Abstract
The Sinai model of a tracer diffusing in a quenched Brownian potential is a much-studied problem exhibiting a logarithmically slow anomalous diffusion due to the growth of energy barriers with the system size. However, if the potential is random but periodic, the regime of anomalous diffusion crosses over to one of normal diffusion once a tracer has diffused over a few periods of the system. Here we consider a system in which the potential is given by a Brownian bridge on a finite interval and then periodically repeated over the whole real line and study the power spectrum of the diffusive process in such a potential. We show that for most of realizations of in a given realization of the potential, the low-frequency behavior is , i.e., the same as for standard Brownian motion, and the amplitude is a disorder-dependent random variable with a finite support. Focusing on the statistical properties of this random variable, we determine the moments of of arbitrary, negative, or positive order and demonstrate that they exhibit a multifractal dependence on and a rather unusual dependence on the temperature and on the periodicity , which are supported by atypical realizations of the periodic disorder. We finally show that the distribution of has a log-normal left tail and exhibits an essential singularity close to the right edge of the support, which is related to the Lifshitz singularity. Our findings are based both on analytic results and on extensive numerical simulations of the process .
- Received 4 July 2016
DOI:https://doi.org/10.1103/PhysRevE.94.032131
©2016 American Physical Society