# Tsallis Distribution as a Λ-Deformation of the Maxwell–Jüttner Distribution

## TL;DR

This paper explores how the Tsallis distribution can be linked to relativistic thermodynamics and the cosmological constant.

## Contribution

It introduces a new connection between Tsallis statistics, de Sitter space, and the cosmological constant Λ.

## Key findings

- The Tsallis distribution is shown to be a de Sitterian deformation of the Maxwell–Jüttner distribution.
- The Tsallis parameter q depends on the cosmological constant Λ as q=1+ℓcΛ/n.
- This links relativistic thermodynamics with quantum statistical considerations and cosmology.

## Abstract

Currently, there is no widely accepted consensus regarding a consistent thermodynamic framework within the special relativity paradigm. However, by postulating that the inverse temperature 4-vector, denoted as β, is future-directed and time-like, intriguing insights emerge. Specifically, it is demonstrated that the q-dependent Tsallis distribution can be conceptualized as a de Sitterian deformation of the relativistic Maxwell–Jüttner distribution. In this context, the curvature of the de Sitter space-time is characterized by Λ/3, where Λ represents the cosmological constant within the ΛCDM standard model for cosmology. For a simple gas composed of particles with proper mass m, and within the framework of quantum statistical de Sitterian considerations, the Tsallis parameter q exhibits a dependence on the cosmological constant given by q=1+ℓcΛ/n, where ℓc=ℏ/mc is the Compton length of the particle and n is a positive numerical factor, the determination of which awaits observational confirmation. This formulation establishes a novel connection between the Tsallis distribution, quantum statistics, and the cosmological constant, shedding light on the intricate interplay between relativistic thermodynamics and fundamental cosmological parameters.