Occupation times and ergodicity breaking in biased continuous time randomwalks

Abstract

Continuous time random walk (CTRW) models are widely used to model diffusion incondensed matter. There are two classes of such models, distinguished by the convergenceor divergence of the mean waiting time. Systems with finite average sojourn time areergodic and thus Boltzmann–Gibbs statistics can be applied. We investigate the statisticalproperties of CTRW models with infinite average sojourn time; in particular, theoccupation time probability density function is obtained. It is shown that in thenon-ergodic phase the distribution of the occupation time of the particle on a given latticepoint exhibits bimodal U or trimodal W shape, related to the arcsine law. The key pointsare as follows. (a) In a CTRW with finite or infinite mean waiting time, the distribution ofthe number of visits on a lattice point is determined by the probability that a member ofan ensemble of particles in equilibrium occupies the lattice point. (b) The asymmetryparameter of the probability distribution function of occupation times is related to theBoltzmann probability and to the partition function. (c) The ensemble average is givenby Boltzmann–Gibbs statistics for either finite or infinite mean sojourn time,when detailed balance conditions hold. (d) A non-ergodic generalization of theBoltzmann–Gibbs statistical mechanics for systems with infinite mean sojourn time isfound.