Deformed random walk: Suppression of randomness and inhomogeneous diffusion.

We study a generalization of the random walk (RW) based on a deformed translation of the unitary step, inherited by the q algebra, a mathematical structure underlying nonextensive statistics. The RW with deformed step implies an associated deformed random walk (DRW) provided with a deformed Pascal triangle along with an inhomogeneous diffusion. The paths of the RW in deformed space are divergent, while those corresponding to the DRW converge to a fixed point. Standard random walk is recovered for q→1 and a suppression of randomness is manifested for the DRW with -1<γ_{q}<1 and γ_{q}=1-q. The passage to the continuum of the master equation associated to the DRW led to a van Kampen inhomogeneous diffusion equation when the mobility and the temperature are proportional to 1+γ_{q}x, and provided with an exponential hyperdiffusion that exhibits a localization of the particle at x=-1/γ_{q} consistent with the fixed point of the DRW. Complementarily, a comparison with the Plastino-Plastino Fokker-Planck equation is discussed. The two-dimensional case is also studied, by obtaining a 2D deformed random walk and its associated deformed 2D Fokker-Planck equation, which give place to a convergence of the 2D paths for -1<γ_{q_{1}},γ_{q_{2}}<1 and a diffusion with inhomogeneities controlled by two deformation parameters γ_{q_{1}},γ_{q_{2}} in the directions x and y. In both the one-dimensional and the two-dimensional cases, the transformation γ_{q}→-γ_{q} implies a change of sign of the corresponding limits of the random walk paths, as a property of the deformation employed.

Reply to "Comment on 'Deformed Fokker-Planck equation: Inhomogeneous medium with a position-dependent mass"'.

It is shown that in the "Comment on 'Deformed Fokker-Planck equation: Inhomogeneous medium with a position-dependent mass,"' three crucial observations have gone unnoticed, thus restricting its conclusion on the legitimacy of the Langevin equation for a position-dependent mass, Eq. (46) of da Costa et al. [Phys. Rev. E 102, 062105 (2020)2470-004510.1103/PhysRevE.102.062105].

Deformed Fokker-Planck equation: Inhomogeneous medium with a position-dependent mass.

We present the Fokker-Planck equation (FPE) for an inhomogeneous medium with a position-dependent mass particle by making use of the Langevin equation, in the context of a generalized deformed derivative for an arbitrary deformation space where the linear (nonlinear) character of the FPE is associated with the employed deformed linear (nonlinear) derivative. The FPE for an inhomogeneous medium with a position-dependent diffusion coefficient is equivalent to a deformed FPE within a deformed space, described by generalized derivatives, and constant diffusion coefficient. The deformed FPE is consistent with the diffusion equation for inhomogeneous media when the temperature and the mobility have the same position-dependent functional form as well as with the nonlinear Langevin approach. The deformed version of the H-theorem permits to express the Boltzmann-Gibbs entropic functional as a sum of two contributions, one from the particles and the other from the inhomogeneous medium. The formalism is illustrated with the infinite square well and the confining potential with linear drift coefficient. Connections between superstatistics and position-dependent Langevin equations are also discussed.

Majorization and Dynamics of Continuous Distributions.

In this work we show how the concept of majorization in continuous distributions can be employed to characterize mixing, diffusive, and quantum dynamics along with the H-Boltzmann theorem. The key point lies in that the definition of majorization allows choosing a wide range of convex functions ϕ for studying a given dynamics. By choosing appropriate convex functions, mixing dynamics, generalized Fokker-Planck equations, and quantum evolutions are characterized as majorized ordered chains along the time evolution, being the stationary states the infimum elements. Moreover, assuming a dynamics satisfying continuous majorization, the H-Boltzmann theorem is obtained as a special case for ϕ ( x ) = x ln x .

Distinguishability notion based on Wootters statistical distance: Application to discrete maps.

We study the distinguishability notion given by Wootters for states represented by probability density functions. This presents the particularity that it can also be used for defining a statistical distance in chaotic unidimensional maps. Based on that definition, we provide a metric d¯ for an arbitrary discrete map. Moreover, from d¯, we associate a metric space with each invariant density of a given map, which results to be the set of all distinguished points when the number of iterations of the map tends to infinity. Also, we give a characterization of the wandering set of a map in terms of the metric d¯, which allows us to identify the dissipative regions in the phase space. We illustrate the results in the case of the logistic and the circle maps numerically and analytically, and we obtain d¯ and the wandering set for some characteristic values of their parameters. Finally, an extension of the metric space associated for arbitrary probability distributions (not necessarily invariant densities) is given along with some consequences. The statistical properties of distributions given by histograms are characterized in terms of the cardinal of the associated metric space. For two conjugate variables, the uncertainty principle is expressed in terms of the diameters of the associated metric space with those variables.