# The Generalized Boltzmann Distribution is the Only Distribution in Which the Gibbs-Shannon Entropy Equals the Thermodynamic Entropy

## TL;DR

This paper proves that the generalized Boltzmann distribution uniquely ensures the equality of Gibbs-Shannon entropy and thermodynamic entropy, highlighting that this equality only occurs at thermodynamic equilibrium.

## Contribution

It establishes a unique characterization of the generalized Boltzmann distribution based on entropy equality, clarifying the conditions for entropy equivalence.

## Key findings

- Gibbs-Shannon entropy equals thermodynamic entropy only at equilibrium.
- The generalized Boltzmann distribution is uniquely characterized by this entropy equality.
- Entropy equality does not hold for non-equilibrium distributions.

## Abstract

We show that the generalized Boltzmann distribution is the only distribution for which the Gibbs-Shannon entropy equals the thermodynamic entropy. This result means that the thermodynamic entropy and the Gibbs-Shannon entropy are not generally equal, but rather than the equality holds only in the special case where a system is in equilibrium with a reservoir.