Random Walks Associated with Nonlinear Fokker–Planck Equations

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.

Non-Arrhenius behavior and fragile-to-strong transition of glass-forming liquids

Characterization of the non-Arrhenius behavior of glass-forming liquids is a broad avenue for research toward the understanding of the formation mechanisms of noncrystalline materials. In this context, this paper explores the main properties of the viscosity of glass-forming systems, considering super-Arrhenius diffusive processes. We establish the viscous activation energy as a function of the temperature, measure the degree of fragility of the system, and characterize the fragile-to-strong transition through the standard Angell's plot. Our results show that the non-Arrhenius behavior observed in fragile liquids can be understood through the non-Markovian dynamics that characterize the diffusive processes of these systems. Moreover, the fragile-to-strong transition corresponds to a change in the spatiotemporal range of correlations during the glass transition process.

Characterization of the non-Arrhenius behavior of supercooled liquids by modeling nonadditive stochastic systems

The characterization of the formation mechanisms of amorphous solids is a large avenue for research, since understanding its non-Arrhenius behavior is challenging to overcome. In this context, we present one path toward modeling the diffusive processes in supercooled liquids near glass transition through a class of nonhomogeneous continuity equations, providing a consistent theoretical basis for the physical interpretation of its non-Arrhenius behavior. More precisely, we obtain the generalized drag and diffusion coefficients that allow us to model a wide range of non-Arrhenius processes. This provides a reliable measurement of the degree of fragility of the system and an estimation of the fragile-to-strong transition in glass-forming liquids, as well as a generalized Stokes-Einstein equation, leading to a better understanding of the classical and quantum effects on the dynamics of nonadditive stochastic systems.

Non-Gaussian Distributions to Random Walk in the Context of Memory Kernels

The investigation of diffusive process in nature presents a complexity associated withmemory effects. Thereby, it is necessary new mathematical models to involve memory conceptin diffusion. In the following, I approach the continuous time random walks in the context ofgeneralised diffusion equations. To do this, I investigate the diffusion equation with exponential andMittag–Leffler memory-kernels in the context of Caputo–Fabrizio and Atangana–Baleanu fractionaloperators on Caputo sense. Thus, exact expressions for the probability distributions are obtained,in that non-Gaussian distributions emerge. I connect the distribution obtained with a rich class ofdiffusive behaviour. Moreover, I propose a generalised model to describe the random walk processwith resetting on memory kernel context.