Elastic Chain in a Random Potential: Simulation of the Displacement Function $<(u(x)-u(0))^2>$ and Relaxation

TL;DR

This paper simulates an elastic chain in a random potential to analyze displacement correlations, finding exponents consistent with theoretical predictions and revealing forbidden regions in well-particle distances.

Contribution

It provides numerical simulation results of displacement functions and exponents in a one-dimensional elastic chain, confirming theoretical predictions and exploring the distribution of pinning well distances.

Findings

Exponent η ≈ 3/5 at intermediate distances

Exponent χ ≈ 2/3 for size-dependent displacement

Development of forbidden regions in well-particle distance distribution

Abstract

We simulate the low temperature behaviour of an elastic chain in a random potential where the displacements $u(x)$ are confined to the {\it longitudinal} direction ($u(x)$ parallel to $x$) as in a one dimensional charge density wave--type problem. We calculate the displacement correlation function $g(x)=< (u(x)-u(0))^2>$ and the size dependent average square displacement $W(L)=<(u(x)-\bar{u})^2>$. We find that $g(x)\sim x^{2\eta}$ with $\eta\simeq3/4$ at short distances and $\eta\simeq3/5$ at intermediate distances. We cannot resolve the asymptotic long distance dependence of $g$ upon $x$. For the system sizes considered we find $g(L/2)\propto W\sim L^{2\chi}$ with $\chi\simeq2/3$. The exponent $\eta\simeq3/5$ is in agreement with the Random Manifold exponent obtained from replica calculations and the exponent $\chi\simeq2/3$ is consistent with an exact solution for the chain with {\it transverse} displacements ($u(x)$ perpendicular to $x$).The distribution of nearest distances between pinning wells and chain-particles is found to develop forbidden regions.