Multi-dimensional q-Gaussian densities describing systems of confined interacting particles with drag

Abstract

Fokker–Planck equations with power-law nonlinearities in the diffusion term are useful for the description of various complex systems in physics and other disciplines. These evolution equations provide an effective representation of overdamped systems of particles interacting through short-range forces and confined by an external potential. It has been recently shown that the nonlinear Fokker–Planck equation admits an embedding within a Vlasov-like mean-field equation that allows to incorporate inertial effects to the associated dynamics. Exact time-dependent solutions of the q-Gaussian form (with compact support) of the Vlasov-like equation have been found for one-dimensional systems with quadratic confining potentials. In the present contribution, we explore the possibility of extending this type of solutions to multi-dimensional systems with N spatial dimensions. We found exact time-dependent q-Gaussian solutions in \(N=2\) and \(N=3\), and investigate their main properties. We also prove that this type of solutions does not exist in systems with spatial dimension \(N>3\).