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E. T. Jaynes
Edwin Thompson Jaynes extended statistical mechanics to connect it to probability theory, Claude Shannon's information theory, and Bayesian statistical inferences.
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He championed the work of J. Willard Gibbs, contrasting it to the earlier work of Ludwig Boltzmann. His 1957 "principle of maximum entropy" says that the probability distribution that best represents the current state of knowledge is the one with largest entropy. In 1964, Jaynes examined the difference between the Boltzmann and Gibbs formulations of the entropy. They differ, he says, because of different treatments of "interparticle forces." The status of the Gibbs and Boltzmann expressions for entropy has been a matter of some confusion in the literature. We show that:Jaynes explains that Gibbs entropy is a conserved quantity, for the same reason as the Louiville theorem that conserves the hyper-volume in phase space of a cloud of particles as it traverses its trajectory.
Boltzmann entropy increases. We can show that this is a consequence of quantal interactions during particle collisions, which deny the claim of microscopic irreversibility and erase the path information in the gas particles that would be needed to support Loschmidt's objection to the Boltzmann In the writer's 1962 Brandeis lectures on statistical mechanics, the Gibbs and Boltzmann expressions for entropy were compared briefly, and it was stated that the Gibbs formula gives the correct entropy, as defined in phenomenological thermodynamics, while the Boltzmann As to the connections between entropy and information, in particular, "subjective human ignorance," Jaynes says, The phase volume W
According to Jaynes (and Gibbs), information is conserved when macroscopic order disappears because it simply changes into microscopic (thus invisible) order as the path information of all the gas particles is preserved. As Boltzmann's mentor Joseph Loschmidt had argued in the early 1870's, if the velocities of all the particles could be reversed at an instant, the future evolution of the gas would move in the direction of decreasing entropy. All the original order would reappear. This is consistent with the idea of Pierre-Simon Laplace's super-intelligent demon and completely deterministic laws of nature. It also follows from the Louville theorem that the hyper-volume of a cloud of points in phase space is a constant as the system evolves. Classical mechanics and physical determinism was shown to be only an approximation for large numbers of particles shortly after Gibbs's death by Albert Einstein and the later "founders" of quantum mechanics. When quantum effects are included in the collision of gas particles, Boltzmann's idea of "molecular disorder" is seen to be correct and path information is destroyed. Nevertheless, Gibbs's idea of the conservation of information is still widely held today by mathematical physicists. And most texts on statistical mechanics still claim that microscopic collisions between particles are reversible. Some explicitly claim that quantum mechanics changes nothing, but that is because they limit themselves to the unitary (conservative and deterministic) evolution of the Schrödinger equation and ignore the collapse of the wave function.
For example, Richard Tolman (p.8) claimed that the “principle of dynamical reversibility” holds also in quantum mechanics in appropriate form, indicating that quantum theory supplies no new kind of element for understanding the actual irreversibility in the macroscopic behavior of physical systems.
This is because both classical and quantum statistical mechanics describe The origin of irreversibility depends on the ontological chance involved in von Neumann's Process 1, Dirac's projection postulate, the "collapse of the wave function," denied by so many interpretations of quantum mechanics and ignored in statistical mechanics texts. In her 2008 book, Carolyne Van Vliet (p.678) says that the theory of non-equilibrium statistical mechanics is incomplete without some kind of randomization at the microscopic level.
Ter Haar, D. 1995. Microscopic physics is irreversible as a consequence of ontological indeterminacy. Jaynes likely did not accept the collapse of the quantum mechanical wave function. He was strongly influenced by Eugene Wigner, who was an early denier of the projection postulate and supporter of the unitary evolution of the universal wave function. He says
I have profited from discussions of these problems, over many years, with Professor E. P. Wigner, from whom I first heard the remark, "Entropy is an anthropomorphic concept." Jaynes' view (and Gibbs') is philosophical determinism, for which information about the universe at one time gives us the information at all times (Laplace's demon).
Boltzmann, like Maxwell before him (and Exner and Schrödinger after him - at least initially) knew that determinism cannot be proven by any experimental results, which are necessarily only Jaynes is correct that there is a strong relationship between the physics of statistical mechanics and the mathematics of information theory.
References
Gibbs vs Boltzmann Entropies,
The Gibbs Paradox, in |