Physics The Generalized Plasma in One Dimension ... - Project Euclid
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Physics The Generalized Plasma in One Dimension ... - Project Euclid
The Generalized Plasma in One Dimension: Evaluation of a Partition Function D. M. Bressoud1'* and I. P. Goulden2 1 Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, USA 2 Department of Mathematics, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
Abstract. Forrester and Jancovici have given sum rules for a two-dimensional generalized plasma with two species of particles interacting through logarithmic potentials with three independent coupling constants. They have also found a specific one-dimensional solvable model which satisfies the analogs of their sum rules. A class of one-dimensional models for which the partition function is evaluable is given as well as a more general result evaluating multidimensional integrals. 1. Introduction
Forrester and Jancovici [1] have given an exactly solved model for a onedimensional generalized plasma with two species of particles (roman and greek), interacting through logarithmic potentials, with three independent coupling constants. This was motivated by their discovery of sum rules for such a generalized plasma in two dimensions (see also Halperin [2] and Girvin [3]) and the desire to at least verify the one-dimensional analogs of these rules. The two-dimensional system with roman and greek particles of density ρR and ρG, respectively, and independent coupling constants gRR,gRG>&GG has Hamiltonian tofy-gGG Σ ln^
(1.1)
where the particle-background and background-background interactions have been chosen in a way which compensates the remote particle-particle interactions * Partially supported by N.S.F. grant
288
D. M. Bressoud and I. P. Goulden
so that we can expect the system to have a well-behaved thermodynamic limit. Forrester and Jancovici's exactly solved model is for the system where all particles lie on a circle of radius R:
£RR= SRG = 2, £GG= 4. We shall consider the more general case of this onedimensional model,
