Random walks with memory profile of the κ-exponential: a first measurement of diffusive regimes

Abstract

A new class of random walks, Elephant Random Walks (ERW) in the memory type, non-Markovian, was proposed by Schütz and Trimper characterized by diffusive regimes typical of anomalous diffusion. Following this line, we proposed a model of random walks guided by the principle of kinetic interaction (KIP), where, according to KIP, the temporal evolution of the distribution function of identical particles subject to binary collisions leads us to an ever-increasing functional with time, satisfying the statement of the irreversibility of the second law of Thermodynamics. We derived a distribution of the exponential  that is discrete. In the limit , the distribution of the -exponential falls into the ordinary exponential distribution, where we found the typical Brownian diffusion. In the upper limit of anti-equilibrium ( ), we found a transition from the ordinary diffusive regime to the super-diffusive regime. Additionally, moving away from the anti-equilibrium region, we found the ERW’s diffusive behavior. Our results are yet another case of anomalous transport in complex systems, which are associated with particle diffusion for which the variance spreads non-linearly with time.