Microscopic expression of the second law of thermodynamics

TL;DR

This paper provides a microscopic derivation of the second law of thermodynamics, showing how entropy differences relate to relative entropy and passivity in canonical ensembles, bridging microscopic and macroscopic descriptions.

Contribution

It introduces a microscopic derivation of the second law based on canonical ensembles and highlights the roles of relative entropy and passivity in entropy increase.

Findings

Entropy difference comprises two positive components.

Relative entropy quantifies the initial and final state difference.

Passivity ensures the positivity of the entropy change.

Abstract

The microscopic derivation of the second law for macroscopic system is given under the phenomenological assumption that both the initial and final states are described by mutually different canonical ensembles. In particular, it is also shown that the entropy difference between the initial and final states is composed of two positive components. One of the components is expressed as the relative entropy between the initial and final states, while the other is positive due to a dynamical stability called passivity of the canonical ensemble.