The microscopic meaning of grand potential: cluster properties of the one-dimensional lattice gas

TL;DR

This paper explores the microscopic interpretation of grand potential through cluster properties of a one-dimensional lattice gas, demonstrating the combinatorial approach's effectiveness across different temperature regimes.

Contribution

It provides a concrete example of the combinatorial method applied to a lattice gas, extending traditional statistical mechanics results with new insights at various temperature limits.

Findings

Closed-form grand partition function at zero temperature

Non-analytic behavior of grand potential confirmed

Alternative proofs for known results at finite temperatures

Abstract

We demonstrate, with a concrete example, how the combinatorial approach to a general system of particles, which was introduced in detail in the earlier paper arXiv:1205.4986, works and where it enters to provide a genuine extension of results obtainable by more traditional methods of statistical mechanics. To this end, an effort is made to study cluster properties of the one-dimensional lattice gas with nearest neighbor interactions. Three cases: the infinite temperature limit, the range of finite temperatures, and the zero temperature limit are discussed separately, yielding some new results and providing alternative proofs of known results. In particular, the closed-form expression for the grand partition function in the zero temperature limit is obtained, which results in the non-analytic behavior of the grand potential, in accordance with the Yang-Lee theory.