Two-dimensional telegraphic processes and their fractional generalizations

Jaume Masoliver; Katja Lindenberg

doi:10.1103/PhysRevE.101.012137

Two-dimensional telegraphic processes and their fractional generalizations

Phys Rev E. 2020 Jan;101(1-1):012137. doi: 10.1103/PhysRevE.101.012137.

Authors

Jaume Masoliver¹, Katja Lindenberg²

Affiliations

¹ Department of Condensed Matter Physics and Institute of Complex Systems (UBICS), University of Barcelona, 08028 Catalonia, Spain.
² Department of Chemistry and Biochemistry and BioCircuits Institute, University of California, San Diego 92093-0340, USA.

PMID: 32069616
DOI: 10.1103/PhysRevE.101.012137

Abstract

We study the planar motion of telegraphic processes. We derive the two-dimensional telegrapher's equation for isotropic and uniform motions starting from a random walk model which is the two-dimensional version of the multistate random walk with a continuum number of states representing the spatial directions. We generalize the isotropic model and the telegrapher's equation to include planar fractional motions. Earlier, we worked with the one-dimensional version [Masoliver, Phys. Rev. E 93, 052107 (2016)PREHBM2470-004510.1103/PhysRevE.93.052107] and derived the three-dimensional version [Masoliver, Phys. Rev. E 96, 022101 (2017)PREHBM2470-004510.1103/PhysRevE.96.022101]. An important lesson is that we cannot obtain the two-dimensional version from the three-dimensional or the one-dimensional one from the two-dimensional result. Each dimension must be approached starting from an appropriate random walk model for that dimension.