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Random Walks

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An Introduction to Anomalous Diffusion and Relaxation

Part of the book series: PoliTO Springer Series ((PTSS))

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Abstract

This chapter deals with the random walk problem and its connections with the diffusion processes. Its first part is dedicated to an elementary approach to the classical random walks or random flights problem. Then, a generalization of the random walk, starting from a nonlinear diffusion equation (or nonlinear Fokker-Planck equation), is investigated, creating the conditions to discuss the central limit theorem and a kind of its generalization. In practice, we deal with a generalization of the simplest random walk, that is, the one of a free particle walking a given distance in a positive or negative direction at each step of time, via a nonlinear Markov chain obtained from the porous media equation. Subsequently, the concepts of anomalous diffusion and of those random walks with space and time continuous, which are called continuous-time random walk, are reviewed, focusing some formal aspects of the anomalous dynamics and pointing towards recent extensions of these methods. It is shown that the continuous-time random walk can be formulated in order to obtain reaction-diffusion equations. These equations describe the evolution of several species undergoing anomalous diffusion, with reactions governed by linear mean-field equations. We discuss also a theory for the intermittent continuous-time random walk and Lévy walks, in which the particles are stochastically reset to a given position at the end of each step of renewal, with a given resetting rate. The link between the formalisms of normal and anomalous diffusion is discussed, motivating the application of the fractional calculus to the whole approach on which the book is based.

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Authors and Affiliations

  1. Departamento de Física, Universidade Estadual de Maringá, Maringá, Paraná, Brazil

    Luiz Roberto Evangelista

  2. Departamento de Física, Universidade Estadual de Ponta Grossa, Ponta Grossa, Paraná, Brazil

    Ervin Kaminski Lenzi

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  1. Luiz Roberto Evangelista
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  2. Ervin Kaminski Lenzi
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Correspondence to Luiz Roberto Evangelista .

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Evangelista, L.R., Lenzi, E.K. (2023). Random Walks. In: An Introduction to Anomalous Diffusion and Relaxation. PoliTO Springer Series. Springer, Cham. https://doi.org/10.1007/978-3-031-18150-4_3

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