Lie symmetries and related group-invariant solutions of a nonlinear Fokker-Planck equation based on the Sharma-Taneja-Mittal entropy

Abstract

In the framework of the statistical mechanics based on the Sharma-Taneja-Mittal entropy we derive a family of nonlinear Fokker-Planck equations characterized by the associated non-increasing Lyapunov functional. This class of equations describes kinetic processes in anomalous mediums where both super-diffusive and subdiffusive mechanisms arise contemporarily and competitively. We classify the Lie symmetries and derive the corresponding group-invariant solutions for the physically meaningful Uhlenbeck-Ornstein process. For the purely diffusive process we show that any localized state asymptotically approaches a bell shape well fitted by a generalized Gaussian which is, in general, a quasi-self-similar solution for this class of purely diffusive equations.

Symmetry Reductions and Invariant Solutions of a Nonlinear Fokker-Planck Equation Based on the Sharma-Taneja-Mittal Entropy

“…In this application, the linear version of the equation is considered appropriate for the description of a wide variety of physical phenomena characterized by short-range interactions and/or short-time memories, typically associated with normal diffusion. The nonlinear Fokker-Planck equation, on the other hand, is associated with anomalous diffusion, generally associated with non-Gaussian distributions Scarfone and Wada, [2].…”

“…In this paper we study a nonlinear Fokker-Planck equation (derived in Scarfone and Wada [2]) in the framework of statistical mechanics based on a twoparameter entropy known as the Sharma-Taneja-Mittal (STM) entropy. The equation is a (1 + 1)-partial differential equation, namely…”

“…We shall refer to this equation as the Sharma-Taneja-Mittal nonlinear Fokker-Planck (STM-NFP) equation. For a detailed account of the context and derivation of the equation the interested reader may consult Scarfone and Wada [2] wherein Equation (1) is studied, along with two well-known special cases, via Lie symmetry analysis. Admitted Lie point symmetries are found and used to construct invariant solutions.…”

“…Admitted Lie point symmetries are found and used to construct invariant solutions. The invariant solutions of the STM-NFP equation reported in Scarfone and Wada [2] are based on two infinitesimal symmetry generators…”

Information Geometry of κ-Exponential Families: Dually-Flat, Hessian and Legendre Structures

“…For instance, such distributions have not finite momenta of any order as in κ-statistics where ⟨x n ⟩ < ∞ only for n < 1 κ − 1 [76]. An expectation defined by means of escort distribution might be more appropriate to solve these and other questions, although other types of nonlinear expectation can also be introduced [77].…”