Boltzmann's Entropy and Time's Arrow

Given the success of Ludwig Boltzmann's statistical approach in explaining the observed irreversible behavior of macroscopic systems in a manner consistent with their reversible microscopic dynamics, it is quite surprising that there is still so much confusion about the problem of irreversibility. (See figure 1.) I attribute this confusion to the originality of Boltzmann's ideas: It made them difficult for some of his contemporaries to grasp. The controversies generated by the misunderstandings of Ernst Zermelo and others have been perpetuated by various authors. There is really no excuse for this, considering the clarity of Boltzmann's later writings. Since next year, 1994, is the 150th anniversary of Boltzmann's birth, this is a fitting moment to review his ideas on the arrow of time. In Erwin Schrodinger's words, "Boltzmann's ideas really give an understanding" of the origin of macroscopic behavior. All claims of inconsistencies that I know of are, in my opinion, wrong; I see no need for alternate explanations. For further reading I highly recommend Boltzmann's works as well as references 2—7. (See also PHYSICS TODAY, January 1992, page 44, for a marvelous description by Boltzmann of his visit to California in 1906.) Boltzmann's statistical theory of time-asymmetric, irreversible nonequilibrium behavior assigns to each microscopic state of a macroscopic system, be it solid, liquid, gas or otherwise, a number SB, the "Boltzmann entropy" of that state. This entropy agrees (up to terms that are negligible for a large system) with the macroscopic thermodynamic entropy of Rudolf Clausius, Se(,, when the system is in equilibrium. It then also coincides with the Gibbs entropy SG, which is defined not for an individual microstate but for a statistical ensemble (a collection of independent systems, all with the same Hamiltonian, distributed in different microscopic states consistent with some specified macroscopic constraints). However, unlike SG, which does not change in time even for ensembles