  In this work we study the transition from normal to anomalous diffusion of
Brownian particles on disordered potentials. The potential model consists of a
series of "potential hills" (defined on unit cell of constant length) whose
heights are chosen randomly from a given distribution. We calculate the exact
expression for the diffusion coefficient in the case of uncorrelated potentials
for arbitrary distributions. We particularly show that when the potential
heights have a Gaussian distribution (with zero mean and a finite variance) the
diffusion of the particles is always normal. In contrast when the distribution
of the potential heights are exponentially distributed we show that the
diffusion coefficient vanishes when system is placed below a critical
temperature. We calculate analytically the diffusion exponent for the anomalous
(subdiffusive) phase by using the so-called "random trap model". We test our
predictions by means of Langevin simulations obtaining good agreement within
the accuracy of our numerical calculations.

Disordered Systems and Neural Networks;Statistical Mechanics