Quantum macrostates, equivalence of ensembles, and an H-theorem

This chapter recounts how Boltzmann reacted to Hermann Helmholtz’s analogy between thermodynamic systems and a special kind of mechanical system (the “monocyclic systems”) by grouping all attempts to relate thermodynamics to mechanics, including the kinetic-molecular analogy, into a family of partial analogies all derivable from what we would now call a microcanonical ensemble. At that time, Boltzmann regarded ensemble-based statistical mechanics as the royal road to the laws of thermal equilibrium (as we now do). In the same period, he returned to the Boltzmann equation and the H theorem in reply to Peter Guthrie Tait’s attack on the equipartition theorem. He also made a non-technical survey of the second law of thermodynamics seen as a law of probability increase.

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The Boltzmann Equation and the H Theorem (1872–1875)

10.1093/oso/9780198816171.003.0004 ◽

2018 ◽

Author(s):

Olivier Darrigol

Keyword(s):

Heat Conduction ◽

Boltzmann Equation ◽

Transport Phenomena ◽

Dilute Gas ◽

The Boltzmann Equation ◽

H Theorem ◽

Irreversible Evolution

This chapter covers Boltzmann’s writings about the Boltzmann equation and the H theorem in the period 1872–1875, through which he succeeded in deriving the irreversible evolution of the distribution of molecular velocities in a dilute gas toward Maxwell’s distribution. Boltzmann also used his equation to improve on Maxwell’s theory of transport phenomena (viscosity, diffusion, and heat conduction). The bulky memoir of 1872 and the eponymous equation probably are Boltzmann’s most famous achievements. Despite the now often obsolete ways of demonstration, despite the lengthiness of the arguments, and despite hidden difficulties in the foundations, Boltzmann there displayed his constructive skills at their best.

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Approach to Equilibrium, the H-Theorem and Irreversibility

10.1093/oso/9780199592357.003.0003 ◽

2018 ◽

Author(s):

Sauro Succi

Keyword(s):

Boltzmann Equation ◽

Detailed Knowledge ◽

Approach To Equilibrium ◽

The Boltzmann Equation ◽

H Theorem ◽

Irreversible Evolution

Like most of the greatest equations in science, the Boltzmann equation is not only beautiful but also generous. Indeed, it delivers a great deal of information without imposing a detailed knowledge of its solutions. In fact, Boltzmann himself derived most if not all of his main results without ever showing that his equation did admit rigorous solutions. This Chapter illustrates one of the most profound contributions of Boltzmann, namely the famous H-theorem, providing the first quantitative bridge between the irreversible evolution of the macroscopic world and the reversible laws of the underlying microdynamics.

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The H-theorem for the entropy of waves

Physics Letters A ◽

10.1016/j.physleta.2021.127315 ◽

2021 ◽

pp. 127315

Author(s):

Eiichirou Kawamori

Keyword(s):

H Theorem

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Generalized information and entanglement entropy, gravitation and holography

International Journal of Modern Physics A ◽

10.1142/s0217751x15300392 ◽

2015 ◽

Vol 30 (16) ◽

pp. 1530039 ◽

Cited By ~ 10

Author(s):

O. Obregón

Keyword(s):

Entanglement Entropy ◽

Ideal Gas ◽

Einstein Equations ◽

Von Neumann Entropy ◽

Nonextensive Statistical Mechanics ◽

Von Neumann ◽

H Theorem ◽

Gauge Gravity ◽

Gravity Duality ◽

Correction Terms

A nonextensive statistical mechanics entropy that depends only on the probability distribution is proposed in the framework of superstatistics. It is based on a Γ(χ2) distribution that depends on β and also on pl. The corresponding modified von Neumann entropy is constructed; it is shown that it can also be obtained from a generalized Replica trick. We further demonstrate a generalized H-theorem. Considering the entropy as a function of the temperature and volume, it is possible to generalize the equation of state of an ideal gas. Moreover, following the entropic force formulation a generalized Newton's law is obtained, and following the proposal that the Einstein equations can be deduced from the Clausius law, we discuss on the structure that a generalized Einstein's theory would have. Lastly, we address the question whether the generalized entanglement entropy can play a role in the gauge/gravity duality. We pay attention to 2d CFT and their gravity duals. The correction terms to the von Neumann entropy result more relevant than the usual UV ones and also than those due to the area dependent AdS3 entropy which result comparable to the UV ones. Then the correction terms due to the new entropy would modify the Ryu–Takayanagi identification between the CFT entanglement entropy and the AdS entropy in a different manner than the UV ones or than the corrections to the AdS3 area dependent entropy.

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