AGING AND DIFFUSION IN LOW-DIMENSIONAL ENVIRONMENTS

We study out-of-equilibrium dynamics and aging for a particle diffusin g in one-dimensional environments, such as the random force Sinai mode l, as a toy model for low dimensional systems. We study fluctuations o f two time (t(w),t) quantities from the probability distribution Q(z,t ,t(w)) of the relative displacement z=x(t) -x(t(w)) in the limit of la rge waiting time t(w)-->infinity using numerical and analytical techni ques. We find three generic large time regimes: (i) a quasiequilibrium regime (finite tau=t-t(w)) where Q(z,tau) satisfies a general fluctua tion dissipation theorem equation, (ii) an asymptotic diffusion regime for large time separation where Q(z)dz similar to (Q) over bar[L(t)/L (t(w))]dz/L(t), and (iii) an intermediate ''aging'' regime for interme diate time separation [h(t)/h(t(w)) finite], with Q(z,t,t')=f(z,h(t)/h (t')). In the unbiased Sinai model we find numerical evidence for regi mes (i) and (ii), and for (iii) with <(Q(z,t,t'))over bar>=Q(0)(z)f(h( t)/h(t')) and h(t)similar to lnt. Since h(t)similar to L(t) in Sinai's model there is a singularity in the diffusion regime to allow for reg ime (iii). A directed model, related to the biased Sinai model, is sol ved 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(z) in landscapes with fast growing barriers. It accounts qualitatively for aging in Sin ai's model. We also identify a mechanism for aging in low dimensional phase space corresponding to an almost degeneracy of barriers. We illu strate this mechanism by introducing an exactly solvable model, with b arriers and wells, which shows clearly diffusion and aging regimes wit h a rich variety of functions h(t).