Thermal state from entropy maximization

Maximum-entropy derivation of the canonical (and generalized Gibbs) distribution from macroscopic constraints.

Thermal state from entropy maximization

A central construction of equilibrium statistical mechanics is: given only partial macroscopic information, choose the least biased microscopic distribution consistent with that information. This principle is implemented by maximizing the Gibbs/Shannon entropy .

Problem (choose a distribution from constraints)

Let $x$ denote a microstate in classical phase space , and let $H(x)$ be the Hamiltonian function . A statistical state is a probability density $\rho(x)\ge 0$ normalized with respect to the phase-space volume element $d\Gamma$ :

\int \rho(x)\,d\Gamma(x) \;=\; 1.

The (dimensionless) entropy functional is

s[\rho] \;=\; -\int \rho(x)\,\log\rho(x)\,d\Gamma(x),

and the physical entropy is $S[\rho]=k_B\,s[\rho]$ with $k_B$ the Boltzmann constant .

Assume we only know the mean energy (a macroscopic constraint)

\int \rho(x)\,H(x)\,d\Gamma(x) \;=\; U.

Solution (canonical Gibbs distribution)

Maximizing $s[\rho]$ subject to normalization and fixed mean energy gives, via Lagrange multipliers, a unique maximizer of the form

\rho_\beta(x) \;=\; \frac{e^{-\beta H(x)}}{Z(\beta)}, \qquad Z(\beta) \;=\; \int e^{-\beta H(x)}\,d\Gamma(x).

The normalizing factor $Z(\beta)$ is the canonical partition function , and the resulting state is exactly the canonical ensemble . The multiplier $\beta$ is identified with the inverse temperature .

This provides a variational (information-theoretic) characterization of thermal equilibrium: among all distributions with the same mean energy, $\rho_\beta$ has maximal entropy.

Generalization (multiple constraints)

If, in addition to the mean energy, one constrains the mean values of other quantities $Q_i(x)$ (often conserved charges),

\int \rho(x)\,Q_i(x)\,d\Gamma(x) \;=\; q_i,

the same entropy-maximization procedure yields

\rho(x) \propto \exp\!\Big(-\lambda_0 - \sum_i \lambda_i Q_i(x)\Big),

which is the basis of the generalized Gibbs ensemble .

Interpretation via relative entropy

Entropy maximization can also be phrased as an optimization over information distance: the maximizer is the distribution that is “closest” to a chosen reference measure while satisfying the constraints, using the Kullback–Leibler divergence . The uniqueness/optimality is underwritten by the Gibbs inequality .

Physical meaning

The constraints encode what is known macroscopically (here, the mean energy).
The maximizer is the least structured microscopic state compatible with those constraints.
When a small system is weakly coupled to a large heat bath, this construction matches the canonical state obtained from deriving the canonical ensemble from the microcanonical picture .