Poincaré's recurrence theorem remained as far as I know, unproven until 1919 when Caratheodóry proved it. Why then did it represent an issue to Boltzmann? Boltzmann died in 1906, did he not know about this? I believe he did, because his ergodic hypothesis is designed exactly to combat this assertion of the poincaré's theorem.

Edit: Poincaré's recurrence Theorem asserts that any Hamiltonian Flow is bound to recurr for almost every initial state, for any degree of accuracy.

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    $\begingroup$ I think the question can be redeemed if it is asking about the apparent inconsistency of the so-called H Theorem and the (eventual) Poincare recurrence of improbable states--en.wikipedia.org/wiki/H-theorem $\endgroup$ – daniel Dec 14 '17 at 14:08
  • $\begingroup$ I think this question is about the historical development of physics (eg did Boltzmann know about PRT, was his hypothesis designed to combat PRT), rather than the current content of physics. History of Science and Mathematics SE would be more suitable. $\endgroup$ – sammy gerbil Dec 14 '17 at 18:01
  • $\begingroup$ @sammygerbil: The question of the title is not included in the question itself. If we ignore the title you may be right. I am not sure how this is normally handled. Can we edit the title into the question or ask the OP to do so? $\endgroup$ – daniel Dec 14 '17 at 18:16
  • $\begingroup$ Possible duplicate of Is entropy related to Poincare recurrence time? $\endgroup$ – sammy gerbil Dec 14 '17 at 18:24
  • $\begingroup$ @daniel I think it is clear that the intention is to ask an historical question. Changing the nature of the question is something which should be left to the asker. Edits by others should be restricted to improving clarity (eg grammar, formatting). $\endgroup$ – sammy gerbil Dec 14 '17 at 18:30

The reason the Poincaré recurrence theorem (also called Zermelo-Poincaré recurrence) posed a problem is that Boltzmann constructed his entire theory with the assumption that time had a direction. To be precise, he defined his now famous H-theorem such that time increased in the "correct" direction. That is, Boltzmann's equation describes a time-irreversible system which is directly in contradiction with Zermelo-Poincaré recurrence. The interesting part is that Boltzmann made a few more assumptions that we now know to be incorrect but we do not fully understand why the failure of these assumptions does not render the predicted results of his equation useless. Those assumptions are:

  1. that the velocities of any two particles in the system described by the particle velocity distribution functions are uncorrelated (this fails the moment two particles collide); and
  2. that the collisions result in a time-irreversible evolution of the equations.

We know that 1 fails the moment two particles collide, which kind of defeats the purpose of having a collision operator (Note that this assumption is now called the propagation of one-sided chaos).

We know that Boltzmann's collision operator assumed that particles collided like billiard balls undergoing elastic collisions, a process that is time-reversible. This originally led to what is called Loschmidt’s paradox – How can an irreversible process result from lots of reversible processes (perhaps a poor paraphrasing?)? This was eventually resolved physicists realized that the reversible microdynamics involved in collisions were not contradictory to irreversible macrodynamics (i.e. thermodynamics) if one uses the right amount of probability to interpret the macroscopic model. One must realize that the initial choice of the distribution function in the derivation of the Boltzmann equation introduces the necessary and sufficient probabilities. Unfortunately, there is no mathematically rigorous argument against Loschmidt’s paradox.

Poincaré's recurrence theorem remained as far as I know, unproven until 1919 when Caratheodóry proved it. Why then did it represent an issue to Boltzmann?

It was an issue because Boltzmann's system was ultimately made of reversible processes yet it predicted irreversible dynamics. If the system is irreversible, then one cannot return to the original state. It turns out that the irreversibility is somewhat artificially introduced in multiple ways, one of which is assuming a continuous distribution function to describe a discrete number of particles, i.e., by assuming a continuous distribution function one loses the ability to follow any individual particle in time, thus one loses information. One can also render a system irreversible through the choice of initial and/or boundary conditions.

Boltzmann died in 1906, did he not know about this? I believe he did, because his ergodic hypothesis is designed exactly to combat this assertion of the poincaré's theorem.

Yes, Boltzmann was aware of Poincaré's theorem. In one of the references below (I believe, it's been several years since I dug through them) one of the authors comments that someone asked Boltzmann why he could assign a sign to his H-theorem, i.e., how could he force time to go in one direction? Boltzmann apparently quipped a remark along the lines of "Show me an example where it does the opposite." I do not recall the exact quote/exchange, but the point is that Boltzmann was aware of this issue.


  • Evans, D.J. "On the entropy of nonequilibrium states," J. Statistical Phys. 57, pp. 745–758, doi:10.1007/BF01022830, 1989.
  • Evans, D.J., and G. Morriss Statistical Mechanics of Nonequilibrium Liquids, 1st Edition, Academic Press, London, 1990.
  • Evans, D.J., and D. J. Searles "Equilibrium microstates which generate second law violating steady states," Phys. Rev. E 50, pp. 1645–1648, doi:10.1103/PhysRevE.50.1645, 1994.
  • Evans, D.J., E.G.D. Cohen, and G.P. Morriss "Viscosity of a simple fluid from its maximal Lyapunov exponents," Phys. Rev. A 42, pp. 5990–5997, doi:10.1103/PhysRevA.42.5990, 1990.
  • Gressman, P.T., and R.M. Strain "Global classical solutions of the Boltzmann equation with long-range interactions," Proc. Nat. Acad. Sci. USA 107, pp 5744–5749, doi:10.1073/pnas.1001185107, 2010.
  • Hoover, W. (Ed.) Molecular Dynamics, Lecture Notes in Physics, Berlin Springer Verlag, 258, 1986.
  • Villani, C., Chapter 2, A review of mathematical topics in collisional kinetic theory, pp. 71–74, North-Holland, Washington, D.C., doi:10.1016/S1874-5792(02)80004-0, 2002.
  • Villani, C. "Entropy production and convergence to equilibrium for the Boltzmann equation," in XIVTH International Congress on Mathematical Physics, Edited by J.-C. Zambrini, pp. 130–144, doi:10.1142/9789812704016_0011, 2006.

The famous H-theorem was (and is still taught in schools of physics around the world) and is a proof that, for a closed system, the entropy should increase or remain constant. This section here https://en.wikipedia.org/wiki/Ludwig_Boltzmann#second_thermodynamics_law_as_a_law_of_disorder is Boltzmann's view on the principle that entropy equals disorder. Take now the Poincare recurrence theorem. It asserts that, if one waits enough number of years, all the molecules of air in one room could be found in only its upper half or packed in just one corner. But this is an entropy decrease. See the conflict?

The Boltzmann's equation and the H-theorem are the roots of the classical kinetic theory of gases for they serve as support/justification for Maxwell's work, for example.

P.S. I deleted one myth of physics which stood in the original version of the post.

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    $\begingroup$ @daniel Post changed to address some points $\endgroup$ – DanielC Dec 14 '17 at 17:55
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    $\begingroup$ -1 Not useful. I don't see how this answers the question about Boltzmann's knowledge of and response to the Poincare Recurrence Theorem. $\endgroup$ – sammy gerbil Dec 14 '17 at 18:18
  • $\begingroup$ Read the OP in the current form and its questions both the one in the title and the ones in the corpus. Then reread what I wrote abd reassess if I addressed them. $\endgroup$ – DanielC Dec 14 '17 at 19:22
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    $\begingroup$ The question is asking about something which happened in history. (Why did Poincaré's theorem... Not why does Poincaré's theorem... ) Note also the point made about dates. The conflict which you point out is already implied within the question. (I believe he did, because his ergodic hypothesis is designed exactly to combat this assertion of the poincaré's theorem.) The question is asking about the historical impact of one concept on another. $\endgroup$ – sammy gerbil Dec 14 '17 at 19:37

Boltzman did know about Poincare's theorem. It was pointed out to him by Zermelo. Boltzmann immediately recognized that his H Theorem was technically wrong, having contained an assumption of molecular chaos. He thus realized that entropy increase was a statistical matter. The H Theorem nevertheless remains a useful component of kinetic theory.


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