Reaction-Diffusion Processes of Hard-Core Particles

TL;DR

This paper analyzes a broad class of reaction-diffusion particle systems, revealing integrable properties, duality relations, and connections to quantum spin models, with exact correlation function equations and spectral equivalences.

Contribution

It introduces a 12-parameter stochastic model with exact correlation equations, duality relations, and links to quantum spin Hamiltonians, expanding understanding of reaction-diffusion processes.

Findings

Correlation functions satisfy linear differential-difference equations.

Average density follows an integrable diffusion equation.

Spectrum of the stochastic Hamiltonian matches that of a quantum spin model.

Abstract

We study a 12-parameter stochastic process involving particles with two-site interaction and hard-core repulsion on a $d$-dimensional lattice. In this model, which includes the asymmetric exclusion process, contact processes and other processes, the stochastic variables are particle occupation numbers taking values $n_{\vec{x}}=0,1$. We show that on a 10-parameter submanifold the $k$-point equal-time correlation functions $\exval{n_{\vec{x}_1} \cdots n_{\vec{x}_k}}$ satisfy linear differential- difference equations involving no higher correlators. In particular, the average density $\exval{n_{\vec{x}}} $ satisfies an integrable diffusion-type equation. These properties are explained in terms of dual processes and various duality relations are derived. By defining the time evolution of the stochastic process in terms of a quantum Hamiltonian $H$, the model becomes equivalent to a lattice model in thermal equilibrium in $d+1$ dimensions. We show that the spectrum of $H$ is identical to the spectrum of the quantum Hamiltonian of a $d$-dimensional, anisotropic spin-1/2 Heisenberg model. In one dimension our results hint at some new algebraic structure behind the integrability of the system.