Ensemble Formula Sheet

Use these formulas to move between equilibrium probabilities, thermodynamic potentials, and fluctuations. The trace space and the variables held fixed matter as much as the partition function itself: a fixed-number state has no number fluctuations, whereas a grand-canonical state generally does.

Required background. Statistical ensembles compares physical controls; thermal density operators constructs Gibbs states; partition functions explains the state sums used here.

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	energy may vary within the shell; $N$ is fixed	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

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

In an energy eigenbasis this is $\langle A\rangle=\sum_n p_n A_{nn}$, whether or not $[A,H_N]=0$. Off-diagonal elements matter in products and time-dependent correlations, but do not enter this one-point trace.

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 $\Omega$	$T,V,\mu$	$\Omega=U-TS-\mu N$	$d\Omega=-S\,dT-P\,dV-N\,d\mu$

These differentials require a differentiable equilibrium state function and the listed work variables. Extensivity is an additional assumption for Euler identities such as $U=TS-PV+\mu N$, not for the differential relations themselves. Include surface or other work terms when their parameters vary. 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

For a temperature-independent scalar $C$, 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 $C$. If $C$ depends on an external parameter, the corresponding free-energy derivative also changes by the derivative of $C$; an omitted constant is then not harmless for forces.

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;
• entropy branches that lack the supporting-line condition, including possible nonconcave branches in bounded-spectrum systems;
• observables dominated by global fluctuations;
• limits taken in different orders.

Agreement of mean densities does not imply equality of fluctuation distributions. Negative temperature alone does not cause ensemble inequivalence: an additive bounded-spectrum system can have a concave entropy branch with negative slope.

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

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. 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

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

3. 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.

References

• R. Balian, From Microphysics to Macrophysics, Volume I, Springer (1991).
• H. B. Callen, Thermodynamics and an Introduction to Thermostatistics, 2nd ed., Wiley (1985).
• 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).
• R. K. Pathria and P. D. Beale, Statistical Mechanics, 3rd ed., Elsevier (2011).