The approach to equilibrium in statistical thermodynamics—do we really need coarse-graining?

Abstract

The technique of coarse-graining is conventionally used in both introductory and more advanced textbooks on statistical mechanics and thermodynamics when discussing a thermodynamic state described by an ensemble approaching equilibrium. In quantum statistical thermodynamics there exists a natural uncertainty-limited phase space volume element, but in the case of classical statistical thermodynamics no such natural phase space volume element exists and coarse-graining by its nature can appear as a rather arbitrary and ad-hoc procedure for students first encountering it. The present work discusses the approach to equilibrium in the ensemble picture, based on observable thermodynamic properties of the system in question. We suggest that a fruitful teaching approach can be based on the perspective that the system has reached thermodynamic equilibrium (distinct from statistical equilibrium) if all its observable macroscopic thermodynamic properties are indistinguishable from those of a canonical ensemble with the same observable macroscopic properties. This differs to an approach based on whether the probability distribution function (specifically the integral of the natural logarithm of the probability distribution) match those of a canonical distribution (which they cannot do in thermally isolated systems, due to the Liouville theorem). These ideas are essentially those of ‘weak convergence’, normally encountered in more mathematically advanced treatments, but whose basic ideas can be motivated with a simple mathematical outline, and is based on a set of common and reasonable assumptions concerning the evolution of probability distribution functions in a thermally isolated system. The approach may help alleviate student concerns over the use of coarse-graining in classical statistical thermodynamics. A relatively small elevation in the mathematical level of the concepts therefore provides a considerable gain in clarity and coherence of the emergent physical picture, and aligns with previous work by Jaynes, recently highlighted by me, on the definition of entropy in the ensemble picture.