Anomalous fluctuations in random walk dynamics

Abstract

The anomalous dispersion of noninteracting particles randomly walking in a network is considered. It is shown that the existence of large dangling branches attached to a backbone induces a “l/f”-like behavior in the current autocorrelation function at low frequencies. The waiting times associated with dangling loops scale liket ^−3/2. The size of the dangling branches provides a lower cutoff to the power law behavior. When the side branches are infinite, self-similar structures, the power law behavior persists up to a zero frequency. The currents we consider are created either by a bias on the random walk or by a current source. We consider both the total current, which is often referred to in the literature, and the current measured at endpoints of a specimen attached to a (model) battery. The differences and similarities between the two corresponding correlations are analyzed. In particular, we find that in the second case “l/f” noise exists only for large bias. When a statistical distribution of dangling branches is considered, we find that the largest power of frequency in the spectrum is 1.13. Much of our results are true when the dangling branches are replaced by “traps” having waiting time distributions that equal those of the branches. The waiting time associated with a power law distribution of dangling loops (m ^−x:m is the length of the loop) scales liket ⁻¹ −(x/2). However, it is shown that geometry alone can be responsible for the appearance of power laws in the spectra. Random geometry can be regarded as a model (or source) of random hopping times.