Over-damped Brownian dynamics in piecewise-defined energy landscapes

We study the over-damped Brownian dynamics of particles moving in piecewise-defined potential energy landscapes U(x), where the height Q of each section is obtained from the exponential distribution p(Q) = aβ exp(−aβQ), where β is the reciprocal thermal energy, and a > 0. The averaged effective diffusion coefficient Deff is introduced to characterise the diffusive motion: <x²> = 2 Deff t. A general expression for Deff in terms of U(x) and p(Q) is derived, and then applied to three types of energy landscape: flat sections, smooth maxima, and sharp maxima. All three cases display a transition between sub-diffusive and diffusive behaviour at a = 1, and a reduction to free diffusion as a → ∞. The behaviour of Deff around the transition is investigated and found to depend heavily upon the shape of the maxima: energy landscapes made up of flat sections or smooth maxima display power-law behaviour, whilst for landscapes with sharp maxima, strongly divergent behaviour is observed. Two aspects of the sub-diffusive regime are studied: the growth of the mean squared displacement with time, and the distribution of mean first-passage times.