Boltzmann's Entropy and Time's Arrow

By Joel L. Lebowitz, in Physics Today, September 1993

p 32

The controversies generated by the misunderstandings of Ernst Zermelo and others have been perpetuated by various authors.
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 [...] with the macroscopic thermodynamic entropy of Rudolf Clausius, Seq, 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 [...]
This separation of scales [...] is exactly what enables us to predict the evolution "typical" of a particular macroscopic system -- where, after all, we actually observe irreversible behavior.
This behavior should be distinguished from the chaotic but time-symmetric behavior of systems with a few degrees of freedom [...]

p 33

The Lorentz gas shows ipso facto [...] not only that there is no conflict between reversible microscopic and irreversible macroscopic behavior but also that the latter follows from the former [...]

p 35

Thus not only did Boltzmann's great insights give a microscopic interpretation of the mysterious thermodynamic entropy of Clausius; they also gave a natural generalization of entropy to nonequilibrium macrostates M, and with it an explanation of the second law of thermodynamics -- the formal expression of the time-asymmetric evolution of macroscopic states occurring in nature.

p 38

The analysis given above in terms of classical mechanics applies also to quantum mechanics [...]
It seems to me that there is no necessity or room in quantum mechanics for such an extra measurement postulate ["wavefunction collapse"].
[...] having results for typical microstates rather that averages is not just a mathematical nicety but is at the heart of understanding the microscopic origin of the observed macroscopic behavior.

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Marc Girod
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