Aging continuous time random walks

We investigate biased and nonbiased aging continuous time random walks (ACTRW), using fractal renewal theory. For example, a biased ACTRW process describes a Montroll-Weiss CTRW process which starts at time -t(a) and then at time t=0 a bias is added to the random walk (i.e., an external field is switched on). Statistical behaviors of the displacement of the random walker r=r(t)-r(0) in the time interval (0,t) are obtained, after aging the random walk in the time interval (-t(a),0). In ACTRW formalism, the Green function P(r,t(a),t) depends on the age of the random walk t(a) and the forward time t. We derive a generalized Montroll-Weiss equation, which yields an exact expression for the Fourier double Laplace transform of the ACTRW Green function. Asymptotic long times t(a) and t behaviors of the Green function are shown to be related to the arc-sine distribution and Levy stable laws. In the limit of t>t(a), we recover the standard nonequilibrium CTRW behaviors, while the important regimes t<t(a) and tsimilar or equal tot(a) exhibit interesting aging effects. Convergence of the ACTRW results towards the CTRW behavior, becomes extremely slow when the diffusion exponent becomes small. In the context of biased ACTRW, we investigate an aging Einstein relation. We briefly discuss aging in Scher-Montroll type of transport in disordered materials. (C) 2003 American Institute of Physics.