Ensemble Formula Sheet

This page collects the equilibrium ensemble formulas used most often in many-body quantum mechanics. It is a lookup sheet, not a replacement for the canonical derivations:

• Statistical Ensembles Overview owns the conceptual comparison, exchange conditions, and selection workflow.
• Microcanonical Ensemble owns energy-shell projectors, state-counting conventions, and thermodynamic derivatives from entropy.
• Thermal Density Operators owns the Gibbs-state construction and operator properties.
• Canonical Ensemble owns fixed-$N$ thermodynamics, reservoir motivation, and energy fluctuations.
• Grand-Canonical Ensemble owns Fock-space traces, chemical potential, number fluctuations, and sector recovery.
• Partition Functions owns general state sums, factorization, cumulant generation, and one-particle versus many-body distinctions.
• Classical Limit of Quantum Statistics owns the dilute thermal-wavelength criterion, fugacity expansion, and leading Bose/Fermi exchange corrections.
• Thermodynamic Potentials owns natural variables, Legendre transforms, Euler and Gibbs–Duhem relations, and stability qualifications.
• Entropy in Quantum Statistical Mechanics owns the relation among thermal, state, entanglement, diagonal, and coarse-grained entropies.
• Fluctuations and Susceptibilities owns the unified static response interpretation, Kubo–Mori correction, ensemble caveats, and scaling analysis.
• Maximum Entropy Principle owns constrained entropy maximization, multiplier matching, and exact-versus-mean constraint logic.
• Thermodynamic Limit owns the definition and scaling of bulk limits.
• Ensemble Equivalence owns equivalence levels, concentration, entropy concavity, and nonequivalence criteria.

Unless otherwise stated,

$$\beta = \frac{1}{k_B T},$$

$H$ has no explicit temperature dependence, and all derivatives hold the variables displayed in their subscripts fixed.

Ensemble at a Glance

Ensemble	Controlled quantities	Fluctuating quantities	Equilibrium operator
microcanonical	$E$, $V$, $N$ within a stated window	none of the controlled extensive quantities	normalized projector onto the energy shell
canonical	$T$, $V$, $N$	energy	$e^{-\beta H_N}/Z_N$
grand canonical	$T$, $V$, $\mu$	energy and particle number	$e^{-\beta(H-\mu N)}/\Xi$

The controls identify the ensemble, but finite-size constraints and conserved sectors remain part of the specification.

Canonical Ensemble

For a fixed-particle-number Hilbert space $\mathcal H_N$,

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

with partition function

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

The trace is over the fixed-$N$ sector. For identical particles, that sector already has the required Bose or Fermi symmetry.

Helmholtz free energy

$$F(T,V,N,\lambda,\ldots) = -k_B T\ln Z_N = -\frac{1}{\beta}\ln Z_N.$$

Thermal expectation value

$$\langle A\rangle_\beta = \operatorname{Tr} \left( \rho_{N,\beta}A \right).$$

If $A$ does not commute with $H_N$, this trace remains the correct equilibrium expectation value. Only its energy-basis evaluation contains off-diagonal matrix elements that vanish when $\rho$ is diagonal and the trace is taken.

Internal energy

For a $\beta$-independent Hamiltonian,

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

Equivalently,

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

Entropy

$$S = -k_B \operatorname{Tr} \left( \rho_{N,\beta}\ln\rho_{N,\beta} \right).$$

For the Gibbs state,

$$S = k_B \left( \ln Z_N+\beta U \right).$$

The thermodynamic derivative is

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

Heat capacity and energy fluctuations

$$C_V = \left( \frac{\partial U}{\partial T} \right)_{V,N}.$$

For a temperature-independent Hamiltonian,

$$C_V = k_B\beta^2 \operatorname{Var}(H_N),$$

where

$$\operatorname{Var}(H_N) = \langle H_N^2\rangle - \langle H_N\rangle^2.$$

Equivalently,

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

Pressure and generalized forces

For a volume parameter with a well-defined thermodynamic derivative,

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

For a Hamiltonian parameter $\lambda$,

$$\left( \frac{\partial F}{\partial\lambda} \right)_{T,V,N} = \left\langle \frac{\partial H_N}{\partial\lambda} \right\rangle.$$

If the physical generalized force is defined by

$$X_\lambda = - \frac{\partial H_N}{\partial\lambda},$$

then

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

The sign depends on how $\lambda$ is coupled in the Hamiltonian.

Differential

For a simple system,

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

At strictly fixed integer $N$ in a finite system, the derivative with respect to $N$ is not literal. The chemical-potential term is a thermodynamic-limit or finite-difference statement.

Grand-Canonical Ensemble

Let $N$ be a conserved number operator satisfying

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

Define the grand Hamiltonian

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

The equilibrium density operator is

$$\rho_{\mathrm{gc}} = \frac{e^{-\beta K}}{\Xi},$$

with grand partition function

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

The trace is over the relevant Fock space or direct sum of number sectors.

Sector decomposition and fugacity

Define the fugacity

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

When $H$ preserves number,

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

Thus $\Xi$ is a generating function for canonical partition functions. Convergence restricts the allowed $\mu$ in some bosonic systems.

Grand potential

$$\Omega(T,V,\mu) = -k_B T\ln\Xi = -\frac{1}{\beta}\ln\Xi.$$

For a homogeneous extensive system in the thermodynamic limit,

$$\Omega = -PV.$$

For a finite or inhomogeneous system, use the derivative definition

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

when a volume variation is well defined. The identity $\Omega=-PV$ requires homogeneity and extensivity; it is not a universal finite-system formula.

Mean particle number

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

Equivalently,

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

and

$$\overline N = z \frac{\partial\ln\Xi}{\partial z}.$$

Grand energy and internal energy

For $\beta$-independent $H$, $N$, and fixed $\mu$,

$$\langle K\rangle = - \left( \frac{\partial\ln\Xi}{\partial\beta} \right)_{\mu,V}.$$

Therefore

$$U \equiv \langle H\rangle = - \left( \frac{\partial\ln\Xi}{\partial\beta} \right)_{\mu,V} + \mu\overline N.$$

If the derivative is instead taken at fixed fugacity $z$, then

$$U = - \left( \frac{\partial\ln\Xi}{\partial\beta} \right)_{z,V}.$$

The distinction between fixed $\mu$ and fixed $z$ matters because $z=e^{\beta\mu}$.

Entropy

$$S = -k_B \operatorname{Tr} \left( \rho_{\mathrm{gc}}\ln\rho_{\mathrm{gc}} \right).$$

For the grand-canonical Gibbs state,

$$S = k_B \left[ \ln\Xi + \beta \left( U-\mu\overline N \right) \right].$$

Equivalently,

$$S = - \left( \frac{\partial\Omega}{\partial T} \right)_{V,\mu}.$$

Number fluctuations

$$\operatorname{Var}(N) = \langle N^2\rangle - \overline N^{\,2}.$$

The fluctuation identity is

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

In terms of fugacity,

$$\operatorname{Var}(N) = z \frac{\partial\overline N}{\partial z}.$$

If $n=\overline N/V$ and the isothermal compressibility is

$$\kappa_T = \frac{1}{n^2} \left( \frac{\partial n}{\partial\mu} \right)_T,$$

then

$$\operatorname{Var}(N) = k_B T\, Vn^2\kappa_T.$$

This relation assumes the derivative and thermodynamic variables use consistent volume and boundary conventions.

Grand-energy fluctuations

For fixed $\mu$,

$$\frac{\partial^2\ln\Xi}{\partial\beta^2} = \operatorname{Var}(K),$$

with

$$\operatorname{Var}(K) = \left\langle (H-\mu N)^2 \right\rangle - \left\langle H-\mu N \right\rangle^2.$$

This is not generally equal to $\operatorname{Var}(H)$ because energy and particle number can be correlated.

Differential

For a simple homogeneous system,

$$d\Omega = -S\,dT - P\,dV - \overline N\,d\mu.$$

The Legendre relation is

$$\Omega = U-TS-\mu\overline N.$$

Microcanonical Ensemble

For definitions, entropy conventions, and isolated-system caveats, see Microcanonical Ensemble.

Choose an energy window

$$[E,E+\Delta E]$$

inside a fixed-$N$ Hilbert space, and let $\Pi_{E,\Delta E}$ project onto the states in that window. Define

$$W(E,\Delta E) = \operatorname{Tr} \Pi_{E,\Delta E}.$$

The normalized microcanonical state is

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

Its shell entropy is

$$S_{\mathrm{mc}} = k_B\ln W(E,\Delta E).$$

The window must be wide enough to contain many levels for thermodynamic use, yet narrow on macroscopic energy scales. Different entropy conventions use a shell count, integrated density of states, or smoothed density of states; the convention must be stated before differentiating.

When a differentiable thermodynamic entropy $S(E,V,N)$ exists,

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

and

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

Thermodynamic Potentials

Potential	Natural variables	Definition	Differential
internal energy $U$	$S,V,N$	$U$	$dU=T\,dS-P\,dV+\mu\,dN$
enthalpy $H_{\mathrm{th}}$	$S,P,N$	$H_{\mathrm{th}}=U+PV$	$dH_{\mathrm{th}}=T\,dS+V\,dP+\mu\,dN$
Helmholtz free energy $F$	$T,V,N$	$F=U-TS$	$dF=-S\,dT-P\,dV+\mu\,dN$
Gibbs free energy $G$	$T,P,N$	$G=U-TS+PV$	$dG=-S\,dT+V\,dP+\mu\,dN$
grand potential $\Omega$	$T,V,\mu$	$\Omega=U-TS-\mu N$	$d\Omega=-S\,dT-P\,dV-N\,d\mu$

The differentials assume the ordinary thermodynamic state variables exist and surface, long-range, finite-size, and nonextensive corrections are under control. Thermodynamic Potentials derives the table and explains when exact finite-system sums reduce to Legendre transforms.

Derivative Identities

Canonical cumulants

For a $\beta$-independent Hamiltonian,

$$\frac{\partial^n\ln Z_N}{\partial\beta^n} = (-1)^n \kappa_n(H_N),$$

where $\kappa_n(H_N)$ is the $n$th energy cumulant. In particular,

$$\kappa_1(H_N) = U, \qquad \kappa_2(H_N) = \operatorname{Var}(H_N).$$

Grand-canonical number cumulants

Using $\alpha=\beta\mu$,

$$\frac{\partial^n\ln\Xi}{\partial\alpha^n} = \kappa_n(N)$$

at fixed $\beta$ and $V$. Thus

$$\frac{\partial\ln\Xi}{\partial\alpha} = \overline N, \qquad \frac{\partial^2\ln\Xi}{\partial\alpha^2} = \operatorname{Var}(N).$$

Covariances from mixed derivatives

For a generalized Gibbs state

$$\rho \propto \exp \left( -\sum_a\lambda_a Q_a \right),$$

with mutually commuting conserved quantities $Q_a$,

$$-\frac{\partial\langle Q_a\rangle} {\partial\lambda_b} = \operatorname{Cov}(Q_a,Q_b),$$

where

$$\operatorname{Cov}(A,B) = \langle AB\rangle - \langle A\rangle\langle B\rangle.$$

For noncommuting operators, derivatives involve imaginary-time ordered or Kubo–Mori correlations rather than the naive covariance. Do not extend the commuting formula without checking the operator ordering.

Energy-Zero Shift

If

$$H' = H+C I,$$

then

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

and

$$F' = F+C.$$

The normalized state is unchanged:

$$\rho' = \frac{ e^{-\beta(H+CI)} }{ e^{-\beta C}Z } = \rho.$$

Therefore expectation values of unchanged observables, entropy, heat capacity, and energy variance are invariant. The numerical internal energy shifts by $C$.

Ensemble Equivalence

Canonical, grand-canonical, and microcanonical ensembles often give the same local thermodynamic predictions when:

• the thermodynamic limit exists;
• interactions are sufficiently short ranged or additive;
• the system is away from singular finite-size regimes;
• the selected state lies in a stable phase;
• conserved sectors and symmetry-breaking limits are handled consistently.

Equivalence can fail or require qualification for:

• finite systems;
• long-range nonadditive interactions;
• phase coexistence and first-order transitions;
• constrained, fragmented, or integrable sectors;
• negative-temperature systems with bounded spectra;
• observables dominated by global fluctuations;
• limits taken in different orders.

Agreement of mean densities does not imply equality of fluctuation distributions.

See Ensemble Equivalence for the supporting-line and large-deviation tests, local quantum-state criterion, and finite-size and coexistence qualifications.

Ensemble Selection Checklist

Use the microcanonical ensemble when total energy and particle number are sharply fixed and the energy shell is the natural macroscopic description.

Use the canonical ensemble when energy exchange establishes a temperature while particle number and external controls are fixed.

Use the grand-canonical ensemble when energy and particles can be exchanged, or when summing over number sectors is a controlled computational convenience and $\mu$ is fixed by the desired mean number.

Before applying a formula, state:

1. the Hilbert space or number sector traced over;
2. the conserved quantities and imposed constraints;
3. $T$, $V$, $N$, $\mu$, and any generalized fields held fixed;
4. whether $H$ depends explicitly on $T$, $\mu$, or another differentiation variable;
5. boundary conditions and finite-volume normalization;
6. whether the thermodynamic limit is being used;
7. whether the system is homogeneous and extensive;
8. the order of zero-temperature, infinite-volume, and source-removal limits.

Common Mistakes

• Omitting the number-sector restriction on the canonical trace.
• Using $\Omega=-PV$ for a finite, trapped, or inhomogeneous system without qualification.
• Taking a $\beta$ derivative while silently allowing $H$ or $\mu$ to depend on $\beta$.
• Confusing fixed fugacity with fixed chemical potential.
• Treating $\mu$ as a one-particle energy rather than a thermodynamic conjugate.
• Assuming a grand-canonical density operator implies number-changing unitary dynamics.
• Using fluctuation identities when the relevant partition function does not converge.
• Applying ordinary covariance formulas to noncommuting generalized charges.
• Treating ensemble equivalence as exact for every finite system and every observable.
• Ignoring surface terms, long-range interactions, or phase coexistence in thermodynamic derivatives.
• Comparing entropies without stating the energy-window or density-of-states convention.
• Forgetting that a temperature-dependent effective Hamiltonian adds derivative terms.

Cross-Links

• Many-Body and Quantum Statistical Mechanics
• Symbols and Conventions
• Quantum Gas Formula Sheet
• Fermi Gas Formula Sheet
• Bose Gas Formula Sheet
• Statistical Ensembles Overview
• Quantum Statistics Overview
• Maxwell–Boltzmann Limit
• Microcanonical Ensemble
• Thermodynamic Potentials
• Entropy in Quantum Statistical Mechanics
• Maximum Entropy Principle
• Chemical Potential
• Ensemble Equivalence
• Common Many-Body Hamiltonians
• Thermal Density Operators
• Canonical Ensemble
• Grand-Canonical Ensemble
• Partition Functions
• Thermodynamic Limit
• Entropy Overview
• Quantum Thermodynamics
• Bose–Einstein Distribution
• Fermi–Dirac Distribution
• Statistical Mechanics Checklist

References

1. R. K. Pathria and P. D. Beale, Statistical Mechanics, 3rd ed., Elsevier (2011).
2. K. Huang, Statistical Mechanics, 2nd ed., Wiley (1987).
3. L. D. Landau and E. M. Lifshitz, Statistical Physics, Part 1, 3rd ed., Butterworth-Heinemann (1980).
4. M. Kardar, Statistical Physics of Particles, Cambridge University Press (2007).
5. R. Balian, From Microphysics to Macrophysics, Volume I, Springer (1991).
6. H. B. Callen, Thermodynamics and an Introduction to Thermostatistics, 2nd ed., Wiley (1985).

Exercises

1. Shift the energy zero. Show that shifting $H\mapsto H+CI$ leaves the canonical entropy and heat capacity unchanged.

Solution

The partition function becomes

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

so

$$\ln Z' = \ln Z-\beta C.$$

The internal energy shifts to

$$U' = -\partial_\beta\ln Z' = U+C.$$

Using

$$S = k_B(\ln Z+\beta U),$$

gives

$$\begin{aligned} S' &= k_B \left[ \ln Z-\beta C + \beta(U+C) \right] \\ &= S. \end{aligned}$$

Because $C$ is temperature independent,

$$C_V' = \frac{\partial(U+C)}{\partial T} = C_V.$$

2. Energy fluctuations. Derive the canonical relation

$$C_V = k_B\beta^2 \operatorname{Var}(H).$$

Solution

For a temperature-independent Hamiltonian,

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

Differentiate once more:

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

Since

$$\frac{d\beta}{dT} = -\frac{1}{k_B T^2} = -k_B\beta^2,$$

the chain rule gives

$$C_V = \frac{\partial U}{\partial T} = \frac{\partial U}{\partial\beta} \frac{d\beta}{dT} = k_B\beta^2 \operatorname{Var}(H).$$

3. Number fluctuations. Starting from $\overline N=\beta^{-1}\partial_\mu\ln\Xi$, derive the grand-canonical fluctuation identity.

Solution

At fixed $\beta$,

$$\frac{\partial\overline N}{\partial\mu} = \frac{1}{\beta} \frac{\partial^2\ln\Xi}{\partial\mu^2}.$$

Differentiating the trace shows

$$\frac{\partial^2\ln\Xi}{\partial\mu^2} = \beta^2 \left( \langle N^2\rangle - \langle N\rangle^2 \right).$$

Therefore

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

Because $\beta^{-1}=k_B T$, larger compressibility produces larger equilibrium number fluctuations.

4. Fixed fugacity versus fixed chemical potential. Explain why

$$U = -\left(\partial_\beta\ln\Xi\right)_z$$

but

$$U-\mu\overline N = -\left(\partial_\beta\ln\Xi\right)_\mu.$$

Solution

Write the sector expansion

$$\Xi = \sum_{N,r} \exp \left[ -\beta E_{Nr} + \beta\mu N \right].$$

At fixed $\mu$, differentiating the exponent gives

$$-E_{Nr}+\mu N,$$

so

$$-\left(\partial_\beta\ln\Xi\right)_\mu = \langle H-\mu N\rangle = U-\mu\overline N.$$

At fixed $z=e^{\beta\mu}$, the factor $z^N$ is held constant while only $e^{-\beta E_{Nr}}$ is differentiated. Therefore

$$-\left(\partial_\beta\ln\Xi\right)_z = U.$$

5. Choose the ensemble. Select an ensemble for each situation: an isolated finite system with sharply fixed energy; a spin sample exchanging heat but not particles; and atoms in a subsystem exchanging particles and energy with a much larger cloud.

Solution

• The isolated system with sharply fixed energy is naturally microcanonical, with a stated finite energy window.
• The spin sample exchanging heat while its degrees of freedom remain fixed is naturally canonical.
• The atomic subsystem exchanging both energy and particles is naturally grand canonical, with $\mu$ determined by equilibrium with the larger cloud.

These choices describe the controls. A different ensemble can sometimes be used as a calculational device, but finite-size corrections and fluctuation predictions must then be checked.