Diffusion with critically correlated traps and the slow relaxation of the longest wavelength mode

TL;DR

This paper investigates how critical correlations in trap distributions on a substrate affect diffusion, revealing that such correlations significantly increase the low-frequency density of states and slow the relaxation of the longest wavelength mode.

Contribution

It introduces a numerical analysis of diffusion on a critically correlated trap substrate and links the density of states with the largest eigenvalue through finite size scaling.

Findings

Critical correlations double the stretch exponent in the density of states.

The largest eigenvalue scaling aligns with the density of states analysis.

Critical trap distribution slows down the relaxation of the longest wavelength mode.

Abstract

We study diffusion on a substrate with permanent traps distributed with critical positional correlation, modeled by their placement on the perimeters of a critical percolation cluster. We perform a numerical analysis of the vibrational density of states and the largest eigenvalue of the equivalent scalar elasticity problem using the method of Arnoldi and Saad. We show that the critical trap correlation increases the exponent appearing in the stretched exponential behavior of the low frequency density of states by approximately a factor of two as compared to the case of no correlations. A finite size scaling hypothesis of the largest eigenvalue is proposed and its relation to the density of states is given. The numerical analysis of this scaling postulate leads to the estimation of the stretch exponent in good agreement with the density of states result.