# Thermodynamics from first principles: correlations and nonextensivity

## TL;DR

This paper derives generalized equilibrium distributions for systems with long-range interactions using fundamental principles, justifying nonextensive thermostatistics like Tsallis distribution through ergodic and maximum entropy approaches.

## Contribution

It provides a theoretical foundation for nonextensive thermodynamics by deriving generalized distributions from first principles, including the Tsallis distribution.

## Key findings

- Derivation of generalized equilibrium distributions for correlated systems.
- Justification of Tsallis distribution within a fundamental framework.
- Clarification of the relationship between ergodic and maximum entropy methods.

## Abstract

The standard formulation of thermostatistics, being based on the Boltzmann-Gibbs distribution and logarithmic Shannon entropy, describes idealized uncorrelated systems with extensive energies and short-range interactions. In this letter, we use the fundamental principles of ergodicity (via Liouville's theorem), the self-similarity of correlations, and the existence of the thermodynamic limit to derive generalized forms of the equilibrium distribution for long-range-interacting systems. Significantly, our formalism provides a justification for the well-studied nonextensive thermostatistics characterized by the Tsallis distribution, which it includes as a special case. We also give the complementary maximum entropy derivation of the same distributions by constrained maximization of the Boltzmann-Gibbs-Shannon entropy. The consistency between the ergodic and maximum entropy approaches clarifies the use of the latter in the study of correlations and nonextensive thermodynamics.