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

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