Lie Symmetries of the Nonlinear Fokker-Planck Equation Based on Weighted Kaniadakis Entropy

Abstract

The paper studies the Lie symmetries of the nonlinear Fokker–Planck equation in one dimension, which are associated to the weighted Kaniadakis entropy. In particular, the Lie symmetries of the nonlinear diffusive equation, associated to the weighted Kaniadakis entropy, are found. The MaxEnt problem associated to the weighted Kaniadakis entropy is given a complete solution, together with the thermodynamic relations which extend the known ones from the non-weighted case. Several different, but related, arguments point out a subtle dichotomous behavior of the Kaniadakis constant k, distinguishing between the cases k∈(−1,1) and k=±1. By comparison, the Lie symmetries of the NFPEs based on Tsallis q-entropies point out six “exceptional” cases, for: q=12, q=32, q=43, q=73, q=2 and q=3.

Entropy- A Tale of Ice and Fire

“…In this section, we recall some notions and results from [49,50,91,102], concerning the nonlinear Fokker-Planck equation (NFPE), including important examples of entropies, needed for our next discussion in Section 3.…”

“…In our papers [50,91], we generalized these results, for the case of weighted STM, Tsallis and Kaniadakis entropies, respectively. In the final classification of symmetries, the Lie algebras differentiate by means of six "exceptional" values of the Tsallis parameter q and by two "exceptional" values of the Kaniadakis parameter k.…”

“…In Section 2, we recall some of our recent results ( [50,91]) concerning the Lie symmetries associated to the nonlinear Fokker-Plank equations derived from weighted Kaniadakis or Tsallis entropies. Six special values of the Tsallis parameters, highlighted by the classification of these symmetries, clearly indicate algebraic and geometric invariants which differentiate the Lie algebras involved.…”

“…The corresponding Lie symmetries were determined in [102] (and were recovered in [50,91], from our similar studies, based on the weighted Tsallis and weighted Kaniadakis entropies). The only such entropies which produce Lie symmetries, apart the generic ones…”

Order Properties Concerning Tsallis Residual Entropy

“…Among the applications of other entropies (Rényi entropy, Varma entropy, Kaniadakis entropy, relative entropy, weighted entropy, etc. ), we can list the following: Markov chains (see [16][17][18]), model selection (see [19,20]), combinatorics (see [21,22]), finance (see [23][24][25]), Lie symmetries (see [26,27]), and machine learning (see [28,29]).…”