Random Walks, Reaction-Diffusion, and Nonequilibrium Dynamics of Spin Chains in One-dimensional Random Environments

TL;DR

This paper investigates the long-time behavior of spin chains in one-dimensional random environments using a real space renormalization group, revealing aging phenomena, distribution properties, and effects of small forces in nonequilibrium dynamics.

Contribution

It provides asymptotically exact results for diffusion, aging, and non-equilibrium correlations in 1D random systems, extending to reaction-diffusion models and the random field Ising model.

Findings

Distribution of particle positions derived

Aging behavior with logarithmic scaling identified

Persistence exponents for reaction-diffusion models calculated

Abstract

Sinai's model of diffusion in one-dimension with random local bias is studied by a real space renormalization group which yields asymptotically exact long time results. The distribution of the position of a particle and the probability of it not returning to the origin are obtained, as well as the two-time distribution which exhibits "aging" with $\frac{\ln t}{\ln t'}$ scaling and a singularity at $\ln t =\ln t'$. The effects of a small uniform force are also studied. Extension to motion of many domain walls yields non-equilibrium time dependent correlations for the 1D random field Ising model with Glauber dynamics and "persistence" exponents of 1D reaction-diffusion models with random forces.