On the nonequilibrium entropy of large and small systems

TL;DR

This paper reevaluates various definitions of nonequilibrium entropy for large and small systems, arguing that physical entropy should depend on microstates rather than subjective probabilities, with implications for nano systems.

Contribution

It clarifies the proper interpretation of entropy in nonequilibrium systems and critiques the use of Gibbs-Shannon entropy as a physical entropy for small systems.

Findings

Gibbs-Shannon entropy is not suitable as a physical entropy for systems in contact with reservoirs.

Physical entropy should depend on the microstate, not on subjective probability distributions.

Interprets the Gibbs-Shannon entropy of nano particles as Boltzmann entropy of a dilute gas.

Abstract

Thermodynamics makes definite predictions about the thermal behavior of macroscopic systems in and out of equilibrium. Statistical mechanics aims to derive this behavior from the dynamics and statistics of the atoms and molecules making up these systems. A key element in this derivation is the large number of microscopic degrees of freedom of macroscopic systems. Therefore, the extension of thermodynamic concepts, such as entropy, to small (nano) systems raises many questions. Here we shall reexamine various definitions of entropy for nonequilibrium systems, large and small. These include thermodynamic (hydrodynamic), Boltzmann, and Gibbs-Shannon entropies. We shall argue that, despite its common use, the last is not an appropriate physical entropy for such systems, either isolated or in contact with thermal reservoirs: physical entropies should depend on the microstate of the system, not on a subjective probability distribution. To square this point of view with experimental results of Bechhoefer we shall argue that the Gibbs-Shannon entropy of a nano particle in a thermal fluid should be interpreted as the Boltzmann entropy of a dilute gas of Brownian particles in the fluid.