Analyticity for Classical Gasses via Recursion

Abstract

In the recent work of [Michelen, Perkins, Comm. Math. Phys. 399:1 (2023)], a new lower bound of eC φ (β) −1 is obtained for the positive activity up to which the pressure of a classical system of particles with repulsive pair interactions is analytic. In this paper, we extend their method to the class of radially symmetric, locally stable, and tempered pair potentials. Our main result is that the pressure of such systems is analytic for positive activities up to βC+1) , where C > 0 is the local stability constant, W (•) the Lambert W -function, and A φ (β) the contribution from the attractive part of the pair potential to the temperedness constant. In the high-temperature limit, our result improves the classical Penrose-Ruelle bound of C φ (β) −1 e −(βC+1) by a factor of e 2 . This proves the absence of phase transitions in these systems in the Lee-Yang sense for activities up to the above threshold.

Cluster expansions, trees, inversions and correlations

“…Using the more recent tree-graph inequality [60] as well as other results [52,14,15,49] one can improve the value of c 0 obtained in the original [61] paper, but it will only be a minor improvement and it will not avoid the singularity present in the expansion of the pressure with respect to the activity despite the fact that it is a direct method. It might be worth investigating how one could use the perturbative method of the cluster expansion around a different point than the ideal gas, as it was also hinted in [47] where the authors could obtain some significant improvement for the analyticity of the pressure (but not for the cluster expansion).…”