Partition Functions

A partition function is a basis-independent trace of an equilibrium statistical weight. In the fixed-particle-number canonical ensemble,

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

In the grand-canonical ensemble,

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

These quantities normalize density operators, but normalization is only their first role. Their logarithms generate connected fluctuations, thermodynamic potentials, and responses. Their factorization records independence. Their analytic and asymptotic structure encodes spectra, phases, and low-energy physics.

The canonical and grand-canonical ensemble pages explain their controls. Here the focus is how to count states, extract fixed-number contributions, and generate moments without confusing one-particle and many-particle sums.

Helpful background. The thermal density operator explains normalization; trace-class operators clarify convergence when the Hilbert space is infinite.

Basis-independent state counting

Let $\{|n\rangle\}$ be any orthonormal basis of the relevant Hilbert space. Then

$$Z = \sum_n \langle n| e^{-\beta H} |n\rangle.$$

If $U$ is unitary and

$$|n'\rangle = U|n\rangle,$$

then cyclicity of the trace gives

$$\begin{aligned} \sum_n \langle n'| e^{-\beta H} |n'\rangle &= \operatorname{Tr} \left( U^\dagger e^{-\beta H}U \right) \\ &= \operatorname{Tr} e^{-\beta H}. \end{aligned}$$

The partition function is therefore independent of the basis used to evaluate it. The energy basis is convenient, not fundamental.

Spectral State Sum

For

$$H = \sum_a E_aP_a,$$

functional calculus gives

$$e^{-\beta H} = \sum_a e^{-\beta E_a}P_a.$$

Taking the trace,

$$Z(\beta) = \sum_a g_a e^{-\beta E_a},$$

where

$$g_a = \operatorname{Tr}P_a$$

is the degeneracy of energy $E_a$.

Dropping $g_a$ is equivalent to discarding states. If energy labels are repeated once per independent eigenvector, the same formula can be written

$$Z(\beta) = \sum_n e^{-\beta E_n}.$$

Dimension and Units

The exponential must have a dimensionless argument:

$$\beta H$$

is dimensionless because

$$[\beta] = \text{energy}^{-1}.$$

The trace of $e^{-\beta H}$ is dimensionless, so $Z$ and $\Xi$ are dimensionless.

A classical phase-space partition function requires a reference cell such as $h^{dN}$ to make the integral dimensionless. In quantum mechanics, the Hilbert-space trace supplies the state count directly.

Existence and Convergence

In finite dimension,

$$0<Z(\beta)<\infty$$

for every finite real $\beta$.

In infinite dimension, the operator

$$e^{-\beta H}$$

must be trace class. Equivalently,

$$\sum_n e^{-\beta E_n} < \infty$$

for a discrete spectrum counted with multiplicity.

A Hamiltonian bounded below is not by itself sufficient. The number of states below energy $E$ must not grow too rapidly relative to the Boltzmann suppression.

Positive inverse temperature

For $\beta>0$, low energies are favored. Confined systems with spectra growing sufficiently rapidly often have finite $Z$.

Infinite temperature

At $\beta=0$,

$$Z(0) = \operatorname{Tr}\mathbb I = \dim\mathcal H.$$

This is finite only in finite dimension.

Negative inverse temperature

For $\beta<0$, high energies are favored. The Hamiltonian must be bounded above and the full trace must converge. An upper bound alone does not suffice on an infinite-dimensional space; infinitely many states in a bounded energy interval already make the trace diverge.

Infinite volume

Even when a finite-volume partition function exists, the total $Z_V$ usually has an exponential dependence on volume (growth or decay depends on the energy zero). The bulk object is often

$$\lim_{V\to\infty} \frac{1}{V} \ln Z_V,$$

or the associated free-energy density, not a finite infinite-volume trace.

Partition Functions Generate Moments

Assume $H$ has no explicit $\beta$ dependence and the required moments exist; this holds inside an open interval of Gibbs-trace convergence. Differentiating gives

$$\frac{\partial Z}{\partial\beta} = -\operatorname{Tr} \left( He^{-\beta H} \right).$$

Therefore

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

A second derivative gives

$$\frac{\partial^2\ln Z}{\partial\beta^2} = \langle H^2\rangle_\beta -\langle H\rangle_\beta^2.$$

Thus

$$\frac{\partial^2\ln Z}{\partial\beta^2} = \operatorname{Var}_\beta(H) \geq 0.$$

The convexity of $\ln Z$ in $\beta$ is a variance statement.

Moment-Generating Interpretation

Within the canonical state at inverse temperature $\beta$, consider

$$\left\langle e^{tH} \right\rangle_\beta.$$

Because $H$ commutes with every function of itself,

$$\begin{aligned} \left\langle e^{tH} \right\rangle_\beta &= \frac{ \operatorname{Tr} \left( e^{tH}e^{-\beta H} \right) }{ Z(\beta) } \\ &= \frac{ Z(\beta-t) }{ Z(\beta) }, \end{aligned}$$

whenever both traces converge.

The cumulant-generating function of energy is

$$K_H(t) = \ln Z(\beta-t) -\ln Z(\beta).$$

Consequently,

$$\kappa_n(H) = (-1)^n \frac{ \partial^n\ln Z }{ \partial\beta^n }.$$

For example,

$$\kappa_1(H) = \langle H\rangle,$$

and

$$\kappa_2(H) = \operatorname{Var}(H).$$

Higher derivatives generate higher connected energy fluctuations. The parameter $t$ here has inverse-energy units and is a generating variable, not physical time.

Source-Dependent Partition Functions

Introduce a source $J$ coupled to an observable $A$:

$$H(J) = H-JA.$$

Define

$$Z(\beta,J) = \operatorname{Tr} \exp \left[ -\beta(H-JA) \right].$$

The first derivative is

$$\frac{\partial\ln Z}{\partial J} = \beta \langle A\rangle_J.$$

This remains true even if

$$[H,A]\neq0.$$

The derivative of a noncommuting operator exponential is handled by the Duhamel formula, and cyclicity of the trace reduces the first derivative to the thermal expectation value.

The second derivative requires more care. For noncommuting $A$ and $H(J)$, it produces an imaginary-time or Kubo–Mori correlation rather than simply

$$\beta^2\operatorname{Var}(A).$$

The ordinary variance is recovered when the relevant operators commute. Fluctuations and Susceptibilities develops the Kubo–Mori covariance, static response interpretation, and thermodynamic examples.

Parameter Derivatives and Generalized Forces

Let

$$H = H(\lambda).$$

Then

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

The Helmholtz free energy

$$F = -\frac{1}{\beta}\ln Z$$

satisfies

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

If the generalized force is defined by

$$X_\lambda \equiv -\frac{\partial H}{\partial\lambda},$$

then

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

Explicit $\beta$ or temperature dependence in an effective Hamiltonian adds derivative terms and must be stated.

Grand Partition Function as a Number Generator

Assume

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

Fock space decomposes as

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

The grand partition function is

$$\Xi(\beta,\mu) = \sum_{N=0}^{\infty} e^{\beta\mu N} Z_N(\beta).$$

With fugacity

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

this becomes

$$\Xi(\beta,z) = \sum_{N=0}^{\infty} z^NZ_N(\beta).$$

Thus $\Xi$ is an ordinary generating function for the fixed-$N$ canonical partition functions.

The probability of particle number $N$ is

$$\Pr(N) = \frac{ z^NZ_N }{ \Xi }.$$

Coefficient Extraction

If $\Xi(z)$ is analytic around the origin,

$$Z_N = [z^N]\Xi(z),$$

where $[z^N]$ denotes the coefficient of $z^N$.

Equivalently,

$$Z_N = \frac{1}{2\pi i} \oint_{\mathcal C} \frac{ \Xi(z) }{ z^{N+1} } \,dz,$$

where $\mathcal C$ encloses the origin inside the domain of analyticity.

This contour formula is exact. Saddle-point evaluation at large $N$ connects fixed-number and fixed-chemical-potential descriptions.

Number Cumulants

It is convenient to define

$$\nu \equiv \beta\mu = \ln z.$$

Then

$$\frac{\partial\ln\Xi}{\partial\nu} = \langle N\rangle,$$

and

$$\frac{\partial^2\ln\Xi}{\partial\nu^2} = \operatorname{Var}(N).$$

More generally,

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

Fixed $\nu=\ln z$ and fixed $z$ are the same constraint; either differs from fixed $\mu$ during a $\beta$ derivative. This $\nu$ is the negative of the alternative multiplier $\alpha=-\beta\mu$ used on the grand-canonical page.

Factorization for Independent Subsystems

Suppose

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

and

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

The two terms commute, so

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

Using

$$\operatorname{Tr}_{A\otimes B} (X_A\otimes Y_B) = \operatorname{Tr}_A X_A \operatorname{Tr}_B Y_B,$$

one obtains

$$Z_{AB} = Z_AZ_B.$$

Therefore

$$\ln Z_{AB} = \ln Z_A+\ln Z_B.$$

The logarithm is additive here because the trace factorizes. This is an exact independence statement; extensivity in a thermodynamic limit additionally requires suitable scaling of the Hamiltonian.

When Factorization Fails

Naive factorization fails when:

• the Hamiltonian contains interactions between $A$ and $B$;
• the Hilbert space does not factor into independent tensor factors;
• a global constraint couples otherwise independent subsystems;
• identical-particle symmetrization links particle labels;
• gauge or superselection constraints restrict the physical subspace;
• the trace is projected onto one total-charge sector.

Weak interactions can sometimes be treated perturbatively, but

$$Z_{AB} \neq Z_AZ_B$$

is the generic interacting case.

Single-Particle Versus Many-Body Partition Functions

Let the one-particle Hamiltonian $h$ act on a one-particle space $\mathfrak h$, with energies $\epsilon_i$. Define

$$q(\beta) \equiv \operatorname{Tr}_{\mathfrak h} e^{-\beta h} = \sum_i e^{-\beta\epsilon_i}.$$

The symbol $q$ is a one-particle partition function. It is not automatically the partition function of $N$ identical quantum particles.

Distinguishable Noninteracting Particles

For $N$ labeled, noninteracting particles,

$$\mathcal H_{\mathrm{dist}} = \mathfrak h^{\otimes N},$$

and

$$H_N = \sum_{a=1}^{N} h_a.$$

The partition function factorizes:

$$Z_N^{\mathrm{dist}} = q(\beta)^N.$$

This formula counts permutations of one-particle occupations as distinct labeled states.

Identical Bosons and Fermions

For identical particles, the trace is restricted to the symmetric or antisymmetric subspace:

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

The plus sign denotes bosons and the minus sign fermions. In general,

$$Z_N^{(\pm)} \neq q^N.$$

For noninteracting identical particles, an exact recurrence is

$$Z_N^{(\eta)} = \frac{1}{N} \sum_{\ell=1}^{N} \eta^{\ell-1} q(\ell\beta) Z_{N-\ell}^{(\eta)},$$

with

$$Z_0^{(\eta)}=1, \qquad \eta= \begin{cases} +1,&\text{bosons},\\ -1,&\text{fermions}. \end{cases}$$

To derive the recurrence, expand the logarithm of the independent-mode product near $z=0$:

$$\ln\Xi_\eta(z)=\sum_{\ell\geq1} \frac{\eta^{\ell-1}z^\ell}{\ell}q(\ell\beta).$$

Differentiate with respect to $z$, multiply by $\Xi_\eta$, and compare coefficients in $z\partial_z\Xi_\eta$. The left coefficient is $N Z_N$; the product on the right gives the displayed convolution. This proves the recurrence without treating identical particles as independent labeled copies. The terms with $\ell>1$ are exchange contributions.

Maxwell–Boltzmann Limit

In the dilute, nondegenerate regime, exchange corrections become small and the identical-particle canonical partition function approaches

$$Z_N^{\mathrm{MB}} \simeq \frac{ q(\beta)^N }{ N! }.$$

The factor $1/N!$ prevents classical overcounting of particle labels. This is an approximation to the quantum bosonic or fermionic result, not an exact formula at arbitrary density.

Classical Limit of Quantum Statistics derives the controlling parameter $n\lambda_T^d/g$, the low-fugacity cycle expansion, and the leading Bose and Fermi virial corrections.

Noninteracting Modes in Fock Space

Suppose

$$H-\mu N = \sum_i (\epsilon_i-\mu)n_i.$$

The grand partition function factorizes over modes.

For bosons,

$$\begin{aligned} \Xi_{\mathrm B} &= \prod_i \sum_{n_i=0}^{\infty} \left( ze^{-\beta\epsilon_i} \right)^{n_i} \\ &= \prod_i \frac{1}{ 1-ze^{-\beta\epsilon_i} }, \end{aligned}$$

provided

$$\left| ze^{-\beta\epsilon_i} \right| < 1$$

for every mode, with convergence of the full product also required if infinitely many modes are present.

For fermions,

$$\begin{aligned} \Xi_{\mathrm F} &= \prod_i \sum_{n_i=0}^{1} \left( ze^{-\beta\epsilon_i} \right)^{n_i} \\ &= \prod_i \left( 1+ze^{-\beta\epsilon_i} \right). \end{aligned}$$

Taking logarithms turns products into sums:

$$\ln\Xi_{\mathrm B} = -\sum_i \ln \left( 1-ze^{-\beta\epsilon_i} \right),$$

and

$$\ln\Xi_{\mathrm F} = \sum_i \ln \left( 1+ze^{-\beta\epsilon_i} \right).$$

Example: Two One-Particle Levels

Let

$$\epsilon_0=0, \qquad \epsilon_1=\Delta,$$

and define

$$x \equiv e^{-\beta\Delta}.$$

The one-particle partition function is

$$q = 1+x.$$

Two distinguishable particles

For two labeled particles,

$$Z_2^{\mathrm{dist}} = q^2 = 1+2x+x^2.$$

The coefficient $2x$ counts which labeled particle occupies the excited level.

Two identical bosons

The allowed occupation states are

$$(n_0,n_1) = (2,0),(1,1),(0,2).$$

Therefore

$$Z_2^{\mathrm B} = 1+x+x^2.$$

Two identical spinless fermions

Pauli exclusion allows only

$$(n_0,n_1) = (1,1).$$

Hence

$$Z_2^{\mathrm F} = x.$$

The three answers differ despite sharing the same one-particle spectrum.

Two particles share one-particle levels $0,\Delta$. The numbers beside the many-body levels give degeneracy $g$. The labeled system has two states at total energy $\Delta$; the bosonic system has one; spinless fermions can occupy only the middle total energy. These counts produce the three distinct partition functions above.

Density-of-States Representation

Define

$$\omega(E) = \operatorname{Tr} \delta(E-H).$$

Then

$$Z(\beta) = \int dE\, \omega(E)e^{-\beta E}.$$

The partition function is the Laplace transform of the density of states.

For a discrete spectrum,

$$\omega(E) = \sum_a g_a\delta(E-E_a),$$

and the integral reduces to the spectral sum.

Formally, the inverse transform is

$$\omega(E) = \frac{1}{2\pi i} \int_{c-i\infty}^{c+i\infty} d\beta\, e^{\beta E}Z(\beta),$$

where the vertical contour lies in a domain of convergence. Inverse Laplace transforms are often numerically ill-conditioned; formal invertibility does not guarantee stable reconstruction from noisy data.

Low-Temperature Expansion

For a discrete spectrum with finite degeneracies and a Gibbs trace convergent at some positive inverse temperature, let the ground energy be $E_0$ with degeneracy $g_0$, and let the next distinct energy be

$$E_1 = E_0+\Delta.$$

Then

$$Z(\beta) = e^{-\beta E_0} \left[ g_0 +g_1e^{-\beta\Delta} +\cdots \right].$$

At large positive $\beta$,

$$\ln Z = -\beta E_0 +\ln g_0 +O \left( e^{-\beta\Delta} \right).$$

The leading exponential reveals the ground energy, while the constant term reveals its degeneracy.

Finite Systems and Analyticity

For a finite spectrum,

$$Z(\beta) = \sum_a g_ae^{-\beta E_a}$$

is analytic for finite real $\beta$ and strictly positive there. Consequently,

$$\ln Z(\beta)$$

has no real-axis singularity at finite system size.

Sharp thermodynamic phase transitions require a limit in which system size diverges. In the complex plane, finite-system partition functions can have zeros. Families of such zeros can approach the physical real axis in a thermodynamic limit, producing nonanalytic bulk free energies.

Imaginary-time traces and exchange

The operator $e^{-\beta H}$ is an imaginary-time propagator of duration $\beta\hbar$. For a confined particle with $H=p^2/(2m)+V(q)$ under the usual Trotter convergence assumptions,

$$Z=\int_{q(\beta\hbar)=q(0)}\mathcal Dq\,e^{-S_{\mathrm E}[q]/\hbar}, \qquad S_{\mathrm E}=\int_0^{\beta\hbar}d\tau \left[\frac m2\dot q^2+V(q)\right].$$

The trace closes coordinate paths. For $N$ identical particles it also sums permutations of their endpoints, with positive weights for bosons and permutation signs for fermions. This is distinct from the periodic or antiperiodic boundary conditions of coherent-state fields in an unprojected Fock-space trace. Fixed-number projection requires an additional constraint and changes the simple field-boundary argument. See thermal path integrals and coherent-state thermal traces for the respective constructions.

Stable spectral sums

For a retained list of distinct energies and degeneracies, define $a_j=\ln g_j-\beta E_j$ and $a_{\max}=\max_j a_j$. Then

$$\ln Z=a_{\max}+\ln\sum_j e^{a_j-a_{\max}},\qquad P(E_j)=\frac{e^{a_j-a_{\max}}}{\sum_k e^{a_k-a_{\max}}}.$$

Every remaining exponential is at most one. This log-sum-exp form handles either sign of temperature in a finite spectrum and accounts for large degeneracies, whereas shifting only by the lowest energy is adapted to $\beta>0$.

For independent modes compute sums of logarithms instead of products. Use log1p for $\ln(1+x)$ when $x$ is small, and a stable softplus or logaddexp for $\ln(1+e^x)$ when $x$ can be large.

Numerical stabilization does not cure an incomplete spectrum. Increase the cutoff until $\ln Z$ and the target observables converge, or bound the omitted Boltzmann tail using known state-count growth. The correct particle-number and symmetry sectors must also be included before summation.

Exercises

Basis independence

Let $U$ be unitary. Show that

$$\operatorname{Tr} e^{-\beta UHU^\dagger} = \operatorname{Tr} e^{-\beta H}.$$

Solution

Functional calculus gives

$$e^{-\beta UHU^\dagger} = Ue^{-\beta H}U^\dagger.$$

Using cyclicity,

$$\begin{aligned} \operatorname{Tr} e^{-\beta UHU^\dagger} &= \operatorname{Tr} \left( Ue^{-\beta H}U^\dagger \right) \\ &= \operatorname{Tr} \left( U^\dagger Ue^{-\beta H} \right) \\ &= \operatorname{Tr} e^{-\beta H}. \end{aligned}$$

The partition function depends on the spectrum with multiplicity, not on the chosen basis.

Energy cumulants

Starting from

$$K_H(t) = \ln Z(\beta-t) -\ln Z(\beta),$$

show that the first two cumulants are the mean and variance of $H$.

Solution

The first derivative at $t=0$ is

$$\left. \frac{\partial K_H}{\partial t} \right|_{t=0} = -\frac{\partial\ln Z}{\partial\beta} = \langle H\rangle.$$

The second derivative is

$$\left. \frac{\partial^2K_H}{\partial t^2} \right|_{t=0} = \frac{\partial^2\ln Z}{\partial\beta^2}.$$

Direct differentiation gives

$$\frac{\partial^2\ln Z}{\partial\beta^2} = \langle H^2\rangle -\langle H\rangle^2.$$

Hence

$$\kappa_1(H) = \langle H\rangle, \qquad \kappa_2(H) = \operatorname{Var}(H).$$

Recover fixed-number sectors

For two fermionic modes with one-particle energies $\epsilon_1$ and $\epsilon_2$, expand

$$\Xi = \left( 1+ze^{-\beta\epsilon_1} \right) \left( 1+ze^{-\beta\epsilon_2} \right)$$

and identify $Z_0$, $Z_1$, and $Z_2$.

Solution

Expanding,

$$\Xi ={} 1 + z \left( e^{-\beta\epsilon_1} +e^{-\beta\epsilon_2} \right) + z^2 e^{-\beta(\epsilon_1+\epsilon_2)}.$$

Comparing with

$$\Xi = \sum_{N=0}^{2} z^NZ_N$$

gives

$$Z_0=1,$$ $$Z_1 = e^{-\beta\epsilon_1} +e^{-\beta\epsilon_2},$$

and

$$Z_2 = e^{-\beta(\epsilon_1+\epsilon_2)}.$$

The $N=2$ sector has one state because Pauli exclusion forces both modes to be occupied.

Factorization test

Let

$$H = H_A\otimes\mathbb I_B +\mathbb I_A\otimes H_B +gV_{AB}.$$

For which value of $g$ does exact factorization follow immediately? Explain why commutation of $V_{AB}$ with the uncoupled Hamiltonian is not by itself enough to give $Z=Z_AZ_B$.

Solution

At

$$g=0,$$

the Hamiltonian is a sum of operators on independent tensor factors, so

$$Z_{AB} = Z_AZ_B.$$

If $g\neq0$ and $V_{AB}$ commutes with the uncoupled Hamiltonian, then

$$e^{-\beta H} = e^{-\beta H_A} \otimes e^{-\beta H_B} e^{-\beta gV_{AB}},$$

with the tensor factors understood on the full space. The extra operator $e^{-\beta gV_{AB}}$ generally couples the trace over $A$ and $B$, so its trace does not split into $Z_AZ_B$.

Commutation simplifies the exponential but does not remove the interaction or restore a product state count.

References

• A. L. Fetter and J. D. Walecka, Quantum Theory of Many-Particle Systems, Dover (2003), chapters 1 and 2.
• R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals, emended ed., Dover (2010), chapters 2 and 10.
• K. Huang, Statistical Mechanics, 2nd ed., Wiley (1987), chapters 7–12.
• M. Kardar, Statistical Physics of Particles, Cambridge University Press (2007), chapters 2–6.
• H. Kleinert, Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets, 5th ed., World Scientific (2009), chapters 2 and 7.
• L. D. Landau and E. M. Lifshitz, Statistical Physics, Part 1, 3rd ed., Butterworth–Heinemann (1980), sections 31–56.
• R. K. Pathria and P. D. Beale, Statistical Mechanics, 4th ed., Elsevier (2021), chapters 3–8.
• C. N. Yang and T. D. Lee, “Statistical Theory of Equations of State and Phase Transitions. I. Theory of Condensation”, Physical Review 87, 404–409 (1952).