Aging and diffusion in low dimensional environments

Original · EN

We study out of equilibrium dynamics and aging for a particle diffusing in one dimensional environments, such as the random force Sinai model, as a toy model for low dimensional systems. We study fluctuations of two times (tw, t) quantities from the probability distribution Q(z,t,tw) of the relative displacement z = x(t) - x(tw) in the limit of large waiting time tw → ∞ using numerical and analytical techniques. We find three generic large time regimes: (i) a quasi-equilibrium regime (finite τ=t-tw) where Q(z,τ) satisfies a general FDT equation (ii) an asymptotic diffusion regime for large time separation where Q(z) dz Q[L(t)/L(tw)] dz/L(t) (iii) an intermediate ``aging'' regime for intermediate time separation (h(t)/h(tw) finite), with Q(z,t,t') = f(z,h(t)/h(t')). In the unbiased Sinai model we find numerical evidence for regime (i) and (ii), and for (iii) with Q(z,t,t') = Q₀(z) f(h(t)/h(t')) and h(t) t. Since h(t) L(t) in Sinai's model there is a singularity in the diffusion regime to allow for regime (iii). A directed model, related to the biased Sinai model is solved and shows (ii) and (iii) with strong non self-averaging properties. Similarities and differences with mean field results are discussed. A general approach using scaling of next highest encountered barriers is proposed to predict aging properties, h(t) and f(x) in landscapes with fast growing barriers. We introduce a new exactly solvable model, with barriers and wells, which shows clearly diffusion and aging regimes with a rich variety of functions h(t).