Kappa Entropy and its Thermodynamic Connection

Adopting a bottom-up perspective, we propose a novel two-parametric nonadditive entropy, $S_{\kappa \mathrm{\ell }}$, associated with a Kappa-type power-law velocity distribution, $F_{\kappa \mathrm{\ell }}(v)$, 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 $\mathrm{\ell }=-5\mathrm{/}2$, regardless of the values assumed by the $\kappa$-parameter.