Domingos H. U. Marchetti

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 This Theorem is then used to resolve an obstructive remark of an unpublished paper (Remark 7.5 of \cite{Guidi-Marchetti}) which, whether the standard decomposition of the Yukawa potential into scales were adopted, would impede a direct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.

The pressure of a gas of particles with a uniformly repulsive pair interaction in a finite container is shown to satisfy (exactly as a formal object) a “viscous” Hamilton–Jacobi (H–J) equation whose solution in power series is recursively given by the variation of constants formula. We investigate the solution of the H–J and of its Legendre transform equation by the Cauchy-majorant method and provide a lower bound to the radius of convergence on the virial series of the fluid which goes beyond the threshold established by Lagrange’s inversion formula. Such results obtained in (On the virial series for a gas of particles with uniformly repulsive pairwise interactions (2014) Preprint) are reviewed and regarded as the first step towards the solution of a problem posed by Kac, Uhlenbeck and Hemmer (J. Math. Phys. 4 (1963) 216–228), questioning on the relation of the approach to the mean field theory with Ursell–Mayer expansion.