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

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 $\eta \simeq 3/4$ at short distances and $\eta \simeq 3/5$ 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 $\chi \simeq 2/3$. The exponent $\eta \simeq 3/5$ is in agreement with the Random Manifold exponent obtained from replica calculations and the exponent $\chi \simeq 2/3$ 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.

19 pages of LaTex, 6 postscript figures available on request, submitted to Journal of Physics A, MAJOR CHANGES