# On the Equivalence of Gibbs, Boltzmann, and Thermodynamic Entropies in Equilibrium and Nonequilibrium Scenarios

## TL;DR

This paper explores when Gibbs, Boltzmann, and thermodynamic entropies are equivalent in both equilibrium and nonequilibrium systems.

## Contribution

It identifies a domain where these entropies are equivalent and Jaynes' maximum entropy principle applies.

## Key findings

- Equivalence holds when entropy change rate is zero or very small.
- Jaynes' principle is valid within this domain.
- Outside this domain, the entropies are not equivalent.

## Abstract

In this presentation, we have identified the domain of equivalence amongst the Boltzmann, Gibbs, and thermodynamic entropies. In this domain, ergodicity is followed even for (i) all nonequilibrium steady states and (ii) those time-dependent nonequilibrium states belonging to it. The condition of this domain is either that the rate of entropy change is zero or its magnitude is exceedingly small. Its implication is that, in this domain, Jaynes’ principle of maximum entropy estimate also holds. Outside this domain, the said equivalence among three entropies is not feasible, and the operation of the Jaynes’ principle of maximum entropy estimate does not remain of practical utility.