On the Relation of the Laws of Thermodynamics to Statistical Mechanics

Much of the philosophical literature on the relations between thermodynamics and statistical mechanics has to do with the process of relaxation to equilibrium. There has been comparatively little discussion of how to obtain what have traditionally been recognized as laws of thermodynamics, the zeroth, first, and second laws, from statistical mechanics. This note is about how to obtain analogues of those laws as theorems of statistical mechanics. The difference between the zeroth and second laws of thermodynamics and their statistical mechanical analogues is that the statistical mechanical laws are probabilistically qualified; what the thermodynamical laws say will happen, their statistical mechanical analogues say will probably happen. For this reason, it is entirely appropriate — indeed, virtually inevitable — for the quantities that are statistical mechanical analogues of temperature and entropy to be attributes of probability distributions. I close with some remarks about the relations between so-called "Gibbsian" and "Boltzmannian" methods in statistical mechanics.