Projection operator formalism and entropy

TL;DR

This paper derives a generalized entropy definition using the Zwanzig projection operator formalism, establishing its relation to invariant measures and extending its validity to non-equilibrium and far-from-equilibrium systems.

Contribution

It provides a deductive derivation of entropy within the projection operator framework, applicable outside the thermodynamic limit and without shell thickness assumptions.

Findings

Entropy relates to invariant measures, especially Liouville measure in classical mechanics.

The derived entropy is valid for non-equilibrium and far-from-equilibrium states.

The entropy expression depends on macroscopic variables and differs subtly from microcanonical definitions.

Abstract

The entropy definition is deduced by means of (re)deriving the generalized non-linear Langevin equation using Zwanzig projector operator formalism. It is shown to be necessarily related to an invariant measure which, in classical mechanics, can always be taken to be the Liouville measure. It is not true that one is free to choose a ``relevant'' probability density independently as is done in other flavors of projection operator formalism. This observation induces an entropy expression which is valid also outside the thermodynamic limit and in far from equilibrium situations. The Zwanzig projection operator formalism therefore gives a deductive derivation of non-equilibrium, and equilibrium, thermodynamics. The entropy definition found is closely related to the (generalized) microcanonical Boltzmann-Planck definition but with some subtle differences. No ``shell thickness'' arguments are needed, nor desirable, for a rigorous definition. The entropy expression depends on the choice of macroscopic variables and does not exactly transform as a scalar quantity. The relation with expressions used in the GENERIC formalism are discussed.