Energy dynamics in the Sinai model

TL;DR

This paper analyzes the energy dynamics of a particle in the Sinai model, deriving exact distributions of energy and position over time, revealing nonanalytic behaviors and extensions to non-zero fields, with implications for spin glass magnetization.

Contribution

It provides the exact large time distribution of the scaled potential energy in the Sinai model using RSRG and extends results to non-zero fields and constrained paths, revealing new nonanalytic behaviors.

Findings

Distribution of scaled potential energy exhibits nonanalyticity at w=1

Joint distribution of energy and position computed explicitly

Differences in behavior with reflecting boundary and drift analyzed

Abstract

We study the time dependent potential energy $W(t)=U(x(0)) - U(x(t))$ of a particle diffusing in a one dimensional random force field (the Sinai model). Using the real space renormalization group method (RSRG), we obtain the exact large time limit of the probability distribution of the scaling variable $w=W(t)/(T \ln t)$. This distribution exhibits a {\it nonanalytic} behaviour at $w=1$. These results are extended to a small non-zero applied field. Using the constrained path integral method, we moreover compute the joint distribution of energy $W(t)$ and position $x(t)$ at time $t$. In presence of a reflecting boundary at the starting point, with possibly some drift in the + direction, the RSRG very simply yields the one time and aging two-time behavior of this joint probability. It exhibits differences in behaviour compared to the unbounded motion, such as analyticity. Relations with some magnetization distributions in the 1D spin glass are discussed.