On Ruelle's construction of the thermodynamic limit for the classical microcanonical entropy

TL;DR

This paper clarifies that Ruelle's method for constructing the thermodynamic limit of classical entropy applies directly to the proper microcanonical measure, simplifying the approach and removing the need for regularization.

Contribution

It demonstrates that Ruelle's construction extends to the proper microcanonical measure without regularization, simplifying the theoretical framework.

Findings

Ruelle's construction applies directly to the proper microcanonical measure.

Regularization of the microcanonical measure is unnecessary with minor proof adjustments.

The key formula remains valid without regularization.

Abstract

In this note we make a very elementary technical observation to the effect that Ruelle's construction of the thermodynamic limit of the classical entropy density defined with a regularized microcanonical measure actually establishes the thermodynamic limit for the entropy density defined with the proper microcanonical measure. At this stage a key formula is still derived from the regularized measures. We also show that with only minor changes in the proof the regularization of the microcanonical measure is actually not needed at all.