The partition function of multicomponent log-gases

Abstract

We give an expression for the partition function of a one-dimensional log-gas comprised of particles of (possibly) different integer charge at inverse temperature β = 1 (restricted to the line in the presence of a neutralizing field) in terms of the Berezin (Grassmann) integral of an associated non-homogeneous alternating tensor. This is the analogue of the de Bruijn integral identities (de Bruijn 1956 J. Indian Math. Soc. 19 133–51 ) (for β = 1 and β = 4) ensembles extended to multicomponent ensembles.