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For small thermodynamic systems in contact with a heat bath, we determine thefree energy by imposing the following two conditions. First, the quasi-staticwork in any configuration change is equal to the free energy difference.Second, the temperature dependence of the free energy satisfies theGibbs-Helmholtz relation. We find that these prerequisites uniquely lead to thefree energy of a classical system consisting of $N$-interacting identicalparticles, up to an additive constant proportional to $N$. The free energy thusdetermined contains the Gibbs factorial $N!$ in addition to the phase spaceintegration of the Gibbs-Boltzmann factor. The key step in the derivation is toconstruct a quasi-static decomposition of small thermodynamic systems.

Quasi-static decomposition and the Gibbs factorial in small thermodynamic system