Random walks with homotopic spatial inhomogeneities

TL;DR

This paper introduces a generalized random walk model called homotopic random walk (HRW), which incorporates position-dependent masses and deformation parameters, leading to a homotopic Fokker-Planck equation with unique diffusion properties and connections to nonextensive statistics.

Contribution

The work develops a novel homotopic random walk framework with an associated inhomogeneous Fokker-Planck equation, unifying and extending existing models like Tsallis and Kaniadakis statistics.

Findings

HRW trajectories show convergence, randomness, or divergence based on parameters.

The derived HFPE exhibits superdiffusion and special cases of known equations.

Stationary entropic density indicates inhomogeneous medium screening.

Abstract

In this work we study a generalization of the standard random walk, an homotopic random walk (HRW), using a deformed translation unitary step that arises from a homotopy of the position-dependent masses associated to the Tsallis and Kaniadakis nonexensive statistics. The HRW implies an associated homotopic Fokker-Planck equation (HFPE) provided with a bi-parameterized inhomogeneous diffusion. The trajectories of the HRW exhibit convergence to a position, randomness as well as divergence, according to deformation and homotopic parameters. The HFPE obtained from associated master equation to the HRW presents the features: a) it results an special case of the van Kampen diffusion equation (5) of Ref. [N. G. van Kampen, \emph{Z. Phys. B Condensed Matter} \textbf{68}, 135 (1987)]; b) it exhibits a superdiffusion in function of deformation and homotopic parameters; c) Tsallis and Kaniadakis deformed FPE are recovered as special cases; d) a homotopic mixtured diffusion is observed; and e) it has a stationary entropic density, characterizing a inhomogeneous screening of the medium, obtained from a homotopic version of the H-Theorem.