Philosophy of Physics Graduate Lunch Seminar (Thursday - Week 4, TT25)

I will discuss the relation between two important quantities in statistical mechanics: the coarse-grained Gibbs entropy and the phase-averaged Boltzmann entropy. These quantities (I will argue) are the best options for reducing thermodynamic entropy available to the Gibbsian and Boltzmannian frameworks respectively. Gibbsians must use the coarse-grained Gibbs entropy since the fine-grain version is a constant of motion, and makes the use of probability distributions hard to justify. Boltzmannians meanwhile ought to use the phase-averaged Boltzmann entropy, since this gives them access to probability distributions, which are an indispensable tool. They also ought not to have a problem with using these distributions, since Boltzmannian qualms with probability distributions fall flat if we are not committed to using the Gibbs entropy to reduce thermodynamic entropy.

The coarse-grained Gibbs entropy and the phase-averaged Boltzmann entropy are related simply - the difference between them is equal to the Shannon entropy of the distribution over macro-states (i.e. the probabilities in the entropy function are probabilities of being in a certain macro-state, not in a certain micro-state). This means that disagreements between the two is much easier to quantify than in the case of the un-averaged Boltzmann entropy and the fine-grain Gibbs entropy; that debate usually requires argumentation around the concept of equilibrium, and the status of probabilities. Specifically, it means that the two frameworks agree to the extent that the distribution used is certain about the macro-state of the system. I will explore this consequence for a variety of ways we could interpret and set the probability function, namely, as a credence function, as the actual distribution of many systems, as the actual state of a single system, and as the distribution of a single system over a period of time.