# Gibbs and Boltzmann Entropy in Classical and Quantum Mechanics

## TL;DR

This paper compares Gibbs and Boltzmann entropies in classical and quantum systems, highlighting their differences in equilibrium and non-equilibrium states, and advocates for Boltzmann entropy as the thermodynamic entropy, especially in quantum thermalization.

## Contribution

It provides a detailed comparison of Gibbs and Boltzmann entropies in classical and quantum contexts, emphasizing the relevance of Boltzmann entropy for non-equilibrium and quantum thermalization.

## Key findings

- Gibbs and Boltzmann entropies agree at equilibrium for macroscopic systems.
- Boltzmann entropy better describes thermodynamic entropy in non-equilibrium.
- Quantum Boltzmann entropy supports the individualist view of thermalization.

## Abstract

The Gibbs entropy of a macroscopic classical system is a function of a probability distribution over phase space, i.e., of an ensemble. In contrast, the Boltzmann entropy is a function on phase space, and is thus defined for an individual system. Our aim is to discuss and compare these two notions of entropy, along with the associated ensemblist and individualist views of thermal equilibrium. Using the Gibbsian ensembles for the computation of the Gibbs entropy, the two notions yield the same (leading order) values for the entropy of a macroscopic system in thermal equilibrium. The two approaches do not, however, necessarily agree for non-equilibrium systems. For those, we argue that the Boltzmann entropy is the one that corresponds to thermodynamic entropy, in particular in connection with the second law of thermodynamics. Moreover, we describe the quantum analog of the Boltzmann entropy, and we argue that the individualist (Boltzmannian) concept of equilibrium is supported by the recent works on thermalization of closed quantum systems.