Universality of kappa distributions

Department of Astrophysical Sciences, Princeton University - 171 Broadmead, Princeton, NJ 08540, USA

Received: 3 April 2024
Accepted: 26 April 2024

Abstract

This paper reveals the universality of the particle energy distribution function, despite the arbitrariness that characterizes the generalized thermodynamic entropic function. We show that the canonical distribution, that is, the distribution function that maximizes this entropy under the constraints of canonical ensemble, is always the same and given by the kappa distribution function. We use the recently developed entropy defect to express the generalized entropic formulation. The entropy defect is a thermodynamic concept that describes the loss of entropy due to the order induced by the presence of correlations. Then we carry out functional analysis to maximize the implicit expression of the generalized entropy. Critically, we show that the Lagrange multipliers have the same exact arbitrariness as the generalized entropic function, allowing us to cancel it out and proving the universality of canonical distribution as the kappa distribution function.