Normal and anomalous diffusion of Brownian particles on disordered potentials

• A model of Brownian particles on disordered potentials exhibiting a transition from normal to anomalous subdiffusion is introduced.
• We obtain the exact effective diffusion coefficient, which allows to exactly determine critical temperature.
• The subdiffusive phase is analytically treated by approximating it by the random trap model.
• The diffusion exponents are obtained approximately in a closed form.

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 a 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 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 is exponentially distributed the diffusion coefficient vanishes when the 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”. Our predictions are tested by means of Langevin simulations obtaining good agreement within the accuracy of our numerical calculations.