Smooth entropy in axiomatic thermodynamics

TL;DR

This paper introduces an axiomatic framework connecting phenomenological and statistical thermodynamics, incorporating errors through smooth entropies, and shows how different entropy measures determine the feasibility of thermodynamic processes.

Contribution

It establishes a new axiomatic approach linking macroscopic and microscopic thermodynamics using smooth entropies, applicable to systems of all sizes.

Findings

Smooth min and max entropies characterize process feasibility with errors.

In the macroscopic limit, a single entropy determines possible state transformations.

For i.i.d. systems, von Neumann entropy is the relevant measure.

Abstract

Thermodynamics can be formulated in either of two approaches, the phenomenological approach, which refers to the macroscopic properties of systems, and the statistical approach, which describes systems in terms of their microscopic constituents. We establish a connection between these two approaches by means of a new axiomatic framework that can take errors and imprecisions into account. This link extends to systems of arbitrary sizes including microscopic systems, for which the treatment of imprecisions is pertinent to any realistic situation. Based on this, we identify the quantities that characterise whether certain thermodynamic processes are possible with entropy measures from information theory. In the error-tolerant case, these entropies are so-called smooth min and max entropies. Our considerations further show that in an appropriate macroscopic limit there is a single entropy measure that characterises which state transformations are possible. In the case of many independent copies of a system (the so-called i.i.d. regime), the relevant quantity is the von Neumann entropy.