditivity property of entropy, but in an incorrect assumption made in the classical counting of
states.
In the above derivation, the particles were treated as distinguishable. This means that exchanging two particles was counted as a new microscopic state. Physically, however, identical particles are indistinguishable, and exchanging them does not lead to a new physical
state.
In quantum mechanics, particle indistinguishability is automatic. Even for a classical dilute
gas (away from very low temperatures), one must still account for this indistinguishability
in counting microstates. Failing to do so leads to violations of fundamental thermodynamic
laws.
To correct for this overcounting, the phase space volume must be divided by a factor of N !.
As a result, the entropy is modified as
S = S distin − N k ln N + N k.
For a classical dilute ideal gas, the correct expression for entropy is therefore
3
V
where σ0 = σ + 1.
S = N k ln + ln T + σ0 ,
N
2
26
With this corrected expression, it is easy to verify that
V1
V2
V
− N2 ln
= 0,
S − (S1 + S2 ) = N ln − N1 ln
N
N1
N2
as required by the second law of thermodynamics.
10
Canonical Ensemble
We now consider a system which is not isolated. The system is in thermal contact with a much
larger system, called a heat reservoir. These two systems can exchange energy. Since the reservoir
is very large, removing or adding a finite amount of energy does not change its temperature. The
system of interest, on the other hand, is finite.
We introduce the following notation:
• Ω(E) denotes the phase space volume of the finite system with energy E.
• ΩR (ER ) denotes the phase space volume of the reservoir with energy ER .
• (q, p) are the phase space coordinates of the finite system.
• (qR , pR ) are the phase space coordinates of the reservoir.
• H(q, p) is the Hamiltonian of the finite system.
• HR (qR , pR ) is the Hamiltonian of the reservoir.
• The total Hamiltonian of the combined system is
H(q, p, qR , pR ) = H(q, p) + HR (qR , pR ).
(10.1)
The average value of any observable F (q, p) over the entire system (finite system + reservoir),
at fixed total energy E0 , is given by
R N N N
d q d p d R q dNR p δ(H(q, p, qR , pR ) − E0 ) F (q, p)
(0)
R
⟨F (q, p)⟩E0 =
dN q dN p dNR q dNR p δ(H(q, p, qR , pR ) − E0 )
R
R
dE dER dN q dN p dNR q dNR p δ(H − E) δ(HR − ER ) δ(E + ER − E0 ) F (q, p)
R
R
=
dE dER dN q dN p dNR q dNR p δ(H − E) δ(HR − ER ) δ(E + ER − E0 )
R
R
dE dER dN q dN p δ(H − E) ΩR (ER ) δ(E + ER − E0 ) F (q, p)
R
R
=
dE dER dN q dN p δ(H − E) ΩR (ER ) δ(E + ER − E0 )
R
R
dER dN q dN p ΩR (ER ) δ(H + ER − E0 ) F (q, p)
R
R
=
dER dN q dN p ΩR (ER ) δ(H + ER − E0 )
R N N
d q d p ΩR (E0 − H) F (q, p)
R
=
.
(10.2)
dN q dN p ΩR (E0 − H)
Since the energy of the finite system is much smaller than the total energy E0 of the reservoir,
we may expand ΩR (E0 − H) around E0 :
1
ΩR (E0 − H) = exp SR (E0 − H)
k
"
!#
1
∂SR
1
SR (E0 ) −
= exp
H(q, p) + O
k
∂E E0
N
1
H(q, p)
= exp SR (E0 ) exp −
,
(10.3)
k
kTR
27
where we have used
∂SR
1
=
.
∂E E0
TR
Substituting this into the expression for the average, we obtain
R N N −H(q,p)/(kT )
R
d qd p e
F (q, p)
(0)
R
⟨F (q, p)⟩E0 =
.
N
N
−H(q,p)/(kT
R)
d qd p e
(10.4)
(10.5)
Thus, averaging over the entire isolated system is equivalent to averaging over the finite system
alone with the phase space density
ρ(q, p) = N e−H(q,p)/(kTR ) .
(10.6)
We may now forget about the reservoir, except for the fact that it fixes the temperature TR . The
ensemble defined by (10.6) is called the canonical ensemble.
Partition Function
The phase space volume occupied by the canonical ensemble is called the partition function:
Z 3N 3N
d q d p −βH(q,p)
ZN (V, T ) ≡
e
,
(10.7)
N ! h3N
where β = 1/(kTR ) and the normalization factor 1/(N !h3N ) has been included.
10.1
Equivalence of Canonical and Microcanonical Ensembles
The average value of a macroscopic observable in the canonical ensemble is
R 3N 3N −βH
d qd p e
F (q, p)
can
R
⟨F (q, p)⟩ =
,
d3N q d3N p e−βH
while in the microcanonical ensemble it is
R
⟨F (q, p)⟩microcan =
d3N q d3N p δ(H − E) F (q, p)
R
.
d3N q d3N p δ(H − E)
The numerator of the canonical average can be written as
Z
Numerator = dE d3N q d3N p e−E/(kTR ) δ(H − E) F (q, p)
Z
= dE e−E/(kTR ) ⟨F (q, p)⟩microcan Ω(E).
(10.8)
(10.9)
(10.10)
The factor e−E/(kTR ) decreases rapidly with E, whereas Ω(E) increases rapidly. Their product
is therefore sharply peaked at some energy E = Ē. The location of the peak is determined by
∂
[−βE + ln Ω(E)]Ē = 0
∂E
1
1 ∂S
⇒
−
+
=0
kTR k ∂E
⇒
T (Ē) = TR .
28
(10.11)
Hence,
Numerator = ⟨F (q, p)⟩microcan
Ē
Z
Denominator =
Z
dE e−βE Ω(E),
dE e−βE Ω(E),
(10.12)
(10.13)
and therefore
⟨F (q, p)⟩can = ⟨F (q, p)⟩microcan ,
with T (Ē) = TR .
(10.14)
Thus, in the thermodynamic limit, the canonical and microcanonical ensembles give identical
results for all thermodynamic quantities.
10.2
Free Energy
The partition function may be written as
Z
1
ZN (V, T ) =
d3N q d3N p dE δ(H − E) e−βE
N !h3N
Z
= dE e−βE Ω(E)
Z
= dE e−βE+S(E)/k .
The integrand is sharply peaked at E = Ē. Expanding around Ē,
1 ∂ 2S
2
−βE+S(E)/k
−β Ē+S(Ē)/k
(E − Ē) .
e
=e
exp
2k ∂E 2 Ē
(10.15)
(10.16)
This gives
1
ZN (V, T ) =
e−β Ē+S(Ē)/k
N !h3N
s
2πk
.
(10.17)
1
= Ē − TR S(Ē) − kTR ln 2πkN T 2 CV
2
= Ē − TR S(Ē) + O(ln N ).
(10.18)
|∂ 2 S/∂E 2 |
The free energy is defined by
F (β) = −kTR ln ZN
Thus,
S(Ē) =
Ē − F
.
TR
(10.19)
In the thermodynamic limit,
F = E − T S,
dF = −p dV − S dT,
(10.20)
(10.21)
from which we obtain
p=−
∂F
∂V
∂F
∂T
S=−
∂F
E = F + TS = F − T
.
∂T V
,
T
29
,
(10.22)
V
(10.23)
Although the canonical and microcanonical ensembles give the same results in the thermodynamic limit, the canonical ensemble corresponds more closely to experimental situations. In
practice, systems are rarely completely isolated, and temperature is the quantity that is most directly controlled in experiments.
10.3
Classical ideal gas in the canonical ensemble
Consider N identical, non-interacting classical particles of mass m in a volume V . For an ideal
gas the Hamiltonian is purely kinetic:
H(q, p) =
N
X
p2
i
2m
i=1
,
(10.24)
and there is no dependence on q except through the restriction that qi ∈ V .
Partition function
The canonical partition function is
Z 3N 3N
d qd p
ZN (V, T ) =
exp(−βH(q, p)) ,
N ! h3N
β=
1
.
kT
(10.25)
Since H is independent of q,
Z
d3N q = V N .
(10.26)
For the momentum part we use the Gaussian integral in 3D,
Z
3/2
p2
2πm
.
d p exp −β
=
2m
β
3
Therefore
Z
3N
d
p exp −β
N
X
p2
!
i
=
2m
i=1
2πm
β
(10.27)
3N/2
.
(10.28)
Putting everything together,
VN
ZN (V, T ) =
N ! h3N
2πm
β
3N
2
=
3N
VN
2 .
(2πmkT
)
N ! h3N
(10.29)
It is often convenient to introduce the (classical) thermal wavelength
λT ≡ √
h
,
2πmkT
(10.30)
so that the partition function becomes
1
ZN (V, T ) =
N!
30
V
λ3T
N
.
(10.31)
Average energy
In the canonical ensemble,
Ē = ⟨H⟩ = −
From
∂
ln ZN .
∂β
2πm
3N
ln
,
ln ZN = N ln V − ln N ! − 3N ln h +
2
β
only the last term depends on β, hence
∂ 3N
2πm
3N 1
3
Ē = −
ln
=
= N kT.
∂β 2
β
2 β
2
(10.32)
(10.33)
(10.34)
Thus
3
Ē = N kT.
2
(10.35)
Pressure and equation of state
From thermodynamics in the canonical ensemble,
∂F
∂
ln ZN
F = −kT ln ZN ,
p=−
= kT
.
∂V T
∂V
T
Since ln ZN contains N ln V ,
∂
ln ZN
∂V
=
T
N
,
V
(10.36)
(10.37)
and therefore
N kT
.
V
(10.38)
pV = N kT,
(10.39)
p=
Equivalently,
which is the classical ideal gas equation of state.
Useful equivalent relations
Using Ē = 23 N kT and pV = N kT , we also get
Ē =
11
3
pV.
2
(10.40)
Grand Canonical Ensemble
In practice, the total number of particles in a system is never known precisely. What is experimentally accessible is only the average number of particles. In the grand canonical ensemble, the
system can exchange both energy and particles with its surroundings, so that the particle number
fluctuates while its average value is fixed by external conditions. This situation is analogous to the
canonical ensemble, where the average energy of the system is determined by the temperature of
the reservoir.
31
Consider a system of volume V0 containing N particles at temperature T . Let us choose a
small sub-volume V such that VR = V0 − V ≫ V . We denote the phase-space coordinates of the
particles in V by (q, p) and those in VR by (qR , pR ). The total Hamiltonian can be written as
H(Q, P, N0 ) = H1 (q, p, N ) + HR (qR , pR , NR ).
The canonical partition function of the whole system is
Z
1
d3N0 Qd3N0 P e−βH(Q,P,N0 ) .
ZN0 (V0 , T ) = 3N0
h N0 !
(11.1)
(11.2)
We may therefore regard subsystem 2 as a reservoir. Subsystem 1 can exchange both energy and
particles with this reservoir.
Let K(q, p, N ) be a function defined on subsystem 1. We wish to compute its expectation
value. As discussed earlier,
⟨K(q, p, N0 )⟩micro
= ⟨K(q, p, N )⟩cano
E0
β .
(11.3)
Considering the total system in the canonical ensemble at inverse temperature β, we may write
R 3N 3N
1
d Qd P K(q, p, N ) e−βH(Q,P,N0 )
3N
h 0 N0 !
⟨K(q, p, N )⟩cano
=
.
(11.4)
β
ZN0 (V0 , T )
Here,
Z
d
3N0
3N0
Qd
Z
P →
d
3N
3N
qd
V
Z
p
d3NR qR d3NR pR .
VR
We now separate the integrations over V and VR . The number of particles N inside V can
range from 0 to N0 . For a fixed N , we do not care which particular particles are inside V1 , only
their number. Therefore, there are N0 CN distinct possibilities.
The partition function can thus be written as
Z
Z
Z
N0
X
N0 !
1
3NR
3NR
d qR d pR
d3N q
ZN0 (V0 , T ) = 3N
h N0 ! VR
N
!(N
0 − N )! V
N =0
Z
× d3N p exp −β H(q, p, N ) + HR (qR , pR , NR )
N0
X
1
=
3N
h N!
N =0
Z
1
h3NR NR !
Z
×
=
N0
X
N =0
d3N qd3N p e−βH(q,p)
V
VR
Z
1
d3NR qR d3NR pR e−βHR (qR ,pR )
h3N N !
d3N qd3N p e−βH(q,p,N ) ZNR (VR , T )
(11.5)
V
From here we define a relative probability that N particles in volume V with coordinates (p, q)
ρ(p, q, N ) =
ZNR (VR , T ) e−βH(q,p,N )
.
ZN0 (V0 , T ) h3N N !
(11.6)
From this definition we have
N0 Z
X
N =0
dq dp ρ(p, q, N ) = 1.
V
32
(11.7)
Now, from the definition of free energy (10.18) : ZN (V, T ) = exp[−βF (β, V, N )
ZNR (VR , T )
= exp [−β(F (β, V0 − V, N0 − N ) − F (β, V0 , N0 ))] .
ZN0 (V0 , T )
(11.8)
Expanding F (β, V0 − V, N0 − N ) about V0 and N0 we find, up to first order,
∂F
∂F
F (β, V0 − V, N0 − N ) − F (β, V0 , N0 ) ≈ −
V −
N.
∂V β,N (V0 ,N0 )
∂N β,V (V0 ,N0 )
Using the thermodynamic identities
∂F
= −P,
∂V β,N
∂F
∂N
= µ,
β,V
we obtain
F (β, V0 − V, N0 − N ) − F (β, V0 , N0 ) ≈ P V − µ N,
where P and µ are evaluated at (β, V0 , N0 ). Hence, we have
ZNR (VR , T )
= e−βP V +βµN .
ZN0 (V0 , T )
(11.9)
Therefore we have
zN
e−βP V −βH(q,p)
(11.10)
h3N N !
where z = eβµ is called the fugacity. Since the surrounding system now enters only through its
macroscopic parameters—temperature (T), pressure (P), and chemical potential µ—its detailed
properties need not be considered explicitly. We may therefore treat it as a reservoir. Taking
the limit in which this external system becomes infinitely large, the number of particles N in the
subsystem is no longer constrained by the reservoir size. Consequently, the allowed range of N
becomes
0 ≤ N < ∞.
ρ(p, q, N ) =
The grand canonical partition function is defined as
Q(z, V, T ) =
∞
X
z N ZN (V, T ).
(11.11)
N =0
It is straightforward to verify that
PV
= ln Q(z, V, T ).
kT
11.1
(11.12)
Equivalence Between Canonical and Grand Canonical Ensembles
The expectation value in the grand canonical ensemble is
R 3N 3N
P∞
zN
qd p K(q, p, N ) e−βH(q,p)
gc
N =0 h3N N ! V d
⟨K(q, p, N )⟩β =
R
P∞
zN
3N qd3N p e−βH(q,p)
N =0 h3N N ! V d
P∞
N
cano
N =0 z ⟨K⟩β,N ZN (V, T )
P∞
=
.
N
N =0 z ZN (V, T )
33
(11.13)
Since ZN (V, T ) ∼ e−F ∼ e−N for large N , the product z N ZN (V, T ) is sharply peaked about
N = N̄ . Thus,
∂ βµN −βF (V,T,N ) e
e
=0
∂N
∂F
⇒ µ=
.
∂N N =N̄
(11.14)
The chemical potential of the system equals that of the reservoir, and
cano
⟨K(q, p, N )⟩gc
β = ⟨K(q, p)⟩β,N̄ .
11.2
(11.15)
Thermodynamics
We have
∞
X
Q(z, V, T ) =
eβµN ZN (V, T ).
(11.16)
N =0
The dominant contribution comes from N = N̄ , so
ln Q ∼ βµN̄ + ln ZN (V, T ) = βµN̄ − βF (V, T, N̄ ).
(11.17)
Using
S=−
∂F
∂T
P =−
,
V,N
∂F
∂V
,
µ=
T,N
∂F
∂N
,
(11.18)
V,T
we obtain
dF = −S dT − P dV + µ dN.
(11.19)
Therefore,
1
dT
(µN
−
F
)
+
µ
dN
+
N
dµ
−
dF
kT 2
kT
N
dT
P
=
− µN + F + T S +
dµ +
dV.
2
kT
kT
kT
d(ln Q) = −
(11.20)
Thus,
11.3
1
1
∂
=
ln Q
(F + T S − µN ) =
(E − µN ),
2
∂T
kT
kT 2
µ,V
(11.21)
∂
N
ln Q
=
,
∂µ
kT
T,V
(11.22)
∂
P
ln Q
=
.
∂V
kT
T,µ
(11.23)
Ideal Gas in the Grand Canonical Ensemble
We consider a classical ideal gas obeying Maxwell–Boltzmann statistics in the grand canonical
ensemble at temperature T , volume V , and chemical potential µ. The grand canonical partition
function is defined as
Q(z, V, T ) =
∞
X
z N ZN (V, T ),
N =0
34
z = eβµ ,
β=
1
.
kT
(11.24)
For an ideal gas, the canonical partition function is
V
λ3
N
h
λ= √
,
2πmkT
(11.25)
N
∞
X
1
V
V
Q(z, V, T ) =
z 3
= exp z 3 ,
N
!
λ
λ
N =0
(11.26)
1
ZN (V, T ) =
N!
,
where λ is the thermal de Broglie wavelength.
Substituting into the definition of Q, we obtain
ln Q(z, V, T ) = z
11.3.1
V
.
λ3
(11.27)
Average Number of Particles
The average particle number is given by
⟨N ⟩ = z
∂
V
ln Q = z 3 .
∂z
λ
(11.28)
Equivalently,
⟨N ⟩λ3
.
V
(11.29)
PV
= ln Q.
kT
(11.30)
z=
11.3.2
Pressure and Equation of State
In the grand canonical ensemble,
Therefore,
kT
kT
⟨N ⟩kT
ln Q = 3 z =
.
V
λ
V
This yields the equation of state of the ideal gas,
P =
P V = ⟨N ⟩kT.
11.3.3
(11.31)
(11.32)
Average Energy
The average energy in the grand canonical ensemble is
⟨E⟩ = −
∂
ln Q + µ ⟨N ⟩.
∂β
(11.33)
Using ln Q = zV /λ3 and the fact that λ−3 ∝ T 3/2 , we obtain
⟨E⟩ =
3
⟨N ⟩kT.
2
(11.34)
Equivalently, the energy density satisfies
⟨E⟩
3 ⟨N ⟩kT
3
=
= P.
V
2 V
2
35
(11.35)
12
Equipartition theorem
The equipartition theorem is a fundamental result of classical statistical mechanics. It provides a
direct connection between temperature and the average energy stored in the degrees of freedom of
a system in thermal equilibrium.
The theorem explains the internal energy of ideal gases, classical specific heats, and also highlights the failure of classical physics at low temperatures.
12.1
Statement of the Theorem
Equipartition Theorem (Classical):
For a classical system in thermal equilibrium at temperature T , each independent quadratic
term in the Hamiltonian contributes
1
kB T
(12.1)
2
to the thermal average of the energy.
12.2
Quadratic Degrees of Freedom
A degree of freedom contributes according to equipartition if it appears quadratically in the Hamiltonian, for example,
p2
H ⊃ aq 2
or
H⊃
,
(12.2)
2m
and is independent of other variables (i.e. no cross terms such as qp).
The theorem applies only in the classical regime and assumes thermal equilibrium.
12.3
Derivation in the Canonical Ensemble
Consider a single classical degree of freedom x with Hamiltonian
H(x) = ax2 ,
a > 0.
The canonical partition function is
Z ∞
2
dx e−βax ,
Z=
β=
−∞
(12.3)
1
.
kB T
(12.4)
The thermal average of the energy is
R
⟨H⟩ =
Using standard Gaussian integrals,
Z
2
dx ax2 e−βax
R
.
dx e−βax2
r
π
,
βa
r
Z
1
π
2 −βax2
dx x e
=
,
2βa βa
−βax2
dx e
(12.5)
=
(12.6)
(12.7)
we obtain
⟨H⟩ =
1
1
= kB T.
2β
2
36
(12.8)
12.4
Generalization
For a Hamiltonian of the form
H=
X
ai qi2 +
X p2j
i
j
2mj
,
(12.9)
each quadratic term contributes independently, giving
1
⟨H⟩ = kB T × (number of quadratic terms).
2
12.5
Applications
12.5.1
Ideal Gas
(12.10)
For a particle of mass m in three dimensions,
p2x + p2y + p2z
H=
.
2m
(12.11)
There are three quadratic momentum terms, so
3
⟨E⟩ = kB T.
2
(12.12)
3
U = N kB T.
2
(12.13)
For N particles,
12.5.2
One-Dimensional Harmonic Oscillator
The Hamiltonian is
1
p2
+ mω 2 x2 .
2m 2
(12.14)
⟨E⟩ = kB T .
(12.15)
H=
There are two quadratic terms, hence
Each of the kinetic and potential energies contributes 12 kB T .
12.5.3
Diatomic Molecule (High-Temperature Limit)
Degrees of freedom:
• 3 translational,
• 2 rotational,
• 1 vibrational mode (kinetic + potential).
Total quadratic terms: 3 + 2 + 2 = 7, giving
7
⟨E⟩ = kB T.
2
37
(12.16)
13
Quantum Statistical Mechanics
Unlike classical mechanics, where the state of a system is represented by a point in phase space, in
quantum mechanics the state of a system is represented by a state vector |ψ(t)⟩ in a Hilbert space.
The time evolution of the system is governed by the Schrödinger equation. A quantum mechanical
ensemble is therefore defined as a collection of state vectors, rather than points in phase space.
13.1
Pure and Mixed Ensembles
Quantum mechanical ensembles are of two types: pure ensembles and mixed ensembles. In a pure
ensemble, all members of the ensemble are described by the same state vector. In contrast, in a
mixed ensemble, different members are described by different state vectors.
As an illustrative example, consider silver atoms emerging from a hot oven and passing through
a Stern–Gerlach apparatus. The spins of these atoms are randomly oriented in all possible directions. There is no preferred direction associated with such an ensemble. How should such an
ensemble be represented?
The most general state of a spin- 21 system can be written as
|ψ⟩ = c+ |+⟩ + c− |−⟩ .
(13.1)
Does this state represent an ensemble of atoms with randomly oriented spins?
The answer is no. The above state corresponds to a spin polarized along a definite direction n̂
(see Fig. ??), with
c+
cos(β/2)
.
(13.2)
= iα
c−
e sin(β/2)
To describe an ensemble with random spin orientations, we must introduce the concept of
fractional population (or probability weight). Such an ensemble may be viewed as one in which
50% of the atoms are in the state |+⟩, and the remaining 50% are in the state |−⟩.
In general, a pure ensemble is defined as a collection of systems in which every member is
characterized by the same state |α⟩. A mixed ensemble consists of several types of states: a
fraction w1 of the ensemble is described by |α(1) ⟩, another fraction w2 by |α(2) ⟩, and so on. These
fractional populations satisfy the normalization condition
P
X
wi = 1.
(13.3)
i=1
Here P denotes the number of distinct state vectors in the ensemble. The states |α(i) ⟩ need not
be orthogonal. For example, one may prepare an ensemble in which 30% of the atoms have spin
aligned along the z-axis, 20% along the negative x-axis, and 50% along the y-axis.
Suppose an observable A is measured on such a mixed ensemble. The ensemble average of A
is defined as
[A] =
=
P
X
wi ⟨α(i) | A |α(i) ⟩
i=1
P X
D
X
2
wi ⟨a|α(i) ⟩ a.
(13.4)
i=1 a=1
Here |a⟩ denotes an eigenstate of A and D is the dimensionality of the vector space. The quantity
⟨α(i) | A |α(i) ⟩ is the usual quantum mechanical expectation value, further weighted by the frac2
tional population wi . Probability enters twice: first through the quantum probability ⟨a⟩ α(i) ,
and second through the ensemble probability wi .
38
13.2
Density Operator
The ensemble average of A can be written using an arbitrary basis {|b⟩} as
[A] =
P
X
wi
i=1
=
D
X
⟨α(i) |a⟩ ⟨a| A |b⟩ ⟨b|α(i) ⟩
a,b=1
D
X
" P
X
a,b=1
i=1
#
wi ⟨b|α(i) ⟩⟨α(i) |a⟩ ⟨a| A |b⟩ .
(13.5)
The quantity inside the square brackets is independent of the observable A. This motivates the
definition of the density operator ρ̂,
ρ̂ =
P
X
wi |α(i) ⟩ ⟨α(i) | .
(13.6)
i=1
The matrix elements of the density operator are
X
⟨b| ρ̂ |a⟩ =
wi ⟨b|α(i) ⟩⟨α(i) |a⟩.
(13.7)
i
The ensemble average may now be written compactly as
[A] =
D
X
⟨b| ρ̂ |a⟩ ⟨a| A |b⟩
a,b=1
= Tr(ρ̂A).
(13.8)
Since the trace is basis-independent, the expectation value does not depend on the choice of basis.
13.2.1
Properties of the Density Operator
The density operator satisfies the following properties:
1. It is Hermitian.
2. Its trace is unity.
Indeed,
Tr(ρ̂) =
XX
i
=
wi ⟨b|α(i) ⟩⟨α(i) |b⟩
b
X
wi ⟨α(i) |α(i) ⟩
i
= 1.
(13.9)
3. For a pure ensemble, wi = 1 for one particular i and zero for all others. Then
ρ̂ = |α⟩ ⟨α| ,
(13.10)
ρ̂2 = ρ̂,
(13.11)
ρ̂(ρ̂ − 1) = 0.
(13.12)
Tr(ρ̂2 ) = 1.
(13.13)
which satisfies
or equivalently,
Thus,
39
13.2.2
Time Evolution of the Density Operator
At time t = t0 , let
ρ̂(t0 ) =
X
wi |α(i) (t0 )⟩ ⟨α(i) (t0 )| .
(13.14)
i
The fractional populations wi remain constant if the system is left undisturbed. Time evolution
therefore arises solely from the evolution of the state vectors, which satisfy the Schrödinger equation
∂
(13.15)
iℏ |α(i) (t)⟩ = H |α(i) (t)⟩ .
∂t
Consequently, the density operator obeys
iℏ
∂ ρ̂
= −[ρ̂, H].
∂t
(13.16)
∂ ρ̂
= 0,
∂t
(13.17)
[ρ̂, H] = 0.
(13.18)
Thermal equilibrium requires
which implies
Thus, the density operator and the Hamiltonian are simultaneously diagonalizable.
13.3
Examples
• For a pure ensemble with all particles in the state |Sz , +⟩,
ρ = |Sz , +⟩⟨Sz , +|.
In the Sz basis {|+⟩, |−⟩},
1 0
ρ=
.
0 0
For spin- 21 , equivalently,
1
ρ = (I + σz ).
2
• For a completely Sx , + polarised beam,
ρ = |Sx , +⟩⟨Sx , +|.
Using
1
|Sx , +⟩ = √ |+⟩ + |−⟩ ,
2
we obtain
1
ρ=
2
1 1
.
1 1
Equivalently,
1
ρ = (I + σx ).
2
40
• For a beam consisting of 75% Sz , + and 25% Sx , +, For p = 43 ,
7 1
ρ = 18 81 .
8
8
Equivalently,
ρ=
1
1
3 I + σx + σz .
2
4
4
ρ=
1
1
3 I + σx + σz ,
2
4
4
Using
the expectation values are
⟨Si ⟩ = Tr(ρSi ),
With Bloch vector
P⃗ =
Si =
1
3
, 0,
4
4
,
we obtain
⟨Sx ⟩ =
ℏ
,
8
ℏ
σi .
2
⟨Sz ⟩ =
3ℏ
.
8
14
Ensembles in Quantum Statistical Mechanics
14.1
Microcanonical Ensemble
In the microcanonical ensemble, the density matrix in the energy eigenbasis is
ρab = δab |Λa |2 ,
where
(
|Λa |2 =
1
,
Γ
0,
E < Eb < E + ∆,
otherwise.
(14.1)
(14.2)
The energy eigenvalues satisfy H |b⟩ = Eb |b⟩, and Γ is the number of states with energies in the
interval E < Eb < E + ∆.
The density operator is therefore
X
1
|b⟩ ⟨b|
Γ E<E <E+∆
b
1X
=
|b⟩ ⟨b| [Θ(Eb − E)Θ(E + ∆ − Eb )]
Γ b
ρ̂ =
(14.3)
We can calculate Tr[ρ̂]
1X
⟨a|b⟩ ⟨b|a⟩ [Θ(Eb − E)Θ(E + ∆ − Eb )]
Γ a,b
1X
=
[Θ(Ea − E)Θ(E + ∆ − Ea )]
Γ a
Tr[ρ̂] =
=
1
× Total number of microstates between E and E + ∆.
Γ
41
(14.4)
Since Tr[ρ̂] = 1 we find,
Γ = Total number of microstates between E and E + ∆
The number of accessible states is
X
ρbb ≡ Γ(E),
(14.5)
S = k ln Γ(E).
(14.6)
b
and the entropy is defined as
14.2
Canonical Ensemble
The derivation of the canonical ensemble closely parallels the classical case. In classical statistical mechanics, the phase space density of a configuration is proportional to e−βH . In quantum
statistical mechanics, we proceed analogously.
Consider a system with Hamiltonian H and energy eigenstates |a⟩ satisfying
H|a⟩ = Ea |a⟩.
In thermal equilibrium, the density matrix is diagonal in the energy basis:
ρab = A e−βEa δab ,
where β = 1/(kT ) and A is a normalization constant.
Normalization of the density matrix, Tr(ρ) = 1, gives
X
X
e−βEa .
ρaa = A
1=
(14.7)
(14.8)
a
a
We therefore define the canonical partition function
X
Z(T, V ) =
e−βEa = Tr(e−βH ),
(14.9)
a
so that
1
.
Z(T, V )
(14.10)
e−βH
.
Z(T, V )
(14.11)
A=
The density operator can thus be written as
ρ̂ =
The ensemble average of an operator A is defined by
⟨A⟩ = Tr(ρ̂ A) =
42
Tr(Ae−βH )
.
Tr(e−βH )
(14.12)
14.2.1
3D harmonic oscillator
For a 3D isotropic quantum harmonic oscillator,
Enx ny nz = ℏω nx + ny + nz + 23 ,
ni = 0, 1, 2, . . .
The partition function factorizes:
3
,
Z3D = Z1D
Z1D =
∞
X
1
e−βℏω(n+ 2 ) =
n=0
Hence
Z3D =
e−βℏω/2
1 − e−βℏω
e−βℏω/2
.
1 − e−βℏω
3
.
The average energy is
∂
ℏω
ℏω
⟨E⟩ = −
ln Z3D = 3
+ βℏω
.
∂β
2
e
−1
High-T expansion (x = βℏω ≪ 1) gives
(ℏω)2
(ℏω)4
⟨E⟩ = 3kT +
+O
.
4kT
(kT )3
Classically, equipartition yields
⟨E⟩cl = 3kT.
Quantum effects are important when
kT ≲ ℏω
(βℏω ≳ 1),
in which case ⟨E⟩ → 23 ℏω as T → 0.
Note To determine when quantum effects become important for a particle in a harmonic potential, it is useful to compare two characteristic length scales: the thermal de Broglie wavelength
and the harmonic oscillator length.
The thermal de Broglie wavelength is defined as
λT = √
h
.
2πmkT
(14.13)
This quantity does not represent the physical size of the particle. Rather, it sets the characteristic
spatial coherence length of a thermal quantum particle. It may be understood heuristically by
noting that thermal equilibrium fixes the typical momentum fluctuations,
√
∆p ∼ mkT ,
which, via the uncertainty principle, imply a spatial spread
∆x ∼
ℏ
ℏ
∼√
∼ λT .
∆p
mkT
Thus λT characterizes the length scale over which the wave-like nature of the particle is relevant.
At high temperatures λT is small, and the system behaves classically, whereas at low temperatures
λT increases and quantum effects become significant.
43
For a harmonic oscillator with potential
1
V (x) = mω 2 x2 ,
2
(14.14)
the natural quantum length scale is
r
ℏ
,
(14.15)
mω
which corresponds to the spatial width of the wavefunction and represents the intrinsic length scale
of the system.
Quantum effects become important when the thermal wavelength is comparable to the confinement scale,
λT ≳ λHO .
(14.16)
λHO =
Substituting the definitions of the two length scales leads to the equivalent energy criterion
kT ≲ ℏω.
(14.17)
Therefore, quantum behaviour is expected when the thermal energy is comparable to or smaller
than the spacing of harmonic oscillator energy levels, while classical behaviour emerges in the
opposite limit kT ≫ ℏω.
14.2.2
Maxwell-Boltzmann gas
Consider an ideal Maxwell–Boltzmann gas in the canonical ensemble. Let the occupation numbers
of single-particle states labelled by p be np . The total energy and particle number are
X
X
np .
ϵp np ,
N=
E({np }) =
p
p
For Maxwell–Boltzmann statistics, the canonical partition function may be written in the occupation number representation as
ZN (T, V ) =
1 X Y N!
exp(−βϵp np ) ,
N!
n
!
p
p
X
{np }
np = N.
p
Using the formula
(x1 + x2 + · · · ) =
N!
xn1 2 xn2 2 · · ·
n !n ! · · ·
n ,n ,··· 1 2
X
1
2
we can write
Z1 (T, V )N
.
N!
Now consider a particle confined to a cubic box of side L (V = L3 ) with periodic boundary
conditions. The allowed wavevectors are
ZN (T, V ) =
ki =
2π
ni ,
L
ni ∈ Z,
and the single-particle energies are
ϵn =
ℏ2 k 2
(2πℏ)2 2
=
(n + n2y + n2z ).
2m
2mL2 x
44
In the large-volume limit, the sum over momentum states becomes
Z
X
V
−→
d3 p.
3
(2πℏ)
p
Therefore,
Z
V
p2
3
.
Z1 (T, V ) ≃
d p exp −β
(2πℏ)3
2m
Evaluating the Gaussian integral,
3/2
Z
2πm
3
−βp2 /2m
d pe
=
,
β
we obtain
Z1 (T, V ) = V
m
2πβℏ2
3/2
.
Introducing the thermal de Broglie wavelength
λ= √
h
,
2πmkT
the single-particle partition function becomes
gV
,
λ3
Z1 (T, V ) =
where g is the internal degeneracy.
Finally, the canonical partition function for the Maxwell–Boltzmann gas is
N
1 gV
ZN (T, V ) =
.
N ! λ3
14.3
Grand Canonical Ensemble
Q(z, V, T ) =
=
∞ X
X
N =0
∞
X
e−βH+βµN
b
z N ZN (T, V ).
(14.18)
N =0
The density operator in the grand canonical ensemble is
ρ̂ =
15
z N e−βH
.
ZN (T, V )
Ideal Bose gas
Consider an ideal Bose gas in a rectangular box of volume
V = Lx Ly Lz .
The single–particle Schrödinger equation is
−
ℏ2 2
∇ ψ(r) = Eψ(r).
2m
45
(14.19)
Periodic boundary conditions : We impose
ψ(x + Lx , y, z) = ψ(x, y, z),
ψ(x, y + Ly , z) = ψ(x, y, z),
ψ(x, y, z + Lz ) = ψ(x, y, z).
The eigenfunctions are plane waves
1
ψk (r) = √ ei(kx x+ky y+kz z) .
V
The allowed wavevectors are quantized as
kx =
2πnx
,
Lx
2πny
,
Ly
ky =
kz =
2πnz
,
Lz
where
nx , ny , nz = 0, ±1, ±2, . . .
The corresponding single–particle energy levels are
ℏ2
εnx ,ny ,nz =
2m
"
2πnx
Lx
2
+
2πny
Ly
2
+
2πnz
Lz
2 #
.
The single–particle states are conveniently labeled by the momentum vector p⃗, with
p⃗ = ℏ⃗k.
where,
px = ℏ
15.1
2πnx
,
Lx
py = ℏ
2πny
,
Ly
pz = ℏ
2πnz
.
Lz
Grand partition function
A many–particle state of the system is specified by the occupation numbers {np⃗ }, where for bosons
np⃗ = 0, 1, 2, . . .
In the grand canonical ensemble the grand partition function is
Q(β, V, µ) =
∞
X
z N ZN (V, T ) =
N =0
∞
X
X
zN
N =0
exp[−βE({np⃗ })] .
{n⃗ }
P p
np
⃗=N
Since the sum over N runs from 0 to ∞, the sum over np⃗ actually becomes unrestricted. Therefore
the grand partition function can be written as,


X
X
Q(β, V, µ) =
exp−β
np⃗ (ϵp⃗ − µ) .
{np⃗ }
Using the identity e
P
ai
=
p
⃗
Q ai
e , the exponential factorizes,


X
Y
exp−β
np⃗ (ϵp⃗ − µ) =
e−βnp⃗ (ϵp⃗ −µ) .
p
⃗
p
⃗
46
Since the occupation numbers for different momentum states are independent, the sum over configurations factorizes into a product of independent sums,


∞
Y X

Q=
e−βnp⃗ (ϵp⃗ −µ)  .
np⃗ =0
p
⃗
Since z = eβµ , each sum becomes a geometric series,
∞
X
−βnp⃗ (ϵp⃗ −µ)
e
=
np⃗ =0
∞
X
(ze−βϵp⃗ )np⃗ =
np⃗ =0
1
.
1 − ze−βϵp⃗
Thus the grand partition function takes the form
Q(β, V, z) =
Y
p
⃗
1
.
1 − ze−βϵp⃗
Equation of state: Taking the logarithm, the equation of state is given by,
X
PV
= ln Q = −
ln 1 − ze−βϵp⃗ .
kT
p
⃗
Average occupation number: The average occupation number of the single–particle state p⃗ is
∞
P
1 X NX
⟨np⃗ ⟩ =
z
np⃗ e−β p⃗ ϵp⃗ np⃗
Q N =0
{np⃗ }
which can be written as,
⟨np⃗ ⟩ = −
ze−βϵp⃗
1
1
1 ∂
ln Q =
.
= −1 βϵp⃗
= β(ϵ −µ)
−βϵ
p
⃗
β ∂ϵp⃗
1 − ze
z e −1
e p⃗
−1
Since ⟨np⃗ ⟩ can not be negative, we must have,
eβ(ϵp⃗ −µ) ≥ 1,
∀ p⃗.
This implies ϵp⃗ ≥ µ for all p⃗. Since, ϵp⃗ > ϵ0 for p⃗ ̸= 0, we have µ ≤ ϵ0 . For free bose gas, ϵ0 = 0,
therefore µ ≤ 0 and hence z ≤ 1.
Average total number of particles: The average total number of particles is
⟨N ⟩ = z
X ze−βϵp⃗
X
∂
ln Q =
=
⟨np⃗ ⟩.
∂z
1 − ze−βϵp⃗
p
⃗
15.2
p
⃗
Thermodynamic limit
Starting from
ln Q = − ln(1 − z) −
X
−βϵp⃗
ln 1 − ze
,
p
̸⃗=0
where the term − ln(1 − z) corresponds to the ground state p⃗ = 0.
47
p2
ϵp⃗ =
,
2m
In the thermodynamic limit, the sum over p⃗ ̸= 0 is replaced by an integral
Z
X
gV
−→
d3 p.
(2πℏ)3
p
̸⃗=0
Why? In the thermodynamic limit V → ∞, the spacing between allowed momentum states
becomes very small and the discrete sum over p⃗ can be replaced by an integral. For a particle in a
box of volume V = L3 with periodic boundary conditions,
px =
2πℏ
nx ,
L
2πℏ
ny ,
L
py =
pz =
2πℏ
nz ,
L
where nx , ny , nz are integers. The spacing between momentum states is therefore
3
2πℏ
(2πℏ)3
∆px ∆py ∆pz =
=
.
L
V
Thus each quantum state occupies a volume (2πℏ)3 /V in momentum space, and the density of
states in momentum space is
V
.
(2πℏ)3
Therefore in the thermodynamic limit we replace
Z
X
V
d3 p f (⃗p).
f (⃗p) −→
3
(2πℏ)
p
⃗
If the function depends only on the magnitude p = |⃗p|, we use spherical coordinates
d3 p = 4πp2 dp,
and obtain
X
p
⃗
V
f (p) −→
(2πℏ)3
Equivalently,
X
p
⃗
Thus
V
f (p) = 2 3
2π ℏ
gV
ln Q = − ln(1 − z) −
(2πℏ)3
Z
4πp2 dp f (p).
Z ∞
p2 f (p) dp.
0
Z
2
d3 p ln 1 − ze−βp /2m .
Using P V = kT ln Q, we obtain
PV
gV
= ln Q = − ln(1 − z) −
kT
(2πℏ)3
Z
2
d3 p ln 1 − ze−βp /2m .
Writing the momentum integral in spherical coordinates,
d3 p = 4πp2 dp,
we obtain
P
1
g
= − ln(1 − z) − 2 3
kT
V
2π ℏ
Z ∞
48
0
−βp2 /2m
p dp ln 1 − ze
.
2
For z ̸= 1 the first term is negligible in the limit of large V .
For the particle number,
⟨N ⟩ = ⟨N0 ⟩ + ⟨Nex ⟩,
⟨N0 ⟩ =
z
.
1−z
The excited particles contribute
⟨Nex ⟩ =
X
p
̸⃗=0
Hence
1
gV
−→
2 /2m
−1
βp
(2πℏ)3
z e
−1
⟨N ⟩
1 z
g
=
+ 2 3
V
V 1 − z 2π ℏ
Z ∞
p2 dp
0
Z
d3 p
1
z −1 eβp2 /2m − 1
1
z −1 eβp2 /2m − 1
.
Bose functions: Introducing the thermal wavelength
λT = √
h
,
2πmkT
and the Bose functions
gν (z) =
∞
X
zℓ
ℓ=1
the results become
and
ℓν
,
PV
gV
= − ln(1 − z) + 3 g5/2 (z),
kT
λT
1 z
g
⟨N ⟩
=
+ 3 g3/2 (z).
V
V 1 − z λT
High temperature limit: Starting from
1
1
P
= 3 g5/2 (z) − log(1 − z),
kT
λ
V
and
1
1
1 z
V
= 3 g3/2 (z) +
,
v= ,
v
λ
V 1−z
N
we now consider the high-temperature limit. As T → ∞, the thermal wavelength
λ=
2πℏ2
mkT
1/2
becomes very small, so that λ3 → 0. Hence the quantity
λ3
N λ3
=
v
V
becomes very small. This is the nondegenerate or classical limit. Now from
1
1
1 z
= 3 g3/2 (z) +
,
v
λ
V 1−z
49
.
for macroscopic V the last term is negligible, so
1
1
≃ 3 g3/2 (z).
v
λ
Therefore
λ3
≃ g3/2 (z).
v
But in the high-temperature limit, λ3 /v ≪ 1, hence
g3/2 (z) ≪ 1.
Since
g3/2 (z) = z +
z2
z3
+
+ ··· ,
23/2 33/2
this is possible only when
z ≪ 1.
Thus in the high-temperature limit the fugacity is very small. For z ≪ 1, we may keep only the
first term in the Bose functions:
g3/2 (z) ≃ z,
g5/2 (z) ≃ z,
and also
− log(1 − z) ≃ z.
Again neglecting the term of order 1/V for a macroscopic system, the pressure equation becomes
1
z
P
≃ 3 g5/2 (z) ≃ 3 .
kT
λ
λ
From the number-density equation,
1
z
1
≃ 3 g3/2 (z) ≃ 3 .
v
λ
λ
Hence
P
1
≃ .
kT
v
Since v = V /N , this gives
P =
kT
N kT
=
,
v
V
or equivalently,
P V = N kT.
Therefore, in the high-temperature limit z ≪ 1, and the Bose gas satisfies the classical ideal-gas
equation of state.
50
Average energy: For the Bose gas the internal energy in the grand ensemble is
U (z, V, T ) = −
∂
ln Q(z, V, T ),
∂β
β=
1
.
kT
The ensemble average can be written as
U (z, V, T ) =
∞
1 X
Q N =0

X
zN

X

{np⃗ }
ϵp⃗ np⃗  e−β
P
⃗ np
⃗
p
⃗ ϵp
.
p
⃗
Using

∂ −β
e
∂β
P
⃗ np
⃗
p
⃗ ϵp
= −

X
ϵp⃗ np⃗  e−β
P
⃗ np
⃗
p
⃗ ϵp
,
p
⃗
we obtain
U (z, V, T ) = −
∂
ln Q(z, V, T ).
∂β
For the Bose gas
V
g5/2 (z),
λ3
∂ V
g5/2 (z) .
U =−
∂β λ3
ln Q =
so that
This gives
U
3 kT
=
g5/2 (z).
V
2 λ3
Thus
U (z, V, T ) =
3 V kT
g5/2 (z).
2 λ3
Since for the Bose gas
P =
we obtain
16
kT
g5/2 (z),
λ3
3
U
= P,
V
2
3
U = P V.
2
Ideal Fermi gas
We consider a gas of non-interacting, spinless fermions confined in a volume V at temperature T ,
in contact with a particle reservoir of chemical potential µ. The system is described in the grand
canonical ensemble. The single-particle energy levels are labeled by momentum p⃗, with energy
ϵp⃗ .
For fermions, the Pauli exclusion principle implies that each state can be occupied by at most
one particle, so that
np⃗ = 0 or 1.
The grand partition function is given by


1
Y X

Q=
e−βnp⃗ (ϵp⃗ −µ)  ,
p
⃗
np⃗ =0
51
where β =
1
.
kB T
Performing the sum over np⃗ , we obtain
1
X
e−βnp⃗ (ϵp⃗ −µ) = 1 + e−β(ϵp⃗ −µ) .
np⃗ =0
Thus the grand partition function becomes
Y
Q=
1 + e−β(ϵp⃗ −µ) .
p
⃗
Taking the logarithm,
ln Q =
X
ln 1 + e−β(ϵp⃗ −µ) .
p
⃗
Equation of state: The grand potential is defined as
Ω = −kB T ln Q.
For a homogeneous system,
Ω = −P V,
and hence
P V = kB T ln Q.
Therefore,
P V = kB T
X
ln 1 + e−β(ϵp⃗ −µ) .
p
⃗
In the thermodynamic limit, the sum can be replaced by an integral,
Z
V
3
−β(ϵp⃗ −µ)
P V = kB T
d
p
ln
1
+
e
.
(2πℏ)3
Average occupancy: The average occupation number of a single-particle state p⃗ is
P1
−βnp⃗ (ϵp⃗ −µ)
⃗e
np⃗ =0 np
⟨np⃗ ⟩ = P1
.
−βnp⃗ (ϵp⃗ −µ)
np⃗ =0 e
Evaluating the numerator and denominator,
1
X
e−βnp⃗ (ϵp⃗ −µ) = 1 + e−β(ϵp⃗ −µ) ,
np⃗ =0
1
X
np⃗ e−βnp⃗ (ϵp⃗ −µ) = e−β(ϵp⃗ −µ) .
np⃗ =0
Thus,
e−β(ϵp⃗ −µ)
1
1
= β(ϵ −µ)
= −1 βϵp⃗
.
⟨np⃗ ⟩ =
−β(ϵp⃗ −µ)
p
⃗
z e +1
1+e
e
+1
The total average number of particles is
⟨N ⟩ =
X
⟨np⃗ ⟩ =
1
X
p
⃗
β(ϵp⃗ −µ)
p
⃗
52
e
+1
.
16.1
High temperature limit
We now consider the high-temperature limit of the free spinless Fermi gas. In this limit the fugacity
z = eβµ
is small, and the thermodynamic quantities may be expanded in powers of z. For a free particle,
ϵp⃗ =
p2
.
2m
Starting from
ln Q =
X
ln 1 + ze−βϵp⃗ ,
p
⃗
and replacing the sum by an integral in the thermodynamic limit,
Z
X
V
→
d3 p,
3
(2πℏ)
p
⃗
we obtain
V
ln Q =
(2πℏ)3
Z
2
d3 p ln 1 + ze−βp /2m .
Using the expansion
ln(1 + x) = x −
we get
Z
V
ln Q =
(2πℏ)3
3
dp
x2 x3
+
− ··· ,
2
3
∞
X
(−1)ℓ−1
ℓ
ℓ=1
2
z ℓ e−ℓβp /2m .
Interchanging sum and integral,
∞
V X (−1)ℓ−1 ℓ
ln Q =
z
(2πℏ)3 ℓ=1
ℓ
Z
Now
Z
3
−ℓβp2 /2m
d pe
=
2
d3 p e−ℓβp /2m .
2πm
ℓβ
3/2
.
Introducing the thermal de Broglie wavelength
h
λ= √
,
2πmkB T
we obtain
∞
ln Q =
V X (−1)ℓ−1 z ℓ
.
λ3 ℓ=1
ℓ5/2
It is convenient to define the Fermi function
fν (z) =
∞
X
(−1)ℓ−1 z ℓ
ℓν
ℓ=1
Equivalently,
1
fν (z) =
Γ(ν)
Z ∞
0
53
.
xν−1 dx
.
z −1 ex + 1
Hence
ln Q =
V
f5/2 (z).
λ3
Since
P V = kB T ln Q,
we get
1
P
= 3 f5/2 (z).
kB T
λ
Next, the average number of particles is
N =z
∂
V d
ln Q = 3 z f5/2 (z).
∂z
λ dz
Using
z
d
f5/2 (z) = f3/2 (z),
dz
we obtain
N=
V
f3/2 (z).
λ3
Therefore, if
v=
V
N
is the volume per particle, then
1
1
= 3 f3/2 (z).
v
λ
For small z, the Fermi functions have the expansions
z3
z2
f3/2 (z) = z − 3/2 + 3/2 − · · · ,
2
3
and
z2
z3
f5/2 (z) = z − 5/2 + 5/2 − · · · .
2
3
Now define
x=
λ3
.
v
Then
x = f3/2 (z) = z −
z2
+ ··· .
23/2
We now invert this relation and write
z = x + ax2 + · · · .
Substituting,
(x + ax2 )2
1
x = (x + ax ) −
+ · · · = x + a − 3/2 x2 + · · · .
23/2
2
2
Hence
a=
1
23/2
54
,
so that
λ3
1
z=
+ 3/2
v
2
We now substitute this into
P
1
= 3
kB T
λ
3 2
λ
+ ··· .
v
z2
z − 5/2 + · · ·
2
.
Using
z =x+
x2
+ ··· ,
23/2
we get
1
x2
P
x2
= 3 x + 3/2 − 5/2 + · · · .
kB T
λ
2
2
Since
1
23/2
this becomes
−
1
25/2
=
1
25/2
,
1
x2
P
= 3 x + 5/2 + · · · .
kB T
λ
2
Recalling that x = λ3 /v, we find
1
1 λ3
P
=
+ ··· .
1 + 5/2
kB T
v
2
v
Equivalently,
1 λ3
P v = kB T 1 + 5/2
+ ···
2
v
1 N λ3
P V = N kB T 1 + 5/2
+ ··· .
2
V
or
Finally, the average occupation number of the state p⃗ is
⟨np⃗ ⟩ =
1
z −1 eβϵp⃗ + 1
.
In the high-temperature limit z ≪ 1, this becomes
⟨np⃗ ⟩ = ze−βϵp⃗ − z 2 e−2βϵp⃗ + · · · .
Thus, to leading order,
⟨np⃗ ⟩ ≃ ze−βϵp⃗ .
Using
1
λ3
+ 3/2
z=
v
2
3 2
λ
+ ··· ,
v
we get, to leading order,
3
λ
⟨np⃗ ⟩ ≃
e−βϵp⃗ .
v
55
16.2
Low temperature limit
We now consider the low-temperature limit of the free spinless Fermi gas. In this limit the temperature is much smaller than the Fermi temperature, and the distribution is sharply peaked around
the chemical potential. The thermodynamic quantities may then be expanded in powers of T .
For a free particle,
p2
.
ϵp⃗ =
2m
The average occupation number is
⟨np⃗ ⟩ =
1
e
β(ϵp⃗ −µ)
+1
.
At very low temperature this approaches a step function:
(
1, ϵp⃗ < µ,
⟨np⃗ ⟩ →
0, ϵp⃗ > µ.
At T = 0, the chemical potential becomes the Fermi energy ϵF . Starting from
X
ln Q =
ln 1 + e−β(ϵp⃗ −µ) ,
p
⃗
and replacing the sum by an integral,
X
p
⃗
we obtain
V
ln Q =
(2πℏ)3
V
→
(2πℏ)3
Z
Z
d3 p,
2
d3 p ln 1 + e−β(p /2m−µ) .
It is convenient to write the particle number as
N=
1
X
e
p
⃗
β(ϵp⃗ −µ)
+1
.
Passing to the continuum,
V
N=
(2πℏ)3
Z
d3 p
1
eβ(p2 /2m−µ) + 1
.
Using spherical coordinates in momentum space,
d3 p = 4πp2 dp,
and changing variable from p to ϵ = p2 /2m, we get
Z ∞
√
V
ϵ1/2
3/2
N=
4π
2
m
dϵ
.
(2πℏ)3
eβ(ϵ−µ) + 1
0
Thus
N
=A
V
Z ∞
dϵ
0
56
ϵ1/2
eβ(ϵ−µ) + 1
,
where
1
A= 2
4π
2m
ℏ2
3/2
.
To evaluate the integral at low temperature, we use the Sommerfeld expansion:
Z µ
Z ∞
π2
ϕ(ϵ)
ϕ(ϵ) dϵ + (kB T )2 ϕ′ (µ) + · · · .
dϵ β(ϵ−µ)
=
e
+1
6
0
0
Taking
1
ϕ′ (ϵ) = ϵ−1/2 ,
2
ϕ(ϵ) = ϵ1/2 ,
we obtain
N
=A
V
Z µ
ϵ
0
Since
Z µ
0
this becomes
π2
2 1
dϵ + (kB T )
+ ··· .
6
2µ1/2
1/2
2
ϵ1/2 dϵ = µ3/2 ,
3
N
2 3/2 π 2
2 −1/2
= A µ + (kB T ) µ
+ ··· .
V
3
12
At T = 0, we define the Fermi energy ϵF by
2 3/2
N
= AϵF .
V
3
Therefore,
ℏ2
(6π 2 n)2/3 ,
2m
Now we determine µ at low temperature. Write
ϵF =
µ = ϵF (1 + δ),
n=
N
.
V
δ ≪ 1.
Substituting into
2 3/2 2 3/2 π 2
ϵ = µ + (kB T )2 µ−1/2 ,
3 F
3
12
2
and keeping terms up to order T , we find
"
#
2
2
π
kB T
µ = ϵF 1 −
+ ··· .
12
ϵF
Next we calculate the pressure. Since
P V = kB T ln Q,
it is convenient to use the relation
P =
2U
3V
for a free non-relativistic gas, where
U=
X
ϵp⃗ ⟨np⃗ ⟩.
p
⃗
57
Passing to the continuum,
Z ∞
U
ϵ3/2
dϵ β(ϵ−µ)
=A
.
V
e
+1
0
Using the Sommerfeld expansion again, now with
3
ϕ′ (ϵ) = ϵ1/2 ,
2
ϕ(ϵ) = ϵ3/2 ,
we get
U
=A
V
Z µ
3/2
ϵ
0
Since
Z µ
0
we obtain
Therefore
That is,
π2
2 3 1/2
dϵ + (kB T ) µ + · · · .
6
2
2
ϵ3/2 dϵ = µ5/2 ,
5
U
2 5/2 π 2
2 1/2
= A µ + (kB T ) µ + · · · .
V
5
4
2 5/2 π 2
2
2 1/2
P = A µ + (kB T ) µ + · · · .
3
5
4
4 5/2 π 2
2 1/2
P =A
µ + (kB T ) µ + · · · .
15
6
Now substitute
"
µ = ϵF
π2
1−
12
kB T
ϵF
2
#
+ ··· .
Keeping terms up to order T 2 , this gives
"
2 #
2
5π
kB T
π2
4 5/2
1/2
+ A (kB T )2 ϵF + · · · .
P = AϵF 1 +
15
12
ϵF
6
Combining terms,
"
#
2
4 5/2
5π 2 kB T
P = AϵF 1 +
+ ··· .
15
12
ϵF
Using
2 3/2
n = AϵF ,
3
we may rewrite this as
"
#
2
2
5π 2 kB T
P = nϵF 1 +
+ ··· .
5
12
ϵF
Equivalently, if v = V /N = 1/n, then
"
#
2
2
5π 2 kB T
P v = ϵF 1 +
+ ··· .
5
12
ϵF
Thus, unlike the high-temperature limit, the pressure does not vanish as T → 0. The finite value
2
P0 = nϵF
5
58
is the degeneracy pressure.
Finally, the average occupation number in the low-temperature limit is
⟨np⃗ ⟩ =
with
"
µ = ϵF
1
e
π2
1−
12
β(ϵp⃗ −µ)
+1
2
kB T
ϵF
,
#
+ ··· .
Therefore, at low temperature,
⟨np⃗ ⟩ ≃
(
1,
ϵp⃗ < ϵF ,
0,
ϵp⃗ > ϵF ,
up to a narrow transition region of width of order kB T around ϵF . A more precise low-temperature
form is
#
"
2
1
π 2 kB T
+ ··· .
⟨np⃗ ⟩ = β(ϵ −µ)
,
µ = ϵF 1 −
12
ϵF
e p⃗
+1
T =0
T >0
1
⟨np⃗ ⟩
0.8
0.6
0.4
0.2
0
17
0
0.2 0.4 0.6 0.8
1
ϵ
1.2 1.4 1.6 1.8
2
Magnetism
The magnetisation per unit volume is defined as the response of the system to an external magnetic
field B,
1 ∂
M=
ln Q.
β ∂H
The magnetic susceptibility is defined as
∂M
χ=
,
∂H T,µ,V
and hence
1 ∂2
ln Ξ.
β ∂H 2
A material is said to be diamagnetic if its magnetic susceptibility χ < 0, and paramagnetic
if χ > 0. To understand diamagnetism in the simplest way, we consider an idealized model of a
physical system.
χ=
59
The magnetic properties of a substance arise primarily from its electrons, which may be either
bound to atoms or nearly free. When an external magnetic field is applied, two important effects
occur:
1. Orbital motion: Electrons (whether free or bound) move in quantized orbits in the presence
of a magnetic field.
2. Spin alignment: The spins of electrons tend to align parallel to the applied magnetic field.
The contribution of atomic nuclei to magnetism is negligible. This is because nuclei are much
more massive than electrons, so their orbital motion is insignificant, and their intrinsic magnetic
moments are typically about 10−3 times smaller than those of electrons. Their influence is mainly
indirect, through modifications of the electronic wavefunctions.
The alignment of electron spins with the external field leads to paramagnetism, while changes
in the orbital motion of electrons give rise to diamagnetism. In real materials, both effects are
present and compete with each other.
We discuss paramagnetism here.
17.1
Paramagnetism
The Hamiltonian of a nonrelativistic free electron in an external magnetic field B is
e 2
1 p + A − µ0 σ · H,
H=
2m
c
eℏ
where µ0 = 2mc
and σ are the Pauli matrices. The first term describes the orbital motion of
the electron and leads to diamagnetism, while the second term represents the interaction of the
electron spin with the magnetic field and gives rise to paramagnetism.
To isolate the paramagnetic contribution, we neglect the orbital term and consider the singleparticle Hamiltonian
p2
− µ0 σ · H.
H=
2m
Since the eigenvalues of σ · H are sH with s = ±1, the single-particle energy levels are
ϵp,s =
p2
− s µ0 H.
2m
For a system of N particles, an energy eigenstate can be specified by occupation numbers np,s
of the single-particle states ϵp,s , where
X
np,s = 0, 1,
np,s = N.
p,s
The total energy is then
E=
X
ϵp,s np,s =
p,s
X p2
p
2m
− µ0 H np,+1 +
p2
+ µ0 H np,−1 .
2m
Defining
np,−1 ≡ n−
p,
np,+1 ≡ n+
p,
and
N+ =
X
p
n+
p,
N− =
X
p
60
n−
p,
N = N+ + N− ,
the total energy can be written as
E=
X
−
(n+
p + np )
p
p2
− µ0 H(N+ − N− ).
2m
The partition function is
ZN =
"
′
X
exp −β
X
−
{n+
p },{np }
− p
(n+
p + np )
2m
p
#
2
+ βµ0 H(N+ − N− ) ,
where the prime on the sum indicates the constraint
X
X
n+
+
n−
p
p = N.
p
p
We evaluate the sum as follows. First fix an integer N+ , and sum over all configurations such
that
X
X
n+
n−
p = N+ ,
p = N − N+ .
p
p
Then sum over all possible values N+ = 0, 1, . . . , N . This gives
! ′′′
!
N
′′
X
X p2
X
X
X p2
ZN =
eβµ0 H(2N+ −N )
exp −β
n+
exp −β
n−
,
p
2m
2m p
+
−
p
p
N =0
{np }
+
{np }
P′′′
P
P
P
over configurations with p n−
where ′′ is over configurations with p n+
p =
p = N+ , and
N − N+ .
(0)
Let ZN denote the partition function of an ideal Fermi gas of N spinless particles of mass m:
!
2
X
X
p
(0)
ZN =
exp −β
np = e−βA(N ) .
2m
P
p
p np =N
Then
ZN = e
−βµ0 HN
N
X
(0)
(0)
e2βµ0 HN+ ZN+ ZN −N+ .
N+ =0
Taking the logarithm,
N
X
1
1
log ZN = −βµ0 H + log
exp [2βµ0 HN+ − βA(N+ ) − βA(N − N+ )] .
N
N
N =0
+
There are N + 1 terms in the sum. For large N , the logarithm is dominated by the largest term,
giving
1
log ZN = βf (N̄+ ),
N
where
f (N̄+ ) = max f (N+ ),
N+
and
f (N+ ) = µ0 H
2N+
1
− 1 − [A(N+ ) + A(N − N+ )] .
N
N
61
Obviously, we can interpret N̄+ as the average number of particles with spin up. If N̄+ is known,
the magnetization per unit volume is
M=
µ0 (2N̄+ − N )
.
V
We now explicitly determine N̄+ . The condition for extremization is
∂f (N+ )
= 0,
∂N+ N+ =N̄+
which gives
2µ0 H −
∂A(N ′ )
∂A(N − N ′ )
−
= 0.
∂N ′ N ′ =N̄+
∂N ′
N ′ =N̄+
Let kT ν(N ) denote the chemical potential of an ideal Fermi gas of N spinless particles:
kT ν(N ) =
Then
and
∂A(N )
.
∂N
∂A(N ′ )
= kT ν(N̄+ ),
∂N ′ N ′ =N̄+
∂A(N − N ′ )
∂A(N − N ′ )
= −kT ν(N − N̄+ ).
=
−
∂N ′
∂(N − N ′ ) N −N ′ =N −N̄+
N ′ =N̄+
Thus the extremization condition becomes
kT ν(N̄+ ) − ν(N − N̄+ ) = 2µ0 H.
This condition states that at a given temperature the average number of particles with spin up is
such that the chemical potential of spin-up particles exceeds that of spin-down particles by 2µ0 H.
Let the Fermi energy of the system be
2 2/3 2
ℏ
3π N
.
ϵF (N ) =
V
2m
In the low-temperature regime (kT ≪ ϵF ), we can use the expansion
"
#
2
π2
kT
kT ν(N ) = ϵF (2N ) 1 −
+ ··· .
12 ϵF (2N )
Thus the condition becomes
π 2 (kT )2
1
1
ϵF (2N̄+ ) − ϵF (2N − 2N̄+ ) −
−
+ · · · = 2µ0 H.
12
ϵF (2N̄+ ) ϵF (2N − 2N̄+ )
Let
r=
2N̄+
− 1,
N
(−1 ≤ r ≤ 1).
Then the above equation becomes
2/3
(1 + r)
− (1 − r)
2/3
π2
−
12
kT
ϵF
2
2µ0 H
(1 + r)−2/3 − (1 − r)−2/3 + · · · =
.
ϵF
62
At absolute zero, this reduces to
(1 + r)2/3 − (1 − r)2/3 =
2µ0 H
.
ϵF
For weak fields (µ0 H ≪ ϵF ), an approximate solution is
r≈
3µ0 H
.
2ϵF
Hence,
N
N̄+ ≈
2
3µ0 H
1+
2ϵF
22/3
.
(1 + r)2/3 − (1 − r)2/3
2µB H
ϵF
Slope = 43
r
−1
+1
−22/3
Thus when H = 0, half the particles have spin up and the other half spin down. When H > 0,
the balance shifts in favor of spin up. From (11.112) and (11.121), we obtain, for absolute zero,
M=
µ0 r
3µ2 H
≈ 0 ,
v
2ϵF v
3µ20
.
2ϵF v
For 0 < kT ≪ ϵF and µ ≪ ϵF , we can solve (11.119) by expanding the left-hand side in
powers of r, and we obtain
"
2 #
3µ0 H
π 2 kT
r≈
1−
,
2ϵF
12 ϵF
"
2 #
π 2 kT
3µ20
1−
.
χ≈
2ϵF v
12 ϵF
χ≈
For high temperatures (kT ≫ ϵF ), we use
ν(N ) ≈ log
N λ3
V
.
Hence (11.115) gives
3
3
λ (1 − r)
2µ0 H
λ (1 + r)
log
− log
=
,
v
v
kT
or
r = tanh
µ0 H
kT
63
≈
µ0 H
.
kT
The magnetic susceptibility per unit volume is then
χ≈
µ20
.
kT v
We thus see a crossover between two regimes:
• At low temperatures: χ ∼ constant (Pauli paramagnetism, quantum regime)
• At high temperatures: χ ∼ T1 (Curie law, classical regime)
18
Blackbody radiation
Consider electromagnetic radiation in thermal equilibrium within a cavity of volume V maintained at temperature T . Such a system is referred to as a blackbody cavity. Experimentally, a
blackbody cavity can be realized by constructing an enclosed cavity within a material, evacuating it completely so that no matter remains inside, and then bringing the surrounding walls to a
fixed temperature. The atoms in the walls continuously emit and absorb electromagnetic radiation. As a result, after sufficient time, the radiation inside the cavity reaches thermal equilibrium
with the walls. In this equilibrium state, the cavity contains only electromagnetic radiation, whose
properties are determined solely by the temperature T . If the cavity is sufficiently large, the thermodynamic properties of the radiation become independent of the detailed nature of the walls.
Consequently, for theoretical analysis, one is free to impose any convenient boundary conditions
on the electromagnetic field (such as periodic or perfectly conducting boundary conditions), without affecting the physical results.
According to quantum theory, electromagnetic radiation is composed of particles called photons. Photons are massless bosons with spin 1. Because they are massless, they always propagate
in free space with the speed of light c. A key consequence of masslessness is that the spin of a
photon cannot take arbitrary orientations. Instead, it can have only two independent components
relative to its direction of motion: either parallel or antiparallel to the momentum. These two possibilities are known as helicity states, labeled +1 and −1. Each helicity state has a direct physical
interpretation. A photon in a definite helicity state corresponds to a plane electromagnetic wave
that is circularly polarized:
• helicity +1: right-circular polarization,
• helicity −1: left-circular polarization.
In practice, it is often more convenient to work with linearly polarized light. Such states can be
obtained by taking appropriate superpositions of the two circular polarization states. In this case,
the photon is no longer in a state of definite helicity (i.e., it is not an eigenstate of spin along the
direction of motion), but it remains a valid physical state. In the following, we shall work with
linearly polarized photon states.
For our purposes, it is sufficient to summarize the basic properties of a photon of angular
frequency ω:
• Energy:
E = ℏω
• Momentum:
p = ℏk,
64
|k| =
ω
c
• Polarization vector:
|ϵ| = 1,
ϵ,
k·ϵ=0
A photon with these properties corresponds to a plane electromagnetic wave. The electric field
of such a wave can be written as
E(r, t) = ϵ ei(k·r−ωt) .
The vector ϵ specifies the direction of the electric field. The condition
k·ϵ=0
expresses the fact that electromagnetic waves are transverse, i.e.,
∇ · E = 0.
Thus, for any given wavevector k, there are only two independent polarization directions.
To determine the allowed modes, we impose periodic boundary conditions on the field inside
a cube of volume V = L3 :
E(r + Lx̂, t) = E(r, t), etc.
This leads to quantization of the wavevector:
k=
2π
n,
L
where n = (nx , ny , nz ), and each component is an integer:
ni = 0, ±1, ±2, . . .
From the periodic boundary conditions, the allowed wavevectors are
k=
2π
n,
L
ni ∈ Z.
Thus each allowed state occupies a volume
(2π)3
V
in k-space. Therefore, the number of allowed states in a volume element d3 k is
V
d3 k.
(2π)3
In spherical coordinates, d3 k = 4πk 2 dk, so the number of allowed momentum states between k
and k + dk is
V
4πk 2 dk.
3
(2π)
Since for each k there are two independent polarization states, the total number of photon states is
2V
4πk 2 dk.
(2π)3
Since atoms can emit and absorb photons, the total number of photons is not a conserved
quantity.
65
The electromagnetic field can be viewed as a collection of independent modes labeled by
(k, ϵ). Each mode behaves like a harmonic oscillator with frequency
ω = c|k|.
If there are nk,ϵ photons in the mode (k, ϵ), then the energy of that mode is
ℏω nk,ϵ .
Hence, the total energy of the electromagnetic field is
X
E{nk,ϵ } =
ℏω nk,ϵ ,
k,ϵ
where
nk,ϵ = 0, 1, 2, . . .
since photons are bosons and any number of photons can occupy the same state.
18.1
Partition function
Since the number of photons is not conserved, there is no restriction on the occupation numbers
{nk,ϵ }. Therefore, the partition function is given by
X
L=
e−βE{nk,ϵ } .
{nk,ϵ }
Using the expression for the total energy,
E{nk,ϵ } =
X
ℏω nk,ϵ ,
k,ϵ
the partition function becomes
!
L=
X
exp −β
{nk,ϵ }
X
ℏω nk,ϵ
k,ϵ
Since different modes (k, ϵ) are independent, the sum factorizes:
L=
∞
YX
e−βℏωn .
k,ϵ n=0
Each sum is a geometric series, which evaluates to
∞
X
e−βℏωn =
n=0
Thus,
1
.
1 − e−βℏω
1
.
−βℏω
1
−
e
k,ϵ
L=
Y
log L = −
X
Taking the logarithm,
log 1 − e−βℏω .
k,ϵ
66
.
Since there are two independent polarization states for each k, this can be written as
X
log L = −2
log 1 − e−βℏω .
k
V
−→
(2π)3
X
k
so that
V
log L = −2
(2π)3
Z
Z
d3 k
d3 k log 1 − e−βℏω .
Using d3 k = 4πk 2 dk and ω = ck, this becomes
Z ∞
Z ∞
V
V
2
−βℏck
log L = −2
4πk dk log 1 − e
=− 2 3
ω 2 dω log 1 − e−βℏω .
3
(2π) 0
π c 0
One can also write this in terms of |⃗k| = k,
Z ∞
V
log L = − 2
k 2 dk log 1 − e−βℏck .
π 0
18.2
Average occupancy and equation of state
Average number of photons with momentum ⃗k is given by
X
1 X
⟨n⃗k ⟩ =
ℏω nk,ϵ
nk exp −β
L
k,ϵ
{nk,ϵ }
!
=−
1 ∂
ln L
βℏ ∂ω
Evaluating the derivative, we obtain
⟨nk ⟩ =
2
eβℏω − 1
,
where the factor of 2 arises from the two possible polarization states.
The internal energy of the radiation field is obtained from the partition function as
U =−
∂
log L.
∂β
Using the explicit form of log L, this can be written as
X
U=
ℏω ⟨nk ⟩,
k
showing that the total energy is the sum over all modes of the average energy per mode.
To compute the pressure, we express the partition function in a form where its dependence on
the volume is explicit. Using
2π
k=
n,
V = L3 ,
L
we obtain
X
−βℏc 2π|n|/V 1/3
log L = −2
log 1 − e
.
n
67
The pressure is given by the thermodynamic relation
P =
1 ∂
log L.
β ∂V
Carrying out the differentiation, we find
P =
1 X
ℏω ⟨nk ⟩.
3V k
Comparing this result with the expression for the internal energy, we obtain the equation of state
1
P V = U.
3
18.3
Planck’s distribution
We now evaluate the internal energy in the thermodynamic limit V → ∞, where the allowed
values of k become continuous. The sum over k can then be replaced by an integral using the
density of states. Thus,
Using
∂
U =−
log L.
∂β
the internal energy is given by
Z ∞
Z
Vℏ
ω3
V ℏc ∞ 3
1
U= 2 3
dω βℏω
= 2
k dk βℏck
.
π c 0
e
−1
π
e
−1
0
Starting with
Z ∞
Vℏ
π 2 c3
dω
0
use
ω=
2πc
.
λ
dω = −
2πc
dλ,
λ2
Then
and
3
ω3
,
eβℏω − 1
ω =
2πc
λ
3
=
8π 3 c3
.
λ3
Substituting,
Vℏ
π 2 c3
Z ∞
0
ω3
Vℏ
dω βℏω
= 2 3
e
−1
π c
Reversing the limits,
Vℏ
= 2 3
π c
Z ∞
dλ
0
Z 0
2πc
8π 3 c3 /λ3
− 2 dλ βℏ(2πc/λ)
.
λ
e
−1
∞
16π 4 c4
1
.
5
2πβℏc/λ
λ
e
−1
Simplifying,
Vℏ
π 2 c3
Z ∞
0
ω3
dω βℏω
= 16π 2 V ℏc
e
−1
68
Z ∞
dλ
0
1
1
5
2πβℏc/λ
λ e
−1
Equivalently,
U
ℏ
= 2 3
V
π c
Z ∞
0
ω3
dω βℏω
=
e
−1
Z ∞
dλ
0
16π 2 ℏc
1
5
2πβℏc/λ
λ
e
−1
It is convenient to define the energy density per unit frequency u(ω, T ) through
Z ∞
Z ∞
U
=
u(ω, T ) dω =
u(λ, T )dλ.
V
0
0
From the above expression, we identify
u(ω, T ) =
ω3
ℏ
π 2 c3 eβℏω − 1
or
1
8πhc
1
16π 2 ℏc
= 5 βhc/λ
5
2πβℏc/λ
λ
e
−1
λ e
−1
This function u(ω, T ) is the spectral energy density of blackbody radiation.
The function u(ω, T ) derived above is Planck’s radiation law. It gives the energy density associated with photons of frequency ω, independent of their polarization and direction of propagation.
The total energy density is obtained by integrating over all frequencies. Evaluating the integral
explicitly, we find
U
π 2 (kT )4
=
.
V
15 (ℏc)3
u(λ, T ) =
The specific heat per unit volume is obtained by differentiating the internal energy with respect
to temperature:
∂U
4π 2 k 4 T 3
.
cV =
=
∂T V
15 (ℏc)3
Thus, the specific heat grows without bound as T → ∞. This reflects the fact that the number of
photons in the cavity is not fixed and increases with temperature.
69