Aging and diffusion in low dimensional environments

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 time (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 quasiequilibrium regime (finite τ=t-tw) where Q(z,τ) satisfies a general fluctuation dissipation theorem equation, (ii) an asymptotic diffusion regime for large time separation where Q(z)dz~Q̄[L(t)/L(tw)]dz/L(t), and (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 regimes (i) and (ii), and for (iii) with Q(z,t,t^')=Q₀(z)f(h(t)/h(t^')) and h(t)~lnt. 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. It accounts qualitatively for aging in Sinai's model. We also identify a mechanism for aging in low dimensional phase space corresponding to an almost degeneracy of barriers. We illustrate this mechanism by introducing an exactly solvable model, with barriers and wells, which shows clearly diffusion and aging regimes with a rich variety of functions $h(t)$.