Three-dimensional telegrapher's equation and its fractional generalization

Abstract

We derive the three-dimensional telegrapher's equation out of a random walk model. The model is a threedimensional version of the multistate random walk where the number of different states form a continuum representing the spatial directions that the walker can take. We set the general equations and solve them for isotropic and uniform walks which finally allows us to obtain the telegrapher's equation in three dimensions. We generalize the isotropic model and the telegrapher's equation to include fractional anomalous transport in three dimensions.

Telegraphic Transport Processes and Their Fractional Generalization: A Review and Some Extensions

“…We review the microscopic model introduced in Ref. [30] for the transport of particles in continuous media. The model is based on a generalization of multistate random walks and assumes a continuum in the number of states [33].…”

“…We have recently solved this problem by obtaining the three-dimensional TE [30] and the two-dimensional TE [31] from random walk models (as we had done previously for the one dimensional case [32]). These models consist of a continuous version of two and three dimensional random walks with a continuum of states [33].…”

“…In a recent work [32] we have presented a derivation of the fractional telegrapher's equation (FTE) in one dimension based on a fractional generalization of the persistent random walk on the line. The continuous multistate model mentioned above allows for a fractional treatment which finally leads to fractional TEs in higher dimensions [30,31].…”

Telegraphic processes with stochastic resetting

“…In this context TE can be derived directly from Maxwell's equations [29,31]. It can also be phenomenologically derived from thermodynamics by a nonlocal generalization of Fick's law called Cattaneo's equation [33][34][35] 1 as well as random-walk theory where the one-dimensional TE is the master equation of the persistent random walk [38,39] (see also [40] for a recent three-dimensional generalization and [41][42][43] for alternative derivations of hyperbolic equations).…”

Particle transport at arbitrary timescales with Poisson-distributed collisions

“…In addition to the changes in direction, one can also consider changes in speed [2,16]. This has proven useful in studies of first-passage times of a persistent random walker [17] and, more recently, in some generalizations of the TE [18]. We pursue the idea by studying situations in which collisions lead to new (random) values of velocity arising from model probability density functions (PDFs).…”