Continuous-time random walk theory of superslow diffusion

¹ Max-Planck-Institut für Physik komplexer Systeme - Nöthnitzer Straße 38, D-01187 Dresden, Germany, EU
² Sumy State University - Rimsky-Korsakov Street 2, UA-40007 Sumy, Ukraine

^a stdenis@pks.mpg.de

Received: 12 September 2010
Accepted: 13 October 2010

Abstract

Superslow diffusion, i.e., the long-time diffusion of particles whose mean-square displacement (variance) grows slower than any power of time, is studied in the framework of the decoupled continuous-time random walk model. We show that this behavior of the variance occurs when the complementary cumulative distribution function of waiting times is asymptotically described by a slowly varying function. In this case, we derive a general representation of the laws of superslow diffusion for both biased and unbiased versions of the model and, to illustrate the obtained results, consider two particular classes of waiting-time distributions.