Reaction-Diffusion Processes, Critical Dynamics and Quantum Chains

TL;DR

This paper explores the connection between non-equilibrium classical processes and quantum chains, revealing new insights into their properties through integrability and algebraic structures.

Contribution

It establishes a framework linking reaction-diffusion processes and critical dynamics to quantum chain models, highlighting applications of integrability and algebraic methods.

Findings

Mapping of classical non-equilibrium problems to quantum chains

Identification of algebraic structures like Hecke algebras in diffusion processes

Analysis of critical exponents and finite-size scaling in these models

Abstract

The master equation describing non-equilibrium one-dimensional problems like diffusion limited reactions or critical dynamics of classical spin systems can be written as a Schr\"odinger equation in which the wave function is the probability distribution and the Hamiltonian is that of a quantum chain with nearest neighbor interactions. Since many one-dimensional quantum chains are integrable, this opens a new field of applications. At the same time physical intuition and probabilistic methods bring new insight into the understanding of the properties of quantum chains. A simple example is the asymmetric diffusion of several species of particles which leads naturally to Hecke algebras and $q$-deformed quantum groups. Many other examples are given. Several relevant technical aspects like critical exponents, correlation functions and finite-size scaling are also discussed in detail.