Statistical Ensembles Overview

A statistical ensemble is a rule for assigning a quantum state when only selected macroscopic constraints are controlled. The ensemble determines a density operator, the Hilbert space over which its trace is taken, the quantities that may fluctuate, and the thermodynamic potential adapted to those controls. Use the Quantum Statistical Mechanics gateway for the chapter-wide dependency order and shorter goal routes.

The three standard equilibrium ensembles answer three different physical questions:

• microcanonical: what state represents an isolated system with energy restricted to a narrow shell?
• canonical: what state represents a system that exchanges energy with a reservoir at fixed temperature?
• grand canonical: what state represents a system that exchanges both energy and a conserved particle number with a reservoir?

These are not three microscopic laws. They are three descriptions built from the same Hamiltonian under different constraints and exchange conditions.

Core Comparison

For a system with volume $V$, Hamiltonian $H$, and conserved number operator $N$, the basic map is:

Ensemble	Controlled quantities	Quantities that fluctuate	Density operator	Trace space	Natural potential
microcanonical	$E,N,V$	energy within the chosen shell	$\rho_{\mathrm{mc}}=\Pi_{E,\Delta}/W$	fixed-$N$ space $\mathcal H_N$	entropy $S(E,N,V)$
canonical	$T,N,V$	energy	$\rho_{\mathrm c}=e^{-\beta H_N}/Z_N$	fixed-$N$ space $\mathcal H_N$	Helmholtz free energy $F(T,N,V)$
grand canonical	$T,\mu,V$	energy and particle number	$\rho_{\mathrm{gc}}=e^{-\beta(H-\mu N)}/\Xi$	Fock space $\mathcal F$	grand potential $\Omega_{\mathrm G}(T,\mu,V)$

Here

$$\beta = \frac{1}{k_{\mathrm B}T}.$$

The symbol $W$ denotes the number of states in the microcanonical shell. The separate notation $\Omega_{\mathrm G}$ is used for the grand potential so that state counting and thermodynamic potentials are not confused.

The ensemble follows the permitted exchanges. An isolated system has no reservoir exchange; a canonical system exchanges energy; a grand-canonical system exchanges both energy and a conserved particle number. Volume and other external controls can be generalized separately.

What an Ensemble Is

In quantum mechanics, an ensemble is represented operationally by a density operator $\rho$. For an observable $A$,

$$\langle A\rangle_\rho = \operatorname{Tr}(\rho A).$$

For a projective measurement with spectral projectors $P_a$,

$$\Pr(a) = \operatorname{Tr}(\rho P_a).$$

The ensemble therefore specifies the state used in the Born rule. It does not replace the Born rule, and it is not merely a classical list of hidden microscopic states.

The word ensemble is used in two compatible ways:

1. as a probability assignment over allowed microscopic states;
2. as an imagined collection of identically prepared systems realizing those probabilities.

The first meaning is fundamental for calculation. A laboratory need not contain infinitely many literal copies of the system.

The Constraint–State–Potential Pattern

An equilibrium calculation has three linked layers.

Constraints

First identify which extensive quantities are fixed exactly and which can be exchanged with an environment. Energy exchange introduces temperature as a control variable. Exchange of a conserved particle number introduces chemical potential.

State

The constraints determine a normalized density operator. For exponential ensembles this has the general form

$$\rho = \frac{ \exp\!\left( -\sum_j\lambda_j Q_j \right) }{ \mathcal Z },$$

where the $Q_j$ are constrained observables and the $\lambda_j$ are their conjugate multipliers.

Potential

The logarithm of the normalization generates equilibrium thermodynamics in the variables held fixed by that ensemble. The appropriate potential is not decorative notation: it is the function whose natural variables match the experimental controls.

This pattern is summarized by

$$\text{constraints} \longrightarrow \rho \longrightarrow \mathcal Z \longrightarrow \text{potential and responses}.$$

The detailed derivative identities live in the pages for each ensemble and in the Ensemble Formula Sheet.

Microcanonical Ensemble

The microcanonical ensemble represents an isolated system whose energy is known only to lie in a narrow interval. At fixed particle number, define the spectral projector

$$\Pi_{E,\Delta} = \mathbf 1_{[E,E+\Delta]}(H_N).$$

The number of states in the shell is

$$W(E,\Delta,N,V) = \operatorname{Tr}_{\mathcal H_N} \Pi_{E,\Delta}.$$

The normalized microcanonical state is

$$\rho_{\mathrm{mc}} = \frac{ \Pi_{E,\Delta} }{ W(E,\Delta,N,V) }.$$

It assigns equal weight to the states in the selected shell:

$$\rho_{\mathrm{mc}}\Pi_{E,\Delta} = \rho_{\mathrm{mc}}.$$

The associated Boltzmann entropy is

$$S_{\mathrm{mc}} = k_{\mathrm B}\ln W.$$

Why an energy shell?

For a finite system, the energy spectrum is discrete. Requiring one exact energy may select a single nondegenerate eigenstate, which is usually too narrow to represent a macroscopic energy specification. A shell width $\Delta$ is chosen so that:

• $\Delta$ is small on the macroscopic energy scale;
• the shell contains many relevant states;
• macroscopic observables are insensitive to modest changes of $\Delta$.

These conditions are limits and modeling choices, not automatic facts for every finite Hamiltonian.

What is fixed?

In the ideal microcanonical construction, $N$, $V$, and the energy shell are fixed. The measured energy can still vary among eigenvalues inside the shell when the shell contains more than one energy. Thus “fixed energy” means restricted to a macroscopically narrow window, not necessarily a delta function at one eigenvalue.

Temperature is derived

Temperature is not an externally imposed variable in the microcanonical ensemble. It is inferred from the entropy:

$$\frac{1}{T} = \left( \frac{\partial S}{\partial E} \right)_{N,V}.$$

Similarly,

$$\frac{P}{T} = \left( \frac{\partial S}{\partial V} \right)_{E,N}, \qquad -\frac{\mu}{T} = \left( \frac{\partial S}{\partial N} \right)_{E,V}.$$

These thermodynamic derivatives require an appropriate smooth large-system description. For a small discrete spectrum, finite differences and shell conventions matter.

Canonical boundary

This overview defines the shell construction only. State counting, entropy definitions, finite-size choices, and isolated-system subtleties belong to the dedicated Microcanonical Ensemble page.

Canonical Ensemble

The canonical ensemble represents a system that exchanges energy with a large thermal reservoir while particle number and external controls remain fixed. Its state is

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

with

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

The trace is taken in one fixed-$N$ Hilbert space. Energy eigenstates receive Boltzmann weights:

$$p_n = \frac{ e^{-\beta E_n} }{ Z_N }.$$

The natural potential is the Helmholtz free energy,

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

For a simple compressible system,

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

The canonical ensemble fixes $T$, not the instantaneous energy. Its energy variance is

$$\operatorname{Var}_{\mathrm c}(H) = k_{\mathrm B}T^2 C_V,$$

when $H$ has no explicit temperature dependence and the derivative defining $C_V$ is taken at fixed $N$ and $V$.

The general Gibbs-operator structure belongs to Thermal Density Operators. Reservoir derivations, fixed-$N$ trace conventions, free-energy identities, and energy fluctuations belong to the Canonical Ensemble.

Grand-Canonical Ensemble

The grand-canonical ensemble represents a system that exchanges both energy and a conserved particle number with a reservoir. Define

$$K \equiv H-\mu N.$$

The equilibrium state is

$$\rho_{\mathrm{gc}} = \frac{ e^{-\beta K} }{ \Xi } = \frac{ e^{-\beta(H-\mu N)} }{ \Xi },$$

where

$$\Xi(\beta,\mu,V) = \operatorname{Tr}_{\mathcal F} e^{-\beta(H-\mu N)}.$$

The trace normally runs over Fock space,

$$\mathcal F = \bigoplus_{N=0}^{\infty} \mathcal H_N.$$

For an equilibrium chemical potential associated with particle number, the standard construction assumes

$$[H,N]=0.$$

The natural potential is

$$\Omega_{\mathrm G} = -k_{\mathrm B}T\ln\Xi.$$

Its differential is

$$d\Omega_{\mathrm G} = -S\,dT -P\,dV -\overline N\,d\mu.$$

The fugacity

$$z \equiv e^{\beta\mu}$$

organizes the fixed-number sectors:

$$\Xi = \sum_{N=0}^{\infty} z^N Z_N.$$

Both energy and particle number fluctuate. In particular,

$$\operatorname{Var}(N) = \frac{1}{\beta} \left( \frac{\partial\overline N}{\partial\mu} \right)_{T,V}.$$

The operator $K=H-\mu N$ controls statistical weights. It is not automatically the physical energy operator that generates real-time evolution. The full Fock-space construction, convergence conditions, sector decomposition, and number fluctuations belong to the Grand-Canonical Ensemble. The derivative, finite-difference, conserved-charge, and sign-convention meanings of $\mu$ belong to Chemical Potential.

One Hamiltonian, Different Trace Domains

Choosing an ensemble involves more than changing an exponential.

In a canonical calculation with exactly $N$ identical particles,

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

where $\mathcal H_N$ is already the bosonic or fermionic $N$-particle space.

In a grand-canonical calculation,

$$\Xi = \operatorname{Tr}_{\mathcal F} e^{-\beta(H-\mu N)},$$

and the trace includes multiple number sectors.

The same formal expression can describe the wrong ensemble if the trace domain is wrong. Writing $e^{-\beta H}$ does not by itself specify whether $N$ is fixed.

Exchange Conditions and Reservoirs

The three ensembles can be organized by placing the system inside a larger isolated “universe.”

Isolated total system

If no energy or particles cross the boundary, the total system is described by a microcanonical shell:

$$E_{\mathrm{tot}}, \qquad N_{\mathrm{tot}}, \qquad V_{\mathrm{tot}} \quad \text{fixed}.$$

Energy reservoir

Let a small subsystem $S$ exchange energy with a much larger bath $B$. For a subsystem energy $E_n$,

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

Expanding the bath entropy to first order gives

$$\begin{aligned} S_B(E_{\mathrm{tot}}-E_n) \simeq{}& S_B(E_{\mathrm{tot}}) \\ &-\frac{E_n}{T}. \end{aligned}$$

Therefore

$$p_n \propto e^{-\beta E_n},$$

which yields the canonical ensemble after normalization.

Energy and particle reservoir

If particles can also move between $S$ and $B$, then the bath entropy expansion includes

$$dS_B = \frac{1}{T}dE_B -\frac{\mu}{T}dN_B.$$

Because

$$dE_B=-dE_S, \qquad dN_B=-dN_S,$$

the subsystem weight becomes

$$p_{n,N} \propto \exp\!\left[ -\beta(E_{n,N}-\mu N) \right].$$

This is the grand-canonical weight.

These reservoir arguments require a bath large enough that its intensive variables change negligibly, weak enough coupling that subsystem energy and particle number are meaningful, and an equilibrium assumption for the composite system.

Maximum Entropy View

The same exponential states follow from constrained entropy maximization. The von Neumann entropy with thermodynamic units is

$$S(\rho) = -k_{\mathrm B} \operatorname{Tr} \left( \rho\ln\rho \right).$$

Maximizing $S(\rho)$ subject to

$$\operatorname{Tr}\rho=1, \qquad \operatorname{Tr}(\rho H)=U$$

gives

$$\rho \propto e^{-\beta H}.$$

Adding the mean-number constraint

$$\operatorname{Tr}(\rho N) = \overline N$$

gives

$$\rho \propto e^{-\beta(H-\mu N)}.$$

This establishes the least-committal equilibrium state consistent with the stated expectation values. It does not by itself prove that a particular isolated system dynamically approaches that state. Thermalization is a separate dynamical question. Maximum Entropy Principle owns the general optimization, multiplier duality, exact-versus-mean constraint distinction, and noncommuting-constraint caveats.

Partition Functions as Generating Objects

The normalization factors contain more than normalization.

For the canonical ensemble,

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

and

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

For the grand-canonical ensemble,

$$\Xi = \operatorname{Tr}_{\mathcal F} e^{-\beta(H-\mu N)},$$

and

$$\frac{1}{\beta} \left( \frac{\partial\ln\Xi}{\partial\mu} \right)_{\beta,V} = \overline N.$$

Second derivatives generate fluctuations and susceptibilities. Derivatives must be taken while holding the correct natural variables fixed. In particular, holding fugacity $z$ fixed is not the same operation as holding chemical potential $\mu$ fixed because

$$z=e^{\beta\mu}.$$

The dedicated Partition Functions page owns the systematic trace, factorization, and generating-function treatment. Fluctuations and Susceptibilities develops the static response interpretation, ensemble dependence, and noncommuting quantum correction.

Ensemble Choice Is a Physical Choice

Convenience matters, but the physical constraints come first.

Use the microcanonical ensemble when

• the system is isolated to the desired accuracy;
• total energy and conserved particle number are sharply constrained;
• state counting or entropy as a function of energy is central;
• the finite-width energy shell is specified.

Use the canonical ensemble when

• the system exchanges energy with a thermal environment;
• temperature is controlled;
• particle number is fixed;
• the Helmholtz free energy is the natural potential.

Use the grand-canonical ensemble when

• both energy and a conserved particle number can be exchanged;
• temperature and chemical potential are controlled;
• occupation-number factorization is useful;
• local subsystems are studied inside a much larger number-conserving system.

Do not force an equilibrium ensemble when

• the preparation is a known pure state;
• the system is undergoing a quench or driven evolution;
• additional conserved quantities remain relevant;
• strong system–bath coupling invalidates the bare subsystem Gibbs state;
• no normalizable partition function exists.

An equilibrium ensemble can still approximate selected observables in some of these situations, but that is a conclusion to test, not a definition to assume.

A Selection Workflow

For a new problem:

1. Write the Hamiltonian and conserved charges. State whether $[H,N]=0$ and identify other exact constraints.
2. Draw the system boundary. Decide whether energy, particles, volume, magnetization, or other quantities cross it.
3. List controlled intensive variables. Examples include $T$, $\mu$, pressure, and external field.
4. Choose the trace space. Distinguish a fixed-$N$ Hilbert space from Fock space.
5. Write the normalized state. Verify positivity, unit trace, and convergence of the normalization.
6. Match the thermodynamic potential. Use entropy, Helmholtz free energy, or grand potential according to the natural variables.
7. Check the finite-size regime. Do not invoke ensemble equivalence before identifying the thermodynamic limit.
8. Validate against the preparation. An ensemble is useful only if its constraints and observables match the physical question.

Examples

A spin in a thermal environment

Consider

$$H = -h\sigma_z.$$

If the spin is weakly coupled to a heat bath at temperature $T$, the canonical state is

$$\rho_{\mathrm c} = \frac{ e^{\beta h\sigma_z} }{ 2\cosh(\beta h) }.$$

The energies fluctuate between $-h$ and $+h$. Calling the state microcanonical merely because the Hamiltonian has discrete eigenvalues would be incorrect; the bath and controlled temperature select the canonical ensemble.

If the spin is isolated and prepared in one exact nondegenerate energy eigenstate, its exact-energy microcanonical projector is that pure state. This finite example shows why the thermodynamic meaning of a “microcanonical shell” is richer in a large many-body spectrum.

A fixed-number optical lattice

Suppose $N$ atoms occupy a lattice described by a number-conserving Hamiltonian. A calculation for the complete isolated cloud at fixed $N$ should use the fixed-number sector:

$$\mathcal H_N.$$

A small region of the cloud can exchange particles with the remainder. Its reduced equilibrium state may be well approximated by a grand-canonical state with an effective $\mu$, even though the total cloud has exactly fixed particle number.

There is no contradiction. Global and subsystem constraints are different.

A quantum dot connected to leads

A quantum dot weakly coupled to large leads at temperature $T$ and chemical potential $\mu$ exchanges both energy and charge. The grand-canonical ensemble is the natural equilibrium reference state.

Under a voltage bias, however, the leads have different chemical potentials. The steady state is nonequilibrium and cannot be represented by one global grand-canonical density operator with a single $\mu$.

Finite Systems and the Thermodynamic Limit

For a finite system, the three ensembles are genuinely different density operators.

In the microcanonical ensemble,

$$\operatorname{Var}(H) \lesssim \Delta^2,$$

with the precise bound determined by the shell convention.

In the canonical ensemble,

$$\operatorname{Var}(H) = k_{\mathrm B}T^2C_V.$$

In the grand-canonical ensemble, particle number also fluctuates:

$$\operatorname{Var}(N)>0$$

unless the state is supported in one number sector.

For regular short-range systems, matching thermodynamic parameters often makes local observables agree in the thermodynamic limit. If

$$\langle H\rangle \propto V, \qquad C_V \propto V,$$

then

$$\frac{ \sqrt{\operatorname{Var}(H)} }{ \langle H\rangle } \propto V^{-1/2}.$$

Relative energy fluctuations vanish even though the canonical energy is not exactly fixed.

What Ensemble Equivalence Does Not Mean

Ensemble equivalence does not say that the density operators become identical in every sense. It usually concerns thermodynamic functions or sufficiently local observables after parameters are matched.

Differences can remain in:

• exact global conservation laws;
• fluctuations of extensive quantities;
• finite-size corrections;
• rare-event probabilities;
• observables that scale with the full system size.

Equivalence can fail or require qualification for:

• finite systems;
• long-range or nonadditive interactions;
• nonconcave entropy functions;
• phase coexistence and first-order transitions;
• constrained or fragmented Hilbert spaces;
• integrable systems with additional conserved charges.

The Thermodynamic Limit page explains the limiting procedure. Ensemble Equivalence owns the precise thermodynamic, macrostate, local-state, fluctuation, and failure criteria.

Equilibrium State Versus Thermalization

An ensemble specifies an equilibrium state. Thermalization asks whether dynamics makes selected observables approach predictions of such a state.

For a closed system evolving unitarily,

$$\rho(t) = e^{-iHt/\hbar} \rho(0) e^{iHt/\hbar},$$

the von Neumann entropy of the complete state is constant:

$$S(\rho(t)) = S(\rho(0)).$$

Nevertheless, local observables can relax, reduced states can become approximately thermal, and coarse-grained entropy can increase. Those statements require dynamical reasoning involving dephasing, typicality, chaos, integrability, or open-system coupling.

Do not infer thermalization solely from the existence of a Gibbs state.

Thermal Entropy Is Not Entanglement Entropy

For a thermal density operator,

$$S_{\mathrm{th}} = -k_{\mathrm B} \operatorname{Tr} \left( \rho\ln\rho \right)$$

is the thermodynamic entropy when the equilibrium identification is valid.

For a pure bipartite state $|\Psi\rangle_{AB}$,

$$S_A = -\operatorname{Tr} \left( \rho_A\ln\rho_A \right)$$

measures entanglement across the partition, with no factor of $k_{\mathrm B}$ in the common information-theory convention.

A global thermal state can have both thermal mixedness and spatial entanglement. The concepts should not be interchanged merely because both use the von Neumann formula.

Beyond the Three Standard Ensembles

Other controls lead to other ensembles:

• fixed temperature and pressure lead to an isothermal–isobaric ensemble;
• fixed temperature and external field lead to magnetic ensembles;
• multiple conserved charges lead to generalized Gibbs ensembles;
• rotation introduces angular-velocity multipliers;
• nonequilibrium steady states require frameworks beyond equilibrium ensembles.

A generalized exponential state may be written

$$\rho_{\mathrm{GGE}} = \frac{ \exp\!\left( -\sum_j\lambda_j Q_j \right) }{ Z_{\mathrm{GGE}} }.$$

When the charges do not commute, derivative and covariance identities require care; ordinary scalar probability manipulations need not transfer unchanged. The standard microcanonical, canonical, and grand-canonical ensembles remain the foundational cases.

Canonical Boundaries

This page owns:

• the conceptual comparison of the three standard ensembles;
• the relation among constraints, exchanges, trace spaces, fluctuations, and potentials;
• a practical ensemble-selection workflow;
• the distinction between ensemble choice, thermalization, and ensemble equivalence.

Other pages own:

• energy shells, state counting, and microcanonical entropy: Microcanonical Ensemble;
• Gibbs operators and their spectral properties: Thermal Density Operators;
• fixed-$N$ free-energy identities and energy fluctuations: Canonical Ensemble;
• particle exchange, fugacity, number fluctuations, and grand-potential identities: Grand-Canonical Ensemble;
• natural variables and systematic Legendre transforms: Thermodynamic Potentials;
• thermal, information-theoretic, entanglement, and coarse-grained entropy distinctions: Entropy in Quantum Statistical Mechanics;
• general constrained-entropy inference and exponential families: Maximum Entropy Principle;
• derivative and cumulant formulas: Partition Functions and Fluctuations and Susceptibilities;
• isolated-system relaxation, ETH, and generalized equilibrium: Nonequilibrium Many-Body Dynamics;
• heat, work, entropy production, and operational thermodynamics: Quantum Thermodynamics.

Common Mistakes

Identifying an ensemble from the exponential alone

The trace domain and exact constraints matter. A fixed-$N$ trace of $e^{-\beta H}$ is canonical; a Fock-space trace of $e^{-\beta(H-\mu N)}$ is grand canonical.

Saying energy is fixed canonically

Temperature is fixed canonically. Energy fluctuates.

Treating chemical potential as a one-particle energy level

Chemical potential is conjugate to a conserved number. It enters the statistical weight as $-\mu N$.

Using the grand Hamiltonian as the real-time Hamiltonian

$H-\mu N$ organizes equilibrium weights. Real-time evolution is generated by the physical Hamiltonian unless a specified rotating-frame or effective description changes the generator.

Assuming ensembles are equivalent for small systems

Equivalence is generally a thermodynamic-limit statement with hypotheses. Finite-system distributions can differ strongly.

Equating maximum entropy with a thermalization proof

Maximum entropy selects a state from constraints. It does not establish the dynamics by which a particular system reaches equilibrium.

Forgetting the shell width

A microcanonical state for a finite quantum spectrum requires an exact energy projector or a stated energy window. The result can depend on that choice.

Mixing thermal and entanglement entropy

The same functional form does not imply the same physical role.

Exercises

Classify four preparations

Identify the most appropriate equilibrium description, or explain why none is exact:

1. a sealed many-body system with sharply specified total energy and particle number;
2. a fixed-number spin sample weakly coupled to a thermostat;
3. a quantum dot weakly coupled to one equilibrium particle reservoir;
4. an isolated pure state immediately after a sudden quench.

Solution

1. The microcanonical ensemble is appropriate when the energy shell and fixed particle number are specified.
2. The canonical ensemble is appropriate: energy can be exchanged, while particle number is fixed.
3. The grand-canonical ensemble is the equilibrium reference: both energy and particle number can be exchanged.
4. The exact state is the unitarily evolving pure state, not an equilibrium ensemble. At late times, selected observables may agree with an equilibrium or generalized ensemble, but that requires a thermalization analysis.

Trace-space diagnostic

A number-conserving Hamiltonian has fixed-sector partition functions

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

Show how the grand partition function is assembled from these sectors and identify the variable conjugate to $N$.

Solution

Fock space is the direct sum

$$\mathcal F = \bigoplus_{N=0}^{\infty} \mathcal H_N.$$

Therefore

$$\begin{aligned} \Xi &= \operatorname{Tr}_{\mathcal F} e^{-\beta(H-\mu N)} \\ &= \sum_{N=0}^{\infty} e^{\beta\mu N} \operatorname{Tr}_{\mathcal H_N} e^{-\beta H_N} \\ &= \sum_{N=0}^{\infty} z^N Z_N, \end{aligned}$$

where

$$z=e^{\beta\mu}.$$

The chemical potential $\mu$ is conjugate to particle number; equivalently, fugacity $z$ counts number sectors in the generating function.

Derive the canonical weight

Let a subsystem have energy $E_n$ and let the bath multiplicity be $W_B(E_{\mathrm{tot}}-E_n)$. Derive the Boltzmann factor to first order in $E_n$.

Solution

Write

$$S_B(E) = k_{\mathrm B}\ln W_B(E).$$

For a bath much larger than the subsystem,

$$\begin{aligned} S_B(E_{\mathrm{tot}}-E_n) \simeq{}& S_B(E_{\mathrm{tot}}) \\ &- E_n \left( \frac{\partial S_B}{\partial E_B} \right)_{E_{\mathrm{tot}}}. \end{aligned}$$

Using

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

gives

$$W_B(E_{\mathrm{tot}}-E_n) \propto \exp\!\left( -\frac{E_n}{k_{\mathrm B}T} \right) = e^{-\beta E_n}.$$

Normalizing over subsystem states produces the canonical probabilities.

Relative energy fluctuations

Assume a canonical short-range system has

$$\langle H\rangle=uV, \qquad C_V=c_VV,$$

where $u$ and $c_V$ remain finite as $V\to\infty$. Determine the scaling of the relative root-mean-square energy fluctuation.

Solution

The canonical fluctuation identity gives

$$\operatorname{Var}(H) = k_{\mathrm B}T^2c_VV.$$

Hence

$$\sqrt{\operatorname{Var}(H)} \propto V^{1/2}.$$

Because $\langle H\rangle\propto V$,

$$\frac{ \sqrt{\operatorname{Var}(H)} }{ \langle H\rangle } \propto V^{-1/2}.$$

Relative fluctuations vanish in the thermodynamic limit, helping explain equivalence for many local thermodynamic predictions. The absolute fluctuation still grows as $V^{1/2}$, and the canonical state never acquires an exactly fixed global energy.

A local grand-canonical state from a fixed total number

Let a total system $A\cup B$ have exactly $N_{\mathrm{tot}}$ particles. Explain how subsystem $A$ can nevertheless be approximately grand canonical.

Solution

Although

$$N_A+N_B = N_{\mathrm{tot}}$$

is fixed, the subsystem number $N_A$ can fluctuate because particles move between $A$ and $B$. If $B$ is much larger and remains near equilibrium, its entropy expansion for a transfer of energy and particles supplies the weight

$$p_{n,N_A} \propto \exp\!\left[ -\beta(E_{n,N_A}-\mu N_A) \right].$$

Thus the reduced equilibrium state of $A$ can be approximately grand canonical. Exact total-number conservation is fully compatible with local number fluctuations.

Ensemble and measurement uncertainty

Suppose a canonical state is diagonal in the energy basis:

$$\rho = \sum_n p_n|E_n\rangle\langle E_n|.$$

For an observable $A$ that does not commute with $H$, write the probability of outcome $a$ and explain the two sources of uncertainty.

Solution

If $P_a$ is the projector for outcome $a$, then

$$\Pr(a) = \operatorname{Tr}(\rho P_a) = \sum_n p_n \langle E_n|P_a|E_n\rangle.$$

The weights $p_n$ express the thermal mixture over energy states. For each energy state, the factor

$$\langle E_n|P_a|E_n\rangle$$

is the Born probability for the measurement of $A$. When $[A,H]\neq0$, an energy eigenstate need not give a definite value of $A$. Statistical uncertainty in the state and quantum measurement uncertainty can therefore appear in the same prediction.

Cross-Links

• Microcanonical Ensemble
• Thermal Density Operators
• Canonical Ensemble
• Grand-Canonical Ensemble
• Partition Functions
• Thermodynamic Potentials
• Entropy in Quantum Statistical Mechanics
• Maximum Entropy Principle
• Chemical Potential
• Ensemble Equivalence
• Ensemble Formula Sheet
• Thermodynamic Limit
• Density Operators
• Trace Rule for Expectation Values
• Entropy Overview
• Quantum Statistics Overview
• Maxwell–Boltzmann Limit
• Bose–Einstein Statistics
• Fermi–Dirac Statistics
• Fock Space and Occupation Number
• Baths and Reservoirs
• Quantum Thermodynamics
• Statistical Mechanics Checklist

References

• R. K. Pathria and P. D. Beale, Statistical Mechanics, 4th ed., Elsevier (2021), especially the ensemble formulation and quantum-statistics chapters.
• M. Kardar, Statistical Physics of Particles, Cambridge University Press (2007), chapters 1–4.
• K. Huang, Statistical Mechanics, 2nd ed., Wiley (1987), chapters 6–9.
• H. B. Callen, Thermodynamics and an Introduction to Thermostatistics, 2nd ed., Wiley (1985), chapters 1–7.
• L. D. Landau and E. M. Lifshitz, Statistical Physics, Part 1, 3rd ed., Butterworth–Heinemann (1980), sections 1–35.
• E. T. Jaynes, “Information Theory and Statistical Mechanics”, Physical Review 106, 620–630 (1957).
• D. Ruelle, Statistical Mechanics: Rigorous Results, W. A. Benjamin (1969), chapters 1–3.
• H. Tasaki, Physics and Mathematics of Quantum Many-Body Systems, Springer (2020), introductory discussion of many-body systems and thermodynamic limits.