Cluster and virial expansions for the multi-species Tonks gas

TL;DR

This paper develops convergence criteria for the pressure expansions of a multi-species non-overlapping rods model, providing explicit formulas, testing known criteria, and exploring phase transitions.

Contribution

It introduces necessary and sufficient convergence conditions for activity and density expansions, generalizes relations to rooted trees, and analyzes phase transitions in the model.

Findings

Virial expansion converges in larger domains than activity expansion.

Explicit formulas relate rods to rooted trees.

Condensation transition occurs for certain activities.

Abstract

We consider a mixture of non-overlapping rods of different lengths $\ell_k$ moving in $\mathbb{R}$ or $\mathbb{Z}$. Our main result are necessary and sufficient convergence criteria for the expansion of the pressure in terms of the activities $z_k$ and the densities $\rho_k$. This provides an explicit example against which to test known cluster expansion criteria, and illustrates that for non-negative interactions, the virial expansion can converge in a domain much larger than the activity expansion. In addition, we give explicit formulas that generalize the well-known relation between non-overlapping rods and labelled rooted trees. We also prove that for certain choices of the activities, the system can undergo a condensation transition akin to that of the zero-range process. The key tool is a fixed point equation for the pressure.