Memory-controlled diffusion

Abstract

Memory effects require for their incorporation into random-walk models an extension of the conventional equations. The linear Fokker-Planck equation for the probability density p(r,t) is generalized by including nonlinear and nonlocal spatial-temporal memory effects. The realization of the memory kernel is restricted due the conservation of the basic quantity p. A general criteria is given for the existence of stationary solutions. In case the memory kernel depends on p polynomially, transport may be prevented. Owing to the delay effects a finite amount of particles remains localized and the further transport is terminated. For diffusion with nonlinear memory effects we find an exact solution in the long-time limit. Although the mean square displacement exhibits diffusive behavior, higher order cumulants offer differences to diffusion and they depend on the memory strength.