Classical diffusion in a random environment

A famous model in statistical physics is the Sinai model of classical diffusion in a random force field (see review in Bouchaud, Comtet, Georges & Le Doussal, Ann. Phys. 1990). Interest in the Sinai model spans many fields of science, ranging from mathematics, statistical physics, polymer physics, finance, population dynamics,…

In a paper with Christian Hagendorf, we have considered the question of the role of absorption in the Sinai model. The motivation is as follows: the free (classical) 1D diffusion is characterised by the return probability P(x,t∣x,0) with a power law decay t-1/2. If a weak concentration of absorbers is introduced, the probability is exponentially suppressed and decays as exp(-t1/3) (Lifshits tail). On the other hand, the effect of a random force field (Sinai problem) uncorrelated in space leads to anomalously slow diffusion characterised by a return probability decay ln-2(t). The understanding of the interplay between these two ingredients with opposite effects is therefore a natural question.

In 2013, we have demonstrated that the return probability generically decays as power law P(x,t∣x,0)~t. The exponent ν=√(μ2+2ρ/g) combines the average drift of the force field (μ), its variance (g) and the density of absorbers (ρ). Remarkably, we have shown that this exponent (first obtained in our paper with C. Hagendorf for ρ→0, α→∞ and μ=0 and by Le Doussal (J.Stat.Mech. 2009) for ρ→0, α→∞ and μ≠0), is completely general for a large class of such models, independent on the distribution of the absorption rates αn or the density ρ.

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