# On the Mayer series of two-dimensional Yukawa gas at inverse temperature in the interval of collapse

## TL;DR

This paper proves a theorem on the minimal energy configurations of a two-dimensional Yukawa gas, resolving issues with the convergence of the Mayer series in the collapse interval, and extends the convergence up to the second threshold.

## Contribution

It introduces a new theorem on minimal energy configurations for Yukawa interactions and applies it to prove convergence of the Mayer series up to 6π, with insights to extend to 8π.

## Key findings

- Proved minimal energy configurations for charged particles with Yukawa potential.
- Established convergence of the Mayer series up to the second collapse threshold 6π.
- Provided mechanisms to extend convergence results up to 8π.

## Abstract

A Theorem on the minimal specific energy for a system with \pm 1 charged particles interacting through the Yukawa pair potential v is proved which may stated as follows. Let v be represented by scale mixtures of d-dimensional Euclid's hat (cutoff at short scale distances) with d\geq 2. For any even number of particles n, the interacting energy U_{n} divided by n, attains an n-independent minimum at a configuration with zero net charge and particle positions collapsed altogether to a point. For any odd number of particles n, the ratio U_{n}/(n-1) attains its minimum value, the same of the previous cases, at the configuration with \pm 1 net charge and particle positions collapsed to a point. This Theorem is then used to resolve an obstructive remark of an unpublished paper (Remark 7.5 of \cite{Guidi-Marchetti}) which, whether the standard decomposition of the Yukawa potential into scales were adopted, would impede a direct proof of the convergence of the Mayer series of the two-dimensional Yukawa gas for the inverse temperature in the whole interval [4\pi ,8\pi ) of collapse. In the present paper, it is proven convergence up to the second threshold 6\pi and its given explanations on the mechanism that allow it to be extend up to 8\pi . The paper distinguishes the matters concerning stability from those related to convergence of the Mayer series. In respect to the latter the paper dedicates to the Cauchy majorante method applied to the density function of Yukawa gas in the interval of collapses. It also dedicates to the proof of the main Theorem and estimates of the modified Bessel functions of second kind involved in both representations of two-dimensional Yukawa potential: standard and scale mixture of the Euclid's hat function.