Equilibrium Ensemble Approach to Disordered Systems I: General Theory, Exact Results

TL;DR

This paper introduces Morita's equilibrium ensemble approach for disordered systems, highlighting its connections to traditional methods, and presents exact solutions for one-dimensional site- and bond-diluted systems with various disorder correlations.

Contribution

It generalizes grand ensemble ideas and links them to low-concentration and perturbation expansions, providing a variational framework and exact solutions for specific disordered models.

Findings

Exact solutions for 1D site-diluted systems

Exact solutions for 1D bond-diluted systems

Connections to conventional disordered system approaches

Abstract

An outline of Morita's equilibrium ensemble approach to disordered systems is given, and hitherto unnoticed relations to other, more conventional approaches in the theory of disordered systems are pointed out. It is demonstrated to constitute a generalization of the idea of grand ensembles and to be intimately related also to conventional low--concentration expansions as well as to perturbation expansions about ordered reference systems. Moreover, we draw attention to the variational content of the equilibrium ensemble formulation. A number of exact results are presented, among them general solutions for site-- and bond-- diluted systems in one dimension, both for uncorrelated, and for correlated disorder.