Extensions and solutions for nonlinear diffusion equations and random walks

Results for Nonlinear Diffusion Equations with Stochastic Resetting

In this study, we investigate a nonlinear diffusion process in which particles stochastically reset to their initial positions at a constant rate. The nonlinear diffusion process is modeled using the porous media equation and its extensions, which are nonlinear diffusion equations. We use analytical and numerical calculations to obtain and interpret the probability distribution of the position of the particles and the mean square displacement. These results are further compared and shown to agree with the results of numerical simulations. Our findings show that a system of this kind exhibits non-Gaussian distributions, transient anomalous diffusion (subdiffusion and superdiffusion), and stationary states that simultaneously depend on the nonlinearity and resetting rate.

Nonlinear Fokker-Planck Equations, H-Theorem and Generalized Entropy of a Composed System

We investigate the dynamics of a system composed of two different subsystems when subjected to different nonlinear Fokker-Planck equations by considering the H-theorem. We use the H-theorem to obtain the conditions required to establish a suitable dependence for the system's interaction that agrees with the thermodynamics law when the nonlinearity in these equations is the same. In this framework, we also consider different dynamical aspects of each subsystem and investigate a possible expression for the entropy of the composite system.

Statistical dynamics of driven systems of confined interacting particles in the overdamped-motion regime

Systems consisting of confined, interacting particles doing overdamped motion admit an effective description in terms of nonlinear Fokker-Planck equations. The behavior of these systems is closely related to the _S power-law entropies and can be interpreted in terms of the _S-based thermostatistics. The connection between overdamped systems and the _S measures provides valuable insights on diverse physical problems, such as the dynamics of interacting vortices in type-II superconductors. The _S-thermostatistical approach to the study of many-body systems described by nonlinear Fokker-Planck equations has been intensively explored in recent years, but most of these efforts were restricted to systems affected by time-independent external potentials. Here, we extend this treatment to systems evolving under time-dependent external forces. We establish a lower bound on the work done by these forces when they drive the system during a transformation. The bound is expressed in terms of a free energy based on the _S entropy and is satisfied even if the driving forces are not derivable from a potential function. It constitutes a generalization, for systems governed by nonlinear Fokker-Planck equations involving general time-dependent external forces, of the H-theorem satisfied by these systems when the external forces arise from a time-independent potential.

Nonlinear Fokker-Planck Equation for an Overdamped System with Drag Depending on Direction

We investigate a one-dimensional, many-body system consisting of particles interacting via repulsive, short-range forces, and moving in an overdamped regime under the effect of a drag force that depends on direction. That is, particles moving to the right do not experience the same drag as those moving to the left. The dynamics of the system, effectively described by a non-linear, Fokker–Planck equation, exhibits peculiar features related to the way in which the drag force depends on velocity. The evolution equation satisfies an H-theorem involving the Sq nonadditive entropy, and admits particular, exact, time-dependent solutions closely related, but not identical, to the q-Gaussian densities. The departure from the canonical, q-Gaussian shape is related to the fact that in one spatial dimension, in contrast to what occurs in two or more spatial dimensions, the drag’s dependence on direction entails that its dependence on velocity is necessarily (and severely) non-linear. The results reported here provide further evidence of the deep connections between overdamped, many-body systems, non-linear Fokker–Planck equations, and the Sq-thermostatistics.