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

TL;DR

This paper derives the free energy of small thermodynamic systems in contact with a heat bath, incorporating the Gibbs factorial, by imposing quasi-static work and Gibbs-Helmholtz conditions, and introduces a quasi-static decomposition method.

Contribution

It introduces a method to determine free energy in small systems using quasi-static decomposition and shows the necessity of the Gibbs factorial in the free energy expression.

Findings

Derived the unique form of free energy for small systems.

Established the role of Gibbs factorial in free energy.

Proposed a quasi-static decomposition approach.

Abstract

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