Exclusive Many-Particle Diffusion in Disordered Media and Correlation Functions for Random Vertex Models

TL;DR

This paper analyzes many-particle diffusion in disordered media by mapping stochastic particle systems to quantum Hamiltonians, deriving exact correlation functions, and exploring models like the six-vertex and random barrier models.

Contribution

It introduces a method to compute two-point density correlations in disordered systems using single-particle probabilities, extending to models with space-dependent interactions.

Findings

Exact correlation functions for disordered media derived.

Connection established between multi-particle and single-particle dynamics.

Application to models like the six-vertex and random barrier models.

Abstract

We consider systems of particles hopping stochastically on $d$-dimensional lattices with space-dependent probabilities. We map the master equation onto an evolution equation in a Fock space where the dynamics are given by a quantum Hamiltonian (continuous time) or a transfer matrix resp. (discrete time). We show that under certain conditions the time-dependent two-point density correlation function in the $N$-particle steady state can be computed from the probability distribution of a single particle moving in the same environment. Focussing on exclusion models where each lattice site can be occupied by at most one particle we discuss as an example for such a stochastic process a generalized Heisenberg antiferromagnet where the strength of the spin-spin coupling is space-dependent. In discrete time one obtains for one-dimensional systems the diagonal-to-diagonal transfer matrix of the two-dimensional six-vertex model with space-dependent vertex weights. For a random distribution of the vertex weights one obtains a version of the random barrier model describing diffusion of particles in disordered media. We derive exact expressions for the averaged two-point density correlation functions in the presence of weak, correlated disorder.