Partition Function · aiwiki.page

A partition function is a sum or integral of statistical weights over the possible states of a system. In statistical mechanics, it normalizes the probabilities of microscopic states and provides a route from their energies to macroscopic properties in thermodynamic equilibrium. Related functions appear in probabilistic modeling, where they convert unnormalized weights into a probability distribution. Its usual symbol is $Z$ , although notation depends on the ensemble and application. (damtp.cam.ac.uk)

Canonical definition

The canonical ensemble describes a system that exchanges energy with a thermal reservoir while its volume $V$ and particle number $N$ remain fixed. At absolute temperature $T$ , its partition function is

Z(T,V,N)=\sum_i e^{-\beta E_i}, \qquad \beta=\frac{1}{k_{\mathrm B}T},

where $E_i$ is the energy of microscopic state $i$ , and $k_{\mathrm B}$ is the Boltzmann constant. The resulting Boltzmann distribution assigns probability

p_i=\frac{e^{-\beta E_i}}{Z}.

Thus $Z$ ensures that $\sum_i p_i=1$ ; it is not itself a probability. States with higher energies receive smaller individual weights at positive temperature. (damtp.cam.ac.uk)

The sum counts states rather than merely distinct energy values. If an energy level $E_a$ contains $g_a$ states, its contribution is $g_a e^{-\beta E_a}$ . Consequently, numerous higher-energy states can collectively outweigh a small number of lower-energy states. For infinite state spaces, the sum must converge for the normalized distribution to exist. (damtp.cam.ac.uk)

Thermodynamic information

The canonical partition function determines the Helmholtz free energy:

F=-k_{\mathrm B}T\ln Z.

Other quantities follow from derivatives. For a Hamiltonian without explicit temperature dependence, the internal energy is

U=\langle E\rangle =-\left(\frac{\partial\ln Z}{\partial\beta}\right)_{V,N}.

The entropy and pressure are

S=k_{\mathrm B}(\ln Z+\beta U), \qquad P=k_{\mathrm B}T \left(\frac{\partial\ln Z}{\partial V}\right)_{T,N}.

The held-fixed variables matter: different constraints lead to different thermodynamic derivatives. These relations make the partition function more than a normalization device—it encodes the equilibrium equation of state and thermodynamic response. (damtp.cam.ac.uk)

Its second derivative measures energy fluctuations:

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

This variance is related to the constant-volume heat capacity by

C_V=\frac{\langle E^2\rangle-\langle E\rangle^2} {k_{\mathrm B}T^2}.

Macroscopic thermal response is therefore directly connected to microscopic fluctuations. (damtp.cam.ac.uk)

Quantum and classical forms

In quantum mechanics, the canonical definition can be written independently of a chosen basis:

Z=\operatorname{Tr}(e^{-\beta\hat H}),

where $\hat H$ is the Hamiltonian operator and $\operatorname{Tr}$ is the trace. Evaluating the trace in an energy eigenbasis recovers the sum over states. The equilibrium density matrix is correspondingly $\rho=e^{-\beta\hat H}/Z$ . (damtp.cam.ac.uk)

For $N$ identical, structureless particles in three dimensions, the classical Maxwell–Boltzmann expression is an integral over phase space:

Z_N=\frac{1}{N!h^{3N}} \int e^{-\beta H(\mathbf q,\mathbf p)} \,d^{3N}q\,d^{3N}p.

Here $h$ is the Planck constant. The phase-space measure divided by $h^{3N}$ makes the statistical count dimensionless, while $N!$ corrects the overcounting caused by permuting identical particles. This classical expression does not replace the appropriate quantum treatment when quantum statistical effects are important. (damtp.cam.ac.uk)

Factorization and a two-state example

When distinguishable subsystems are independent and their energies add, their partition functions multiply:

Z_{\mathrm{total}}=\prod_a Z_a.

This simplifies calculations because $\ln Z_{\mathrm{total}}$ becomes a sum, and the corresponding free energies are additive. Interactions generally prevent this simple factorization. (damtp.cam.ac.uk)

For a single system with two nondegenerate levels, $0$ and $\epsilon>0$ ,

Z_1=1+e^{-\beta\epsilon}, \qquad U_1=\frac{\epsilon}{1+e^{\beta\epsilon}}.

At low positive temperature, the lower state dominates; at temperatures large compared with $\epsilon/k_{\mathrm B}$ , the two probabilities approach equality. For $N$ independent distinguishable copies, $Z_N=(Z_1)^N$ . This example illustrates how state counting and thermal weighting are combined in one expression. (damtp.cam.ac.uk)

Other ensembles

The grand canonical ensemble allows exchange of particles as well as energy with a reservoir. Its grand partition function is

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

where $\mu$ is the chemical potential. The factor $e^{\beta\mu}$ is called fugacity. The associated grand potential is

\Omega=-k_{\mathrm B}T\ln\Xi,

and the mean particle number follows from $\langle N\rangle=\beta^{-1}(\partial\ln\Xi/\partial\mu)_{T,V}$ . Choosing an ensemble determines which states are included and how they are weighted. (damtp.cam.ac.uk)

Probability models and computation

In machine learning, an energy-based model often takes the form

p_\theta(x)=\frac{e^{-E_\theta(x)}}{Z(\theta)}, \qquad Z(\theta)=\int e^{-E_\theta(x)}\,dx,

with a sum replacing the integral for discrete states. Here “energy” can be a learned score rather than physical energy, and any inverse-temperature factor may be absorbed into its definition. A finite, positive $Z(\theta)$ is required for normalization. (deeplearningbook.org)

Computing this function can be difficult because it aggregates weights across a high-dimensional state space. In maximum likelihood estimation, derivatives of $\ln Z(\theta)$ introduce model expectations. Markov chain Monte Carlo can approximate those expectations without directly evaluating the normalization constant. Estimating $Z$ itself is a separate task; annealed importance sampling addresses it by connecting a tractable reference distribution to the target through intermediate distributions. (deeplearningbook.org)