Convergence of the Mayer Series via Cauchy Majorant Method with Application to the Yukawa Gas in the Region of Collapse

TL;DR

This paper develops a method to analyze the convergence of the Mayer series for the Yukawa gas using Cauchy majorant functions, providing explicit solutions and conditions for convergence in the collapse region.

Contribution

It introduces a novel approach using nonlinear differential equations and Lambert W-function to establish convergence criteria for Mayer series in the Yukawa gas.

Findings

Derived explicit convergence domain for Mayer series

Established conditions for convergence in the collapse region

Numerical evidence supports the stability condition

Abstract

We construct majorant functions $\Phi (t,z)$ for the Mayer series of pressure satisfying a nonlinear differential equation of first order which can be solved by the method of characteristics. The domain $| z| <(e\tau) ^{-1}$ of convergence of Mayer series is given by the envelop of characteristic intersections. For non negative potentials we derive an explicit solution in terms of the Lambert $W$ --function which is related to the exponential generating function $T$ of rooted trees as $T(x)=-W(-x)$. For stable potentials the solution is majorized by a non negative potential solution. There are many choices in this case and we combine this freedom together with a Lagrange multiplier to examine the Yukawa gas in the region of collapse. We give, in this paper, a sufficient condition to establish a conjecture of Benfatto, Gallavotti and Nicol\'{o}. For any $\beta \in \lbrack 4\pi, 8\pi)$, the Mayer series with the leading terms of the expansion omitted (how many depending on $\beta $) is shown to be convergent provided an improved stability condition holds. Numerical calculations presented indicate this condition is satisfied if few particles are involved.