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

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 a fixed composition and prescribe to subtract 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, showing 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 proportions in the original compositions and finally address the challenging problem of chemical reactions.