The Partition Function of Multicomponent Log-Gases

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 {\beta} = 1 (restricted to the line in the presence of a neutralizing field) in terms of the Berezin integral of an associated non- homogeneous alternating tensor. This is the analog of the de Bruijn integral identities [3] (for {\beta} = 1 and {\beta} = 4) ensembles extended to multicomponent ensembles.