On the role of composition entropies in the statistical mechanics of polydisperse systems

TL;DR

This paper introduces a new concept of composition entropy for polydisperse systems, clarifies its role in thermodynamics, and provides formulas for mixing entropy, addressing key issues in the statistical mechanics of complex fluids.

Contribution

It defines a non-ambiguous composition entropy for polydisperse systems, distinct from mixing entropy, and derives general expressions for mixing entropy based on probability distribution distances.

Findings

Composition entropy does not contribute to thermodynamics at fixed composition.

Subtracting ln N! from free energy accounts for polydispersity.

Derived formulas relate mixing entropy to probability distribution metrics.

Abstract

Polydisperse systems are commonly encountered when dealing with soft matter in general or any non simple fluid. Yet their treatment within the framework of statistical thermodynamics is a delicate task as the latter has been essentially devised for simple --- non fully polydisperse --- systems. In this paper, we address the issue of defining a non ambiguous combinatorial entropy for these systems. We do so by focusing on the general property of extensivity of the thermodynamic potentials and discussing a specific mixing experiment. This leads us to introduce the new concept of composition entropy for single phase systems that we do not assimilate to a mixing entropy. We then show that they do not contribute to the thermodynamics of the system at fixed composition and prescribe to substract $\ln N!$ from the free energy characterizing a system however polydisperse it can be. We then re-derive general expressions for the mixing entropy between any two polydisperse systems and interpret them in term of distances between probability distributions and show that one of these metrics relates naturally to a recent extension of Landauer's principle. We then propose limiting expressions for the mixing entropy in the case of mixing with equal proportion in the original compositions and finally address the challenging problem of chemical reactions.