Random walk in a two-dimensional self-affine random potential: Properties of the anomalous diffusion phase at small external force

Abstract

We study the dynamical response to an external force F for a particle performing a random walk in a two-dimensional quenched random potential of Hurst exponent H=1/2 . We present numerical results on the statistics of first-passage times that satisfy closed backward master equations. We find that there exists a zero-velocity phase in a finite region of the external force 0<F<F{c} , where the dynamics follows the anomalous diffusion law x(t)∼ξ(F)t^{μ(F)} . The anomalous exponent 0<μ(F)<1 and the correlation length ξ(F) vary continuously with F . In the limit of vanishing force F→0 , we measure the following power laws: the anomalous exponent vanishes as μ(F)∝F^{a} with a≃0.6 (instead of a=1 in dimension d=1 ), and the correlation length diverges as ξ(F)∝F^{-ν} with ν≃1.29 (instead of ν=2 in dimension d=1 ). Our main conclusion is thus that the dynamics renormalizes onto an effective directed trap model, where the traps are characterized by a typical length ξ(F) along the direction of the force, and by a typical barrier 1/μ(F) . The fact that these traps are "smaller" in linear size and in depth than in dimension d=1 , means that the particle uses the transverse direction to find lower barriers.