Strong Anomaly in Diffusion Generated by Iterated Maps

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Abstract
We investigate the diffusion generated deterministically by periodic iterated maps that are defined by x(t+1) = x(t)+ax(z)(t)exp[-(b/x(t))(z-1)], z>1. It is shown that the obtained mean squared displacement grows asymptotically as sigma(2)(t) approximately ln (1/(z-1))(t) and that the corresponding propagator decays exponentially with the scaling variable |x|/square root of (sigma(2)(t))]. This strong diffusional anomaly stems from the anomalously broad distribution of waiting times in the corresponding random walk process and leads to a behavior obtained for diffusion in the presence of random local fields. A scaling approach is introduced which connects the explicit form of the maps to the mean squared displacement.