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

### Daniel Fisher 1, Pierre Le Doussal 2, Cecile Monthus 3

#### Physical Review Letters 80 (1998) 3539-3542

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 äging\' 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.

• 1. Lyman Laboratory of Physics,
University of Harvard
• 2. Laboratoire de Physique Théorique de l'ENS (LPTENS),
CNRS : UMR8549 – Université Paris VI - Pierre et Marie Curie – Ecole Normale Supérieure de Paris - ENS Paris
• 3. Service de Physique Théorique (SPhT),
CNRS : URA2306 – CEA : DSM/SPHT