Relationship between Entropy and Diffusion: A statistical mechanical derivation of Rosenfeld expression for a rugged energy landscape

TL;DR

This paper derives the Rosenfeld diffusion-entropy scaling relation for rugged energy landscapes using a statistical mechanical approach, demonstrating its validity across dimensions and identifying conditions where it breaks down.

Contribution

It provides a rigorous derivation of the Rosenfeld scaling from first principles and extends its applicability to higher dimensions via an effective medium approximation.

Findings

Exact excess entropy for Gaussian rugged landscapes

Rosenfeld scaling derived for any dimension > 1

Scaling breaks down with spatial correlations

Abstract

Diffusion-a measure of dynamics, and entropy-a measure of disorder in the system, are found to be intimately correlated in many systems, and the correlation is often strongly non-linear. We explore the origin of this complex dependence by studying diffusion of a point Brownian particle on a model potential energy surface characterized by ruggedness. If we assume that the ruggedness has a Gaussian distribution then for this model, one can obtain the excess entropy exactly for any dimension. By using the expression for the mean first passage time (MFPT), we present a statistical mechanical derivation of the well-known and well-tested scaling relation proposed by Rosenfeld between diffusion and excess entropy. In anticipation that Rosenfeld diffusion-entropy scaling (RDES) relation may continue to be valid in higher dimensions (where the mean first passage time approach is not available), we carry out an effective medium approximation (EMA) based analysis of the effective transition rate, and hence of the effective diffusion coefficient. We show that the EMA expression can be used to derive the RDES scaling relation for any dimension higher than unity. However, RDES is shown to break down in the presence of spatial correlation among the energy landscape values.