On the derivation of the mean field equation of the Gibbs distribution function for equilibrium vortices in an external field (Regularity and Asymptotic Analysis for Critical Cases of Partial Differential Equations)

Abstract

Motivated by several experimental facts, we are interested in the linear response of equilibrium vortices. In order to study the phenomenon, here we investigate the mean field limit of equilibrium vortices perturbed by an external field and derive the mean field equation of the Gibbs distribution function. Similar limits for classical point particles with bounded interactions were studied by Messer-Spohn [14] and later the results were extended to the system of vortices, which interact via the singular logarithmic potential, by Caglioti et al [2] and Kiessling [10]. In this paper, we start with the review of these results in some detail and extend their arguments to the case for vortices perturbed by an external field.