Cluster and virial expansions for the multi-species Tonks gas

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.

24 pages in J. Stat. Phys. online 2015

References in corpus (5)

• Cluster expansion for abstract polymer models. New bounds from an old approach
• Abstract cluster expansion with applications to statistical mechanical systems
• The analyticity region of the hard sphere gas. Improved bounds
• Multispecies Virial Expansions
• On the Virial Series for a Gas of Particles with Uniformly Repulsive Pairwise Interaction

Cited by in corpus (8)

• Cluster expansions for Gibbs point processes
• Disagreement percolation for the hard-sphere model
• Large deviations and gradient flows for the Brownian one-dimensional hard-rod system
• Revisiting Groeneveld's approach to the virial expansion
• Thermodynamics of a hierarchical mixture of cubes
• Cluster expansions: Necessary and sufficient convergence conditions
• Shearer's point process and the hard-sphere model in one dimension
• Logarithms of Catalan generating functions: A combinatorial approach