Kappa Entropy and its Thermodynamic Connection
J. A. S. Lima, M. H. Benetti
TL;DR
The paper addresses reconciling fat-tailed Kappa velocity distributions with thermodynamics by introducing a two-parameter nonadditive entropy $S_{\kappa\ell}$ and its linked distribution $F_{\kappa\ell}(v)$. Using a deformed logarithm $\ln_{\kappa\ell}$ and its inverse $e_{\kappa\ell}$, it derives a deformed Boltzmann counting framework and an equilibrium distribution that reduces to Boltzmann–Maxwell in the limit $\kappa \to \infty$. The key result is that thermodynamic consistency is achieved only for the subclass with $\ell = -\frac{5}{2}$, where the standard relations $dQ_{rev} = T dS$ and $P V = N k_B T$ hold under standard averaging. This work provides a bottom-up bridge between kinetic theory and thermodynamics for power-law systems and identifies the precise parameter regime where conventional thermodynamics applies.
Abstract
Adopting a bottom-up perspective, we propose a novel two-parametric nonadditive entropy, $S_{κ\ell}$, associated with a Kappa-type power-law velocity distribution, $F_{κ\ell}(v)$, recently derived in the literature. By formulating an extended Neo-Boltzmannian microstate counting procedure and employing standard averaging techniques, we demonstrate that the fundamental laws of thermodynamics are preserved within this generalized power-law framework only whether $\ell=-5/2$, regardless of the values assumed by the $κ$-parameter.