Classical diffusion of a particle in a one-dimensional random force field

Abstract

We present a comprehensive study of the motion of a damped Brownian particle evolving in a static, one dimensional gaussian random force field. We provide both a clear physical picture of the process and a variety of analytical techniques. As the average bias μ is increased, a succession of different “diffusion laws” is observed: Sinai's diffusion ( μ = 0, x 2 ⋍ ln 4 t ), anomalous drift ( x ⋍ t μ, μ < 1 ), anomalous dispersion ( x − Vt ⋍ ±t 1 μ , 1 < μ < 2 ), and finally normal diffusion ( μ > 2), apart from algebraic tails outside the scaling region. We show that all those results can be understood in simple terms through a large scale description of the problem as a directed walk among traps characterized by a broad distribution of release time. From this analysis, the full asymptotic probability distributions (averaged over disorder) are precisely determined, in terms of Lévy stable laws. Sample to sample fluctuations are discussed. The probability of presence at the initial point is more specifically adressed. It amounts to computing the density of states of a Schrödinger equation with a special type of random potential. We obtain exactly the average over disorder of this quantity, using the following two different approaches: the “Dyson-Schmidt” technique, and the replica method. Both reveal interesting technical features, and the latter can be used to obtain information on the full probability distribution (and Green function). Some physical applications of this model are discussed.