Does the first part of the second law also imply its second part?

TL;DR

This paper demonstrates that the first part of the second law of thermodynamics, concerning the existence of entropy as a state function, actually implies the second part about entropy non-decrease in isolated systems.

Contribution

It reveals that the commonly held view of the second part being independent from the first part is incorrect, showing a logical implication between the two.

Findings

First part of the second law implies the second part

Entropy as a state function leads to non-decreasing entropy in isolated systems

Clarifies the logical relationship within the second law of thermodynamics

Abstract

Sommerfeld called the first part of the second law to be the entropy axiom, which is about the existence of the state function entropy. It was usually thought that the second part of the second law, which is about the non-decreasing nature of entropy of thermally isolated systems, did not follow from the first part. In this note, we point out the surprise that the first part in fact implies the second part.