Constructing free energies from partition functions

Partition functions are normalization constants for equilibrium ensembles, but their deeper role is that their logarithms produce the thermodynamic potentials governing macroscopic behavior (see statistical free energy and thermodynamic equilibrium ).

Canonical free energy from the canonical partition function

In the canonical ensemble (fixed $T,V,N$), the partition function $Z_N(\beta,V)$ is constructed by the canonical partition-function construction .

The Helmholtz free energy (as a function of $(T,V,N)$) is defined by

with $\beta$ the inverse temperature . Equivalently, using $T$ explicitly and the Boltzmann constant $k_B$,

This agrees (in the thermodynamic limit and under standard regularity assumptions) with the thermodynamic Helmholtz free energy .

Grand potential from the grand partition function

In the grand canonical ensemble (fixed $T,V,\mu$), the grand partition function $\Xi(\beta,\mu,V)$ is constructed by the grand-canonical partition-function construction .

The corresponding thermodynamic potential is the grand potential

matching the thermodynamic grand potential .

Why the logarithm?

The use of $\log Z$ (and $\log \Xi$) reflects extensivity and additivity:

• For (approximately) independent subsystems, partition functions multiply, while free energies add: if $Z \approx Z^{(1)} Z^{(2)}$, then $-\beta^{-1}\log Z \approx -\beta^{-1}\log Z^{(1)} - \beta^{-1}\log Z^{(2)}$.
• In the thermodynamic limit , one expects $\log Z$ to scale like volume, so $F$ scales like an extensive quantity.

Connection to Legendre structure

Switching which variables are held fixed corresponds to Legendre transforms between potentials. For example, the grand potential is the Legendre transform of $F$ with respect to particle number, as explained in the construction from $F$ to $\Omega$ . This mirrors the conjugacy between $N$ and the chemical potential $\mu$.

Physical interpretation

• $F$ measures the “useful work” available at fixed $(T,V,N)$ once entropic effects are included; minimizing $F$ at fixed $(T,V,N)$ characterizes equilibrium.
• $\Omega$ is the potential adapted to particle exchange; minimizing $\Omega$ at fixed $(T,V,\mu)$ characterizes equilibrium in the presence of a particle reservoir.
• Derivatives of $\log Z$ and $\log \Xi$ generate observable averages and fluctuations; see constructing observables from log partition functions .