Theorem H

The theorem H is a theorem shown by Boltzmann in 1872 within the framework of the kinetic Théorie of the gases, when a gas out of balance checks its equation. According to this theorem, there exists a certain size $H (T)$ which varies in a monotonous way during time, while the gas release towards the state of balance characterized by the distribution of Maxwell.

Historical aspects

The kinetic Théorie of the gases, which is based on the application of the traditional Mécanique to the molecules constituting the Gaz on a microscopic scale, developed starting from work founders of Maxwell (1850) in parallel with the macroscopic Thermodynamique. Boltzmann contributed in a way outstanding to maturation of the kinetic theory of gases.

He seemed trying to identify the size $H (T)$ , which varies in a monotonous way during time, with the Entropie (with the sign near) introduced in thermodynamics by Clausius (1850) and which, for an isolated system, can only grow according to the second principle. This identification would have made it possible to deduce the second principle, macroscopic, starting from the laws of the dynamics of the molecules, microscopic, in accordance with the reductionistic approach of Nature.

Quickly however, Loschmidt, then Zermelo, formulated virulent criticisms against the theorem H, Boltzmann being shown to practice “doubtful mathematics”. This charge does not hold any more since one rigorous theorem shown by Lanford in 1973 (to read below).

The paradox of the reversibility

The paradox of Loschmidt (1876)

Loschmidt wonders how the size $H (T)$ can it vary in a monotonous way during time whereas the equations of traditional mechanics are reversible ? Indeed, if the function $H (T)$ were decreasing and that at a given moment, one reverses exactly all speeds of molecules, then the new evolution is done with back, with $H (T)$ starting by growing. The answer of Boltzmann was brêve: “Go ahead, reverse them! ”, meaning the impossibility practices of such an exact inversion.

Statistical interpretation of the entropy (Boltzmann-1877)

See also: detailed Contenu=Article: [[Entropie#Définition of the entropy according to statistical physics]], [[Entropy]]

In 1877, Boltzamnn proposed a new definition of the entropy:

S \ = \ k_B \ \ ln \ \ Omega

where $k_B$ is the Boltzmann constant, and $\ Omega$ the “number of complexings”, i.e. the number of different microphone-states which are compatible with the thermodynamic macro-state given.

The growth of the entropy was to then be interpreted like a statistical phenomenon : the entropy grows because the system evolves in general of an improbable initial state ( $\ small Omega_i$ ) to a final state much more probable ( $\ Omega_f > \ Omega_i$ ). Local fluctuations are well-sure possible, but their relative size tends towards zero when the number NR of molecules tends towards the infinite one, so that the entropy of a macroscopic system seems to us to grow in a monotonous way.

Inversion of speeds & sensitivity to the initial conditions

With the discovery of the phenomenon of Sensitivity to the initial conditions characteristic of the chaotic systems, we know today that an inversion approchéee speeds quickly will involve a deviation compared to the reversed exact initial orbit, and this as small as are the errors introduced on the initial conditions. Digital simulations show whereas after an approached inversion, the function H (T) starts well by decreasing as Loschmidt predicted it, but that it very quickly recovers to grow again and this for almost all the approximate initial conditions, the real orbit of the system differing from the reversed exact initial orbit.

The paradox of Zermelo

The paradox of Zermelo (1896)

In 1890, whereas he studies the problem with 3 bodies in Celestial mechanics, Poincaré shows a very general theorem: the Theorem of recurrence. This theorem says that, for almost all the initial conditions, a dynamic system conservative whose Espace of the phases is of finished volume will also pass by again during the time close which one wants of his initial condition, and this in a repeated way.

Zermelo then points out to Boltzmann into 1896 that the theorem of recurrence of Poincaré seems to contradict the fact that a dynamic size can vary in a monotonous way, like $H (T)$ the fact. The answer of Boltzmann consists in considering the time of recurrence average: for a macroscopic gas containing $N \ gg 1$ molecules, Boltzmann estimates this one of order $10^N$ , one duration which is largely higher than old of the universe when $NR \ sim \ mathcal {NR} _A = 6.02 \ 10^ {+23}$ ; the recurrences are thus invisible on our scale.

The model of the ballot boxes of Ehrenfest (1907)

The “model of the ballot boxes” is a stochastic model introduced in 1907 by the husbands Paul & Tatiana Ehrenfest to clarify the preceding paradoxes appeared at the end of the 19th century in the bases of statistical mechanics. This model is sometimes also called the “model of the dogs & the chips”. The mathematician Mark Kac wrote in this connection that it was:

“… probably one of the most instructive models of all physics …”

This model is exactly soluble; in particular, one can calculate the average time of recurrence of each state, like his variance for certain interesting states.

The theorem of Lanford (1973)

Lanford showed rigorously that a gas of hard spheres diluted in $\ mathbb {R} ^3$ obeys the equation of Boltzmann within the limit of Boltzmann-Grad, at least for a very short time, equal only to one fifth of the average run time of an atom.

In spite of this restriction over the duration, this rigorous mathematical theorem is very important conceptually, since the equation of Boltzmann involves the theorem H. It is thus acquired today that mathematics of Boltzmann is not “doubtful”!

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