Random walks with homotopic spatial inhomogeneities

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.

Figures (6)

• Figure 1: The walker's path after $n=100$ steps, starting at $x=0$, with $p=1-p=1/2$.
• Figure 2: Paths of the homotopic walker after $n=100$ steps starting at $x=0$ with $p=1/2$ for the standard (dashed black line), Kaniadakis (solid black line), mixed (dashed gray line) and Tsallis (solid gray line) homotopic classes of the random walk. Convergence to the position $x=-1/\gamma$ is reported only for the Tsallis class, in agreement with Theorem (IV). According to Theorem (V), we see that randomness is preserved for Kaniadakis class as in the standard case.
• Figure 3: Evolution of the dimensionless homotopic probability density distribution given by \ref{['free-HFPE-2']} for the Kaniadakis, mixture, Tsallis and standard (Boltzmann) cases with delta initial condition $P(x,t=0)=\delta(x)$ and $\gamma \xi=0.5$.
• Figure 4: Evolution of the dimensionless homotopic probability density distribution given by \ref{['free-HFPE-2']} for the Kaniadakis, mixture, Tsallis and standard (Boltzmann) cases with delta initial condition $P(x,t=0)=\delta(x)$ and $\gamma \xi=-0.5$.
• Figure 5: Evolution of the dimensionaless HMSD given by \ref{['free-HFPE-2']} and \ref{['HMSD']} for the Kaniadakis, mixture, Tsallis and standard (Boltzmann) cases with delta initial condition $P(x,t=0)=\delta(x)$ and $|\gamma \xi|=0.5$. Mixture case between Kaniadakis and Tsallis classes indicates the presence of the homotopy.