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The origin of power-law distributions in deterministic walks: the influence of the space geometry

, , , , , , and . Abstract Book of the XXIII IUPAP International Conference on Statistical Physics, Genova, Italy, (9-13 July 2007)


As the usual stochastic case, the so called deterministic walks describe the movement of a walker in a certain medium, which can or cannot have a random character. However, the rule of locomotion is always taken from some purely deterministic model, instead from a probability distribution. Deterministic walks usually present the technical difficulties common to nonlinear dynamical systems. Moreover, they can give rise to superdiffusive processes. But in contrast to purely random walks, it seems that for determinism walks there are no general guidelines indicating when the evolution would originate power-law distributions for the dynamical variables. In the present contribution we study how superdiffusive behavior can emerge in a deterministic walk. We extend a recently proposed model, in which the walker searches for randomly distributed targets in a large, but finite, 2D region, following a ``go to the closest target site'' rule. For certain very particular parameter conditions, this type of dynamics surprisingly exhibits power law distribution of step lengths. Here we reveal the mechanisms leading to such behavior, showing that the crossover is due to a trapping effect associated to specific spatial configurations of the search environment. The onset of this phenomenon resembles a critical point in thermodynamics, even thought there is no real phase transition in the system.

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