Entropy and the Time Evolution of Macroscopic Systems ,
In his delightful little book, Entropy and the Time Evolution of Macroscopic Systems, Walter T. Grandy Jr aims to bring a wider appreciation for the meaning of entropy from the probability-based point of view. Grandy is no newcomer to the topic: His books Foundations of Statistical Mechanics (Reidel, 1987) and Maximum-Entropy and Bayesian Methods in Inverse Problems (Reidel, 1985) first appeared more than 20 years ago. The deep insights he brings to his latest text draw on those books and a long career spent pondering the subtleties of the entropy concept.
Grandy is a student of E. T. Jaynes, who developed the principle of maximum entropy, or maxent. In this latest book, Grandy carefully develops his mentor’s formalism and then explores how an entropy function defined via the maxent principle can be used to describe time evolution and entropy production in a macroscopic system. Assuming that the reader has had a prior course in statistical physics or thermodynamics, Grandy focuses on the meaning of entropy.
The author takes the reader on a tour of various descriptions of irreversible macroscopic processes, including the Navier-Stokes equations in fluid mechanics, the Boltzmann equation in statistical mechanics, and Onsager’s flux–force equations in irreversible thermodynamics. For each process, Grandy considers whether the procedure of maximizing the entropy based on the available information suffices to unambiguously define an entropy function. The topics that he discusses, although fraught with subtlety, are vitally important for this generation of physicists (also chemists, engineers, and even biologists).
Filled with many examples, Grandy’s thought-provoking exposition will be of interest for years to come. He presents many of the standard debates at greater length than one is likely to find elsewhere, and he augments them with insights afforded by the maxent point of view. For example, he gives an excellent account of the usual mischaracterization of entropy as disorder and a lengthy discussion regarding whether entropy can be assumed to be extensive. The book’s style, and to some extent the subject matter, are reminiscent of Brian Pippard’s classic, Elements of Classical Thermodynamics for Advanced Students of Physics (Cambridge University Press, 1957), which explained the more subtle aspects of thermodynamics to a previous generation.
Grandy’s subtleties, though, are not those of Pippard. The concern of the present monograph is exclusively entropy and its meaning as derived by probability theory. And Grandy’s obtaining meaningful entropies for some of the systems he discusses is an impressive feat. The examples on which his presentation relies are often surprisingly complex. Consider the familiar box divided by a partition—one side is filled with gas, the other side evacuated. Grandy discusses whether a valid expression can be given for the entropy of that system in the time interval between the instant when the partition is removed and the time when the gas has equilibrated in the full volume. In a real achievement for the book, Grandy shows that the maxent approach can find a meaningful entropy for any relaxation process, including the partition example.
While impressed with the derivations, I confess to some annoyance with the book’s repeated questioning of the “existence” of entropy. Even though I am aware that by “existence” Grandy means “existence in the maxent formalism,” I still find that usage irksome. Since Grandy seems to accept the density matrix ρ as an always-valid description of the state of the system, I cannot help but wonder why he objects to John von Neumann’s expression, S = −k Trace[ρ log(ρ)], which guarantees that the entropy exists at each instant. Admittedly, we might not be able to explicitly write down an expression for the density matrix, but its existence and well-definedness are not in question.
That peccadillo and the associated tone of a zealot notwithstanding, Entropy and the Time Evolution of Macroscopic Systems is a scholarly and refreshing text that explores in detail the subtleties of carrying out a maximum entropy approach to describe irreversible processes. Written in an accessible style, the book makes numerous little-known connections between macroscopic and microscopic expressions for entropy and entropy production. I would recommend it to any serious student of statistical physics.