  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.
