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

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 $\ln t/\ln t^{\prime }$ scaling and a singularity at $x\left(t\right)=x\left(t^{\prime }\right)$. The effects of a small uniform force are also studied. Extension to motion of many domain walls yields nonequilibrium time dependent correlations for the 1D random field Ising model with Glauber dynamics and “persistence” exponents of 1D reaction-diffusion models with random forces.