Canonical Ensemble

The canonical ensemble describes a system that can exchange energy with a thermal environment while its particle number and external control parameters are fixed. Its standard macroscopic controls are

$$T,\qquad V,\qquad N,$$

with Hamiltonian $H_N(V,\lambda,\ldots)$ acting on the fixed-$N$ Hilbert space.

The equilibrium state is

$$\rho_{N,\beta} = \frac{ e^{-\beta H_N} }{ Z_N },$$

where

$$Z_N(\beta,V,\lambda,\ldots) = \operatorname{Tr}_{\mathcal H_N} e^{-\beta H_N}.$$

The subscript $N$ is often suppressed, but the trace domain is part of the ensemble definition. The general operator properties of this state live in Thermal Density Operators. Here the focus is fixed-$T,N,V$ thermodynamics.

What Is Fixed and What Fluctuates?

Quantity	Canonical status
temperature $T$	externally fixed
volume $V$ and other controls	externally fixed
particle number $N$	fixed exactly
energy $E$	fluctuates
density operator	$\rho_{N,\beta}$
thermodynamic potential	Helmholtz free energy $F$

The ensemble describes a probability distribution over energy eigenstates in one fixed-particle-number sector. It does not imply that particles are exchanged with the environment. Particle exchange belongs to the grand-canonical ensemble.

Reservoir Motivation

Consider a small system $S$ weakly coupled to a much larger reservoir $B$. The combined system is isolated with total energy

$$E_{\mathrm{tot}} = E_n+E_B.$$

If $S$ occupies an energy eigenstate with energy $E_n$, the reservoir can occupy

$$\Omega_B(E_{\mathrm{tot}}-E_n)$$

compatible states. Therefore

$$p_n \propto \Omega_B(E_{\mathrm{tot}}-E_n).$$

Using

$$S_B(E) = k_{\mathrm B}\ln\Omega_B(E),$$

expand the reservoir entropy:

$$\begin{aligned} S_B(E_{\mathrm{tot}}-E_n) ={}& S_B(E_{\mathrm{tot}}) - \frac{E_n}{T} \\ &- \frac{E_n^2}{2T^2C_B} + \cdots. \end{aligned}$$

The thermodynamic identity

$$\frac{\partial S_B}{\partial E_B} = \frac{1}{T}$$

then gives

$$\Omega_B(E_{\mathrm{tot}}-E_n) \propto e^{-E_n/(k_{\mathrm B}T)} = e^{-\beta E_n}.$$

Normalizing over system states yields the canonical probabilities.

This argument assumes:

• the reservoir is much larger than the system;
• the coupling energy is negligible at leading order;
• the reservoir temperature changes negligibly across relevant system energies;
• the combined system explores or is assigned the appropriate energy shell;
• the fixed particle-number and external-parameter constraints are respected.

At strong system–bath coupling, the reduced equilibrium state need not be the bare Gibbs state of $H_S$.

Quantum Density Operator

For a spectral decomposition

$$H_N = \sum_a E_aP_a,$$

the canonical state is

$$\rho_{N,\beta} = \frac{1}{Z_N} \sum_a e^{-\beta E_a}P_a.$$

If $g_a=\operatorname{Tr}P_a$, the energy probability is

$$\Pr(E_a) = \frac{ g_a e^{-\beta E_a} }{ Z_N }.$$

The partition function is

$$Z_N = \sum_a g_a e^{-\beta E_a}.$$

The sum is over the fixed-$N$ spectrum. Including states with other particle numbers silently changes the ensemble.

The Partition Function as a Generating Object

The canonical partition function is not only a normalization. Its logarithm generates equilibrium thermodynamics. Partition Functions owns the general trace, cumulant, factorization, quantum-statistics, and path-integral structure; this section records the identities needed for fixed-$T,N,V$ thermodynamics.

Define

$$F(T,V,N,\lambda,\ldots) = -k_{\mathrm B}T\ln Z_N.$$

Equivalently,

$$\beta F = -\ln Z_N.$$

The internal energy is

$$U = \langle H_N\rangle = -\frac{\partial}{\partial\beta} \ln Z_N,$$

for a Hamiltonian with no explicit temperature or $\beta$ dependence.

Another useful identity is

$$U = \frac{\partial}{\partial\beta} \left( \beta F \right).$$

The entropy follows from

$$S = -\left( \frac{\partial F}{\partial T} \right)_{V,N,\lambda}$$

and equals

$$S = k_{\mathrm B} \left( \ln Z_N+\beta U \right).$$

These relations use the natural logarithm.

Helmholtz Free Energy

The Helmholtz free energy is the thermodynamic potential appropriate to fixed $T$, $V$, and $N$:

$$F = U-TS.$$

At equilibrium, the Gibbs state minimizes the state functional

$$\mathcal F_T(\rho) = \operatorname{Tr}(\rho H_N) - TS(\rho)$$

over normalized density operators on $\mathcal H_N$.

The equilibrium value is

$$\mathcal F_T(\rho_{N,\beta}) = F = -k_{\mathrm B}T\ln Z_N.$$

Free energy balances energy against entropy. A low-energy state is favored energetically; a large set of accessible states is favored entropically.

Generalized Forces

Let the Hamiltonian depend on an external parameter $\lambda$. Even when $H$ and $\partial_\lambda H$ do not commute, cyclicity of the trace gives

$$\frac{\partial F}{\partial\lambda} = \left\langle \frac{\partial H}{\partial\lambda} \right\rangle_\beta.$$

If the physical generalized force is defined by

$$\mathcal X_\lambda = -\frac{\partial H}{\partial\lambda},$$

then

$$\langle\mathcal X_\lambda\rangle = -\frac{\partial F}{\partial\lambda}.$$

Examples include:

$$P = -\left( \frac{\partial F}{\partial V} \right)_{T,N}$$

for pressure and

$$M = -\left( \frac{\partial F}{\partial B} \right)_{T,N}$$

for magnetization when the Hamiltonian contains the appropriate magnetic-field coupling.

Sign conventions follow the definition of the Hamiltonian term. They should be checked rather than memorized independently of $H$.

Differential Form

For a simple system,

$$dF = -S\,dT - P\,dV + \mu\,dN.$$

The canonical ensemble holds $N$ fixed during fluctuations, but $\mu$ can still be defined by comparing neighboring fixed-$N$ systems:

$$\mu = \left( \frac{\partial F}{\partial N} \right)_{T,V}$$

in a thermodynamic or smoothly interpolated description.

At finite integer $N$, addition and removal chemical potentials are naturally finite differences:

$$\begin{aligned} \mu_+ &= F_{N+1}-F_N, \\ \mu_- &= F_N-F_{N-1}. \end{aligned}$$

They need not coincide in a finite system or across a gap.

Energy Fluctuations

The second $\beta$ derivative of $\ln Z_N$ is

$$\frac{\partial^2}{\partial\beta^2} \ln Z_N = \langle H_N^2\rangle - \langle H_N\rangle^2.$$

Therefore

$$\operatorname{Var}(E) = -\frac{\partial U}{\partial\beta}.$$

For a temperature-independent Hamiltonian,

$$C_V \equiv \left( \frac{\partial U}{\partial T} \right)_{V,N} = k_{\mathrm B}\beta^2 \operatorname{Var}(E).$$

Thus

$$C_V\geq0$$

in the canonical ensemble under these assumptions.

This does not prove that every definition of heat capacity in every ensemble is nonnegative. Microcanonical and nonadditive long-range systems can require separate analysis.

Energy Distribution

The canonical probability of energy $E_a$ is

$$P_\beta(E_a) = \frac{ g_a e^{-\beta E_a} }{ Z_N }.$$

Its mean and variance are

$$\begin{aligned} \overline E &= U, \\ \overline{ (E-U)^2 } &= k_{\mathrm B}T^2 C_V. \end{aligned}$$

For ordinary additive systems away from criticality,

$$U \sim N, \qquad \operatorname{Var}(E) \sim N.$$

Hence the relative energy width scales as

$$\frac{ \sqrt{\operatorname{Var}(E)} }{ U } \sim \frac{1}{\sqrt N}.$$

Macroscopic energy density becomes sharply concentrated even though total energy still fluctuates.

Near criticality, long-range correlations can modify simple scaling. At first-order coexistence, the energy distribution can become bimodal. Finite-size data should be inspected rather than forced into a Gaussian assumption.

Density-of-States Representation

Let $\Omega_N(E)$ count fixed-$N$ states near energy $E$. The partition function is the Laplace transform

$$Z_N(\beta) = \int dE\, \Omega_N(E)e^{-\beta E},$$

or the corresponding discrete sum.

Writing

$$S_N(E) = k_{\mathrm B}\ln\Omega_N(E)$$

gives

$$Z_N(\beta) = \int dE\, \exp \left[ \frac{S_N(E)}{k_{\mathrm B}} - \beta E \right].$$

For a large system, a saddle point $E_*$ satisfies

$$\left. \frac{\partial S_N}{\partial E} \right|_{E_*} = \frac{1}{T}.$$

At leading order,

$$F \approx E_*-TS_N(E_*).$$

This is the Legendre relation between microcanonical entropy and canonical free energy. Its validity can fail or require convexification in nonadditive systems and coexistence regions.

Factorization for Independent Subsystems

Suppose

$$H = H_A\otimes\mathbb I_B + \mathbb I_A\otimes H_B$$

on

$$\mathcal H = \mathcal H_A\otimes\mathcal H_B.$$

The terms commute, so

$$e^{-\beta H} = e^{-\beta H_A} \otimes e^{-\beta H_B}.$$

Therefore

$$Z_{AB} = Z_AZ_B,$$

and

$$F_{AB} = F_A+F_B.$$

This factorization assumes a tensor-product decomposition with independent Hamiltonians. Identical-particle symmetrization, global particle-number constraints, or interactions can prevent a naive product of one-particle partition functions.

Canonical Ensemble and Identical Particles

For $N$ identical particles, the trace must be taken over the appropriate symmetric or antisymmetric $N$-particle subspace:

$$Z_N^{(\pm)} = \operatorname{Tr}_{\mathcal H_N^{(\pm)}} e^{-\beta H_N}.$$

One should not begin with an $N$-fold labeled-particle trace and forget to impose exchange symmetry.

For a dilute classical gas, a factor of $1/N!$ emerges in the appropriate limit. For a quantum-degenerate gas, Bose–Einstein or Fermi–Dirac statistics must be treated directly.

The canonical ensemble fixes $N$ exactly. It can still describe identical particles, occupation numbers, and exchange correlations.

Finite-Size Analyticity

For a finite-dimensional Hamiltonian $H(g)$ analytic in a real parameter $g$,

$$Z_N(\beta,g) = \operatorname{Tr} e^{-\beta H(g)} > 0$$

at finite positive temperature. Therefore

$$F_N(\beta,g) = -\frac{1}{\beta} \ln Z_N(\beta,g)$$

is analytic under the stated assumptions.

Finite systems can have sharp peaks, rapid crossovers, and bimodal distributions. A genuine thermal nonanalyticity requires an infinite-system or another singular limit.

Zero temperature is different: taking $\beta\to\infty$ can expose a finite-system level crossing in the ground-state energy.

Ensemble Equivalence and Its Limits

The canonical ensemble allows energy fluctuations; the microcanonical ensemble fixes an energy shell. For short-range additive systems in a regular thermodynamic limit, local observables often agree between the two ensembles when their mean energies are matched.

The reason is concentration:

$$\frac{\Delta E}{U} \sim N^{-1/2}.$$

Equivalence is not automatic for:

• finite systems;
• long-range nonadditive interactions;
• phase coexistence;
• observables sensitive to global fluctuations;
• constrained sectors;
• nonconcave entropy regions.

“Thermodynamic limit” should not be used as a universal spell that erases ensemble definitions. Ensemble Equivalence gives the full concentration, entropy-concavity, local-state, coexistence, and long-range analysis.

Temperature-Dependent Effective Hamiltonians

The familiar identity

$$U = -\partial_\beta\ln Z$$

assumes $H$ has no explicit $\beta$ dependence. If an effective Hamiltonian is written as $H(\beta)$, then

$$-\frac{\partial}{\partial\beta} \ln Z = \left\langle H + \beta \frac{\partial H}{\partial\beta} \right\rangle.$$

The extra term cannot be dropped.

Temperature-dependent effective Hamiltonians can arise after integrating out degrees of freedom or fitting phenomenological parameters. Their thermodynamic consistency must be derived from the underlying model rather than assumed from formal resemblance.

Example: One Two-Level System

Let

$$H = \Delta|1\rangle\langle1|, \qquad \Delta>0.$$

With

$$x = \beta\Delta,$$

the partition function is

$$Z_1 = 1+e^{-x}.$$

The free energy and internal energy are

$$\begin{aligned} F_1 &= -k_{\mathrm B}T \ln(1+e^{-x}), \\ U_1 &= \frac{\Delta}{ e^x+1 }. \end{aligned}$$

The energy variance is

$$\operatorname{Var}(E) = \Delta^2 \frac{e^x}{ (1+e^x)^2 }.$$

Therefore

$$\frac{C_1}{k_{\mathrm B}} = x^2 \frac{e^x}{ (1+e^x)^2 }.$$

The heat capacity vanishes as $T\to0$ and $T\to\infty$ and has a finite-temperature Schottky peak.

Example: Independent Two-Level Sites

For $L$ distinguishable noninteracting copies,

$$H_L = \sum_{j=1}^{L} \Delta |1\rangle_j\langle1|.$$

Factorization gives

$$Z_L = (1+e^{-x})^L.$$

Consequently,

$$\begin{gathered} F_L = L F_1, \\ U_L = L U_1, \\ C_L = L C_1. \end{gathered}$$

The energy standard deviation grows as $\sqrt L$, while the mean energy grows as $L$ at fixed finite temperature. Relative energy fluctuations therefore scale as $L^{-1/2}$.

Example: Harmonic Oscillator

For

$$H = \hbar\omega \left( a^\dagger a+\frac12 \right),$$

define

$$x = \beta\hbar\omega.$$

The partition function is

$$Z = \frac{ e^{-x/2} }{ 1-e^{-x} }.$$

The internal energy is

$$U = \hbar\omega \left( \frac12 + \frac{1}{e^x-1} \right).$$

The heat capacity is

$$\frac{C}{k_{\mathrm B}} = x^2 \frac{e^x}{ (e^x-1)^2 }.$$

As $T\to0$, $U\to\hbar\omega/2$ and $C\to0$. At high temperature, $U$ approaches $k_{\mathrm B}T$ and $C$ approaches $k_{\mathrm B}$.

Canonical Boundaries

This page owns:

• the fixed-$T,V,N$ ensemble;
• the reservoir motivation for Boltzmann weights;
• $Z_N$ as a fixed-sector trace;
• Helmholtz free energy and its derivatives;
• canonical energy fluctuations and heat capacity;
• the density-of-states transform and large-system saddle;
• factorization and ensemble-equivalence caveats.

Other pages own:

• the general operator structure and temperature limits of the Gibbs state: Thermal Density Operators;
• partition functions as general spectral and generating objects: Partition Functions;
• natural variables, Legendre transforms, Gibbs free energy, and stability identities: Thermodynamic Potentials;
• thermal entropy versus information and entanglement entropy: Entropy in Quantum Statistical Mechanics;
• the general constrained-entropy derivation and multiplier duality: Maximum Entropy Principle;
• the unified static fluctuation–response dictionary and noncommuting correction: Fluctuations and Susceptibilities;
• variable-particle-number traces and number fluctuations: Grand-Canonical Ensemble;
• finite-system addition and removal chemical potentials: Chemical Potential;
• isolated energy shells and microcanonical entropy: Microcanonical Ensemble;
• open-system heat, work, and entropy production: Quantum Thermodynamics;
• Bose–Einstein and Fermi–Dirac occupation laws: Quantum Statistics and Ideal Gases.

Common Mistakes

• Forgetting that the trace is restricted to fixed $N$.
• Calling a variable-particle-number calculation canonical.
• Treating $Z$ as only a normalization and missing its derivative structure.
• Using $\log_{10}$ instead of the natural logarithm.
• Dropping degeneracy factors in an energy-level sum.
• Multiplying one-particle partition functions for identical quantum particles without exchange corrections.
• Inferring a preparation or thermalization mechanism from the equilibrium ensemble.
• Assuming the reservoir derivation remains exact at strong coupling.
• Using $U=-\partial_\beta\ln Z$ when $H$ depends explicitly on $\beta$.
• Forgetting what is held fixed in a thermodynamic derivative.
• Assuming every generalized-force sign convention is the same.
• Concluding that canonical heat capacity can be negative for a temperature-independent finite Hamiltonian.
• Calling a finite-size peak a phase transition without scaling.
• Assuming canonical and microcanonical ensembles are always equivalent.
• Confusing total energy fluctuations with uncertainty in a pure-state energy measurement.

Exercises

Two-level heat capacity

Starting from

$$Z=1+e^{-\beta\Delta},$$

derive the heat capacity and show its low- and high-temperature limits.

Solution

The mean energy is

$$U = \frac{\Delta}{ e^{\beta\Delta}+1 }.$$

Let $x=\beta\Delta$. Differentiating with respect to $T$ gives

$$\frac{C}{k_{\mathrm B}} = x^2 \frac{e^x}{ (1+e^x)^2 }.$$

For $x\to\infty$,

$$\frac{C}{k_{\mathrm B}} \sim x^2e^{-x} \longrightarrow 0.$$

For $x\to0$,

$$\frac{C}{k_{\mathrm B}} \sim \frac{x^2}{4} \longrightarrow 0.$$

Shift the energy zero

If $H'=H+C\mathbb I$, determine how $\rho_\beta$, $Z$, $F$, $U$, $S$, and $C_V$ change.

Solution

The partition function changes by

$$Z' = e^{-\beta C}Z.$$

The normalized state is unchanged:

$$\rho_\beta' = \rho_\beta.$$

The free energy and internal energy shift:

$$F'=F+C, \qquad U'=U+C.$$

The entropy is

$$S' = \frac{U'-F'}{T} = S,$$

and $C_V=\partial U/\partial T$ is unchanged when $C$ is temperature independent.

Independent-site factorization

For $L$ independent two-level sites with energies $0$ and $\Delta$, derive $Z_L$, $F_L$, and the probability that exactly $m$ sites are excited.

Solution

Each site has

$$Z_1 = 1+e^{-\beta\Delta}.$$

Independence gives

$$Z_L = Z_1^L, \qquad F_L = L F_1.$$

There are $\binom{L}{m}$ configurations with $m$ excitations, each of energy $m\Delta$. Hence

$$\Pr(m) = \binom{L}{m} \frac{ e^{-\beta m\Delta} }{ (1+e^{-\beta\Delta})^L }.$$

This is a binomial distribution with single-site excitation probability $1/(e^{\beta\Delta}+1)$.

Fluctuation identity

Show that

$$\operatorname{Var}(E) = \frac{\partial^2}{\partial\beta^2} \ln Z.$$

Solution

First,

$$\frac{\partial}{\partial\beta} \ln Z = -\langle H\rangle.$$

Differentiating again,

$$\begin{aligned} \frac{\partial^2}{\partial\beta^2} \ln Z &= \frac{1}{Z} \operatorname{Tr} \left( H^2e^{-\beta H} \right) \\ &\quad- \left[ \frac{1}{Z} \operatorname{Tr} \left( He^{-\beta H} \right) \right]^2 \\ &= \langle H^2\rangle - \langle H\rangle^2. \end{aligned}$$

Canonical versus grand canonical

A calculation sums over states with $N=0,1,2,\ldots$ and weights each state by $e^{-\beta(E-\mu N)}$. Is it canonical? Identify the correct ensemble and explain what must change to make the calculation canonical.

Solution

It is grand canonical because particle number varies and the chemical potential appears in the weight.

To make the calculation canonical, choose one particle number $N$ and restrict the trace to $\mathcal H_N$:

$$Z_N = \operatorname{Tr}_{\mathcal H_N} e^{-\beta H_N}.$$

The chemical-potential term is then unnecessary because $N$ is fixed. Different fixed-$N$ partition functions can be compared afterward to define addition or removal chemical potentials.

Cross-Links

• Statistical Ensembles Overview
• Microcanonical Ensemble
• Grand-Canonical Ensemble
• Partition Functions
• Thermodynamic Potentials
• Entropy in Quantum Statistical Mechanics
• Maximum Entropy Principle
• Chemical Potential
• Ensemble Equivalence
• Thermal Density Operators
• Ensemble Formula Sheet
• Thermodynamic Limit
• Trace Rule for Expectation Values
• Entropy Overview
• Density of States
• Quantum Thermodynamics
• Statistical Mechanics Checklist

References

• R. K. Pathria and P. D. Beale, Statistical Mechanics, 3rd ed., Elsevier (2011).
• K. Huang, Statistical Mechanics, 2nd ed., Wiley (1987).
• M. Kardar, Statistical Physics of Particles, Cambridge University Press (2007).
• L. D. Landau and E. M. Lifshitz, Statistical Physics, Part 1, 3rd ed., Butterworth–Heinemann (1980).
• H. B. Callen, Thermodynamics and an Introduction to Thermostatistics, 2nd ed., Wiley (1985).
• L. E. Reichl, A Modern Course in Statistical Physics, 4th ed., Wiley–VCH (2016).