Editing H-theorem (section)

== Gibbs' ''H''-theorem ==

[[File:Hamiltonian flow classical.gif|frame|Evolution of an ensemble of [[Hamiltonian mechanics|classical]] systems in [[phase space]] (top). Each system consists of one massive particle in a one-dimensional [[potential well]] (red curve, lower figure). The initially compact ensemble becomes swirled up over time.]]

[[Josiah Willard Gibbs]] described another way in which the entropy of a microscopic system would tend to increase over time.<ref name="gibbs12">Chapter XII, from {{cite book |last=Gibbs |first=Josiah Willard |author-link=Josiah Willard Gibbs |title=Elementary Principles in Statistical Mechanics |year=1902 |publisher=[[Charles Scribner's Sons]] |location=New York|title-link=Elementary Principles in Statistical Mechanics }}</ref> Later writers have called this "Gibbs' ''H''-theorem" as its conclusion resembles that of Boltzmann's.<ref name="tolman">{{cite book | last=Tolman |first=R. C. | author-link = Richard C. Tolman | year=1938 | title=The Principles of Statistical Mechanics | publisher=[[Dover Publications]] | isbn = 9780486638966}}</ref> Gibbs himself never called it an ''H''-theorem, and in fact his definition of entropy—and mechanism of increase—are very different from Boltzmann's. This section is included for historical completeness.

The setting of Gibbs' entropy production theorem is in [[statistical ensemble (mathematical physics)|ensemble]] statistical mechanics, and the entropy quantity is the [[Gibbs entropy]] (information entropy) defined in terms of the probability distribution for the entire state of the system. This is in contrast to Boltzmann's ''H'' defined in terms of the distribution of states of individual molecules, within a specific state of the system.

Gibbs considered the motion of an ensemble which initially starts out confined to a small region of phase space, meaning that the state of the system is known with fair precision though not quite exactly (low Gibbs entropy). The evolution of this ensemble over time proceeds according to [[Liouville's theorem (Hamiltonian)|Liouville's equation]]. For almost any kind of realistic system, the Liouville evolution tends to "stir" the ensemble over phase space, a process analogous to the mixing of a dye in an incompressible fluid.<ref name="gibbs12"/> After some time, the ensemble appears to be spread out over phase space, although it is actually a finely striped pattern, with the total volume of the ensemble (and its Gibbs entropy) conserved. Liouville's equation is guaranteed to conserve Gibbs entropy since there is no random process acting on the system; in principle, the original ensemble can be recovered at any time by reversing the motion.

The critical point of the theorem is thus: If the fine structure in the stirred-up ensemble is very slightly blurred, for any reason, then the Gibbs entropy increases, and the ensemble becomes an equilibrium ensemble. As to why this blurring should occur in reality, there are a variety of suggested mechanisms. For example, one suggested mechanism is that the phase space is coarse-grained for some reason (analogous to the pixelization in the simulation of phase space shown in the figure). For any required finite degree of fineness the ensemble becomes "sensibly uniform" after a finite time. Or, if the system experiences a tiny uncontrolled interaction with its environment, the sharp coherence of the ensemble will be lost. [[Edwin Thompson Jaynes]] argued that the blurring is subjective in nature, simply corresponding to a loss of knowledge about the state of the system.<ref name="jaynes1965">E.T. Jaynes; Gibbs vs Boltzmann Entropies; American Journal of Physics,391,1965</ref> In any case, however it occurs, the Gibbs entropy increase is irreversible provided the blurring cannot be reversed.
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| footer   = Quantum phase space dynamics in the same potential, visualized with the [[Wigner quasiprobability distribution]]. The lower image shows the equilibrated (time-averaged) distribution, with an entropy that is +1.37''k'' higher.
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The exactly evolving entropy, which does not increase, is known as ''fine-grained entropy''. The blurred entropy is known as ''coarse-grained entropy''.
[[Leonard Susskind]] analogizes this distinction to the notion of the volume of a fibrous ball of cotton:<ref>[[Leonard Susskind]], Statistical Mechanics Lecture 7 (2013). [https://www.youtube.com/watch?v=sg15UClUY48&t=83m0s Video] at [[YouTube]].</ref> On one hand the volume of the fibers themselves is constant, but in another sense there is a larger coarse-grained volume, corresponding to the outline of the ball.

Gibbs' entropy increase mechanism solves some of the technical difficulties found in Boltzmann's ''H''-theorem: The Gibbs entropy does not fluctuate nor does it exhibit Poincare recurrence, and so the increase in Gibbs entropy, when it occurs, is therefore irreversible as expected from thermodynamics. The Gibbs mechanism also applies equally well to systems with very few degrees of freedom, such as the single-particle system shown in the figure. To the extent that one accepts that the ensemble becomes blurred, then, Gibbs' approach is a cleaner proof of the [[second law of thermodynamics]].<ref name="jaynes1965"/>

Unfortunately, as pointed out early on in the development of [[quantum statistical mechanics]] by [[John von Neumann]] and others, this kind of argument does not carry over to quantum mechanics.<ref name="GoldsteinLebowitz2010">{{cite journal|last1=Goldstein|first1=S.|last2=Lebowitz|first2=J. L.|last3=Tumulka|first3=R.|last4=Zanghì|first4=N.|title=Long-time behavior of macroscopic quantum systems|journal=The European Physical Journal H|volume=35|issue=2|year=2010|pages=173–200|issn=2102-6459|doi=10.1140/epjh/e2010-00007-7|arxiv=1003.2129|s2cid=5953844}}</ref> In quantum mechanics, the ensemble cannot support an ever-finer mixing process, because of the finite dimensionality of the relevant portion of Hilbert space. Instead of converging closer and closer to the equilibrium ensemble (time-averaged ensemble) as in the classical case, the [[density matrix]] of the quantum system will constantly show evolution, even showing recurrences. Developing a quantum version of the ''H''-theorem without appeal to the ''Stosszahlansatz'' is thus significantly more complicated.<ref name="GoldsteinLebowitz2010"/>

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