Microcanonical entropy for classical systems

TL;DR

This paper introduces a new definition of microcanonical entropy that resolves existing debates, aligns with Boltzmann entropy for large systems, and offers better descriptions for small systems.

Contribution

A novel microcanonical entropy definition that addresses extensivity issues and differentiates between macroscopic and microscopic system descriptions.

Findings

Reproduces Boltzmann entropy results for large systems

Fixes extensivity problems in the caloric equation

Provides improved description for small systems

Abstract

The entropy definition in the microcanonical ensemble is revisited. We propose a novel definition for the microcanonical entropy that resolve the debate on the correct definition of the microcanonical entropy. In particular we show that this entropy definition fixes the problem inherent the exact extensivity of the caloric equation. Furthermore, this entropy reproduces results which are in agreement with the ones predicted with standard Boltzmann entropy when applied to macroscopic systems. On the contrary, the predictions obtained with the standard Boltzmann entropy and with the entropy we propose, are different for small system sizes. Thus, we conclude that the Boltzmann entropy provides a correct description for macroscopic systems whereas extremely small systems should be better described with the entropy that we propose here.