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.

Automated summary

The study focuses on a one-dimensional many-body system where particles interact through repulsive short-range forces and move in an overdamped regime with a drag force depending on direction. The dynamics are effectively described by a non-linear Fokker-Planck equation, showing peculiar features related to the drag force's velocity dependence. The results highlight the deep connections between overdamped many-body systems, non-linear Fokker-Planck equations, and Sq-thermostatistics.