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Free Will
Mental Causation
James Symposium
J. Willard Gibbs

Josiah Willard Gibbs earned the first American Ph.D. in Engineering from Yale in 1863. He went to France where he studied with the great Joseph Louville, who formulated the theorem that the phase-space volume of a system evolving under a conservative Hamiltonian function is a constant along the system's trajectory.

He then travelled to Germany, where in Heidelberg he learned about the work of Hermann von Helmholtz and Gustav Kirchhoff in physics, and Robert Bunsen in chemistry.

Back in New Haven, in the 1870's he published a long monograph that began with the simplest translation into English of Clausius' great first and second laws of thermodynamics.

"The energy of the world is constant. The entropy of the world tends towards a maximum."
But it was his short text Principles in Statistical Mechanics published the year before his death in 1903 that brought him the most fame. In it, he coined the term phase space and the name for his field - statistical mechanics. Earlier he named the chemical potential and the statistical ensemble.

Gibbs formalized the earlier work in "kinetic gas theory" by Ludwig Boltzmann, making it more mathematically rigorous. But he lacked the physical insight of Boltzmann.

Perhaps inspired by the examples of other conservation laws in physics discovered during his lifetime, Gibbs disagreed with Boltzmann's view that information is "lost" when the entropy increases.

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.

Microscopic physics is irreversible as a consequence of ontological indeterminacy.

More details
Gibbs thought that as disorder increased, information was not being "lost" as Boltzmann claimed. He thought that the path information needed to reverse the momenta of all particles is still there in all the particles, so Loschmidt's objection still held for Gibbs.

By seeing that physics is fundamentally statistical (the "classical" world is the result of the law of large numbers averaging over quantum effects), I believe that Boltzmann had the greater physical insight.

There is in fact only one world - the quantum world. When quantum physics is considered in modern statistical mechanics texts, they usually stop with the deterministic Schrödinger equation and conclude that quantum mechanics changes nothing about irreversibility,

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.

And D. ter Haar (p. 292) said “The transition from classical to statistical mechanics does not introduce any fundamental changes.”

This is because both classical and quantum statistical mechanics describe ensembles of systems. Such systems are in “mixed states,” disregarding the interference terms in the density matrix of the “pure states” density operator. This is the basis for decoherence theories.

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. Elements of Statistical Mechanics, Third Edition. Oxford: Butterworth-Heinemann.

Tolman, Richard C. 2010. The Principles of Statistical Mechanics. New York: Dover Publications.

Van Vliet, Carolyne M. 2008. Equilibrium and Non-Equilibrium Statistical Mechanics/. Singapore ; Hackensack, NJ: World Scientific Publishing Company.

Doyle, Robert O. 2014. "The Origin of Irreversibility".

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