Expoente de Hurst e diagrama de fase para persistência induzida amnesticamente em processos não-markovianos.

Abstract

Nowadays there has been a growing interest in anomalous diffusion: the super difusive and sub-difusive processes. The problem about normal diffusion already well established whereas many problems still exist in anomalous diffusion. Several mathematical models and computational techniques have been developed to model such processes. In this work we studied a non-Markovian Random Walk (RW), in one dimension in which the development of the process is governed by decisions taken in the distant past. We used as tool of analysis, analytical and numerical procedures (Monte Carlo method). In this problem, the walker takes its decisions (go right or left) at a given time t, based on the decisions taken in the past, namely in a fraction f of the total time. As far as the decision making process is considered only the distant past is taken into account. This loss of recent memory leads the probability density function of the position to change from Gaussian to non-Gaussian and leads to the emergence of log-periodic oscillations in position, besides producing a change in the behavior of non-persistent to persistent, causing anomalous diffusion. This change is characterized by the Hurst exponent, and is found, surprisingly, in a region where there is negative feedback. The diagram of phases depending on the parameters f and p (fraction of old memory and feedback), shows the following phases: classical non persistence, classical persistence, log-periodic non persistence, log-periodic persistence, Gaussian and non Gaussian with respect to the position of the walker.