At the end of 19th century there was a lively discussion about the nature of the second law of thermodynamics, and its relation to Hamiltonian dynamics. Boltzmann developed a position that the second law is statistical in nature, "ordered" states are less probable, and the entropy is "highly unlikely" to decrease. As a confirmation he offered the H-theorem, a claim that entropy increases in a nearly ideal gas of particles.

But first Loschmidt objected that Boltzmann's "proof" of the H-theorem had a gap in neglecting correlations due to collisions, and in 1896 Zermelo disputed all of the above. He pointed out that statistical equations of Hamiltonian dynamics (Liouville equations) are just as time reversible as equations for individual particles, and therefore so are the equations for probabilities. Hence entropy is not "likely" to decrease or increase, it has to stay constant (up to deviations due to finite if large number of particles, so-called thermal fluctuations). As an illustration Zermelo brought up the Poincare recurrence theorem, which says that every state of a system nearly recurs infinitely many times in its evolution. He concluded that the second law is mathematically inconsistent with Hamiltonian dynamics, and no amount of probability theory can fix that. Poincare expressed the same opinion earlier (1889).

It is interesting that contemporaries thought that Boltzmann won the debate, but Boltzmann felt backed into a corner and "lost faith in himself" (according to Popper's Unended Quest). His response was the fluctuation hypothesis that our vicinity is a giant fluke in the universe, where the second law happens to hold. Zermelo in his turn suggested that the second law might be a selection rule on initial states rather than a "law of nature", but commented that such fix contradicts the "spirit of mechanics", and would not "satisfy anybody for long". Nonetheless, according to Wikipedia "much effort in the field is attempting to understand why the initial conditions early in the universe were those of low entropy, as this is seen as the origin of the second law". How selection of initial states resolves the issue of the entropy staying constant I am not sure. Soon after 1896 the attention of physicists shifted to relativity and quantum mechanics, and the second law dropped out of sight.

Was this debate picked up by others later? If so who advocated what position and how? Who was closer to the truth from the modern point of view, Zermelo or Boltzmann?

EDIT: Here is a modern exposition of the ideas of Ehrenfests and Mark Kac mentioned in the Alexandre's answer. It turns out that the missing ingredient was coarse-graining (macroscopic averaging) introduced by Gibbs in 1902 in a direct follow-up to the debate. Time reversible microscopic systems have evolutions in which coarse-grained entropy appears to grow for astronomically long periods of time, in the spirit of Boltzmann. Of course, they also have evolutions, where the opposite happens. We only observe the former but not the latter, so there is a selection of initial states suggested by Zermelo.

It seems that they were both half right, but lacked the idea of coarse-graining which reconciles their positions. The nature of selection is apparently still unclear and is traced to low entropy "early in the universe", which attracts "much effort in the field", see details here.


2 Answers 2


Bolzman was closer to the truth. On the modern state of this question, I recommend the classics:

Ehrenfest, Paul; Ehrenfest, Tatiana The conceptual foundations of the statistical approach in mechanics. Translated from the German by Michael J. Moravcsik. With a foreword by M. Kac and G. E. Uhlenbeck. Reprint of the 1959 English edition. Dover Publications, Inc., New York, 1990.

And excellent papers of Mark Kac on the subject, one collection of them is Probability and related topics in physical sciences. With special lectures by G. E. Uhlenbeck, A. R. Hibbs, and B. van der Pol. Lectures in Applied Mathematics. Proceedings of the Summer Seminar, Boulder, Colo., 1957, Vol. I Interscience Publishers, London-New York 1959.

Here is a popular (and freely available) article on this discussion: http://ergodic.ugr.es/statphys_grado/bibliografia/zermelo_boltzmann.pdf

The subject seems somewhat too techincal for an explanation here, so I only recommend the literature. It seems to be a general agreement that probability CAN explain the second law of thermodynamics.

However, much remains to be done in this area. In modern mathematics this is called "ergodic theory", and proving that concrete physical systems are indeed ergodic is a difficult mathematical problem.

  • $\begingroup$ The explanation that only initial states inducing entropy increase occur because they are "more probable" seems odd to me. They are only "more probable" in a macroscopic sense introduced by the observer. It seems that cosmologists trace selection all the way back to Big Bang, and that somehow filters through to everything we observe today. $\endgroup$
    – Conifold
    Nov 25, 2014 at 0:40
  • $\begingroup$ @Conifold: This is a history site, not a physics site:-) So we are not discussing here why entropy increases. But I strongly recommend the books of Kac. This is what I used to understand this matter. $\endgroup$ Nov 25, 2014 at 0:56
  • $\begingroup$ Surprised you didn't mention the Flucutation Theroems from which one can actually derive the 2nd law now. Are they controversial? $\endgroup$
    – Mike Wise
    Dec 12, 2021 at 11:20

By choosing "a nearly ideal gas" as a model Boltzmann in fact killed thermodynamics - second law violations can only take place in structured systems. Thermodynamics is long dead - nowadays perpetual motion machines of the second kind are being published in prestigious journals but there is no reaction at all from the scientific community:

http://www.researchgate.net/publication/258157913_Electricity_Generated_from_Ambient_Heat_across_a_Silicon_Surface/file/3deec5272604889353.pdf Electricity generated from ambient heat across a silicon surface, Guoan Tai, Zihan Xu, and Jinsong Liu, Appl. Phys. Lett. 103, 163902 (2013): "We report generation of electricity from the limitless thermal motion of ions across a two-dimensional (2D) silicon (Si) surface at room temperature. (...) ...limitless ambient heat, which is universally present in the form of kinetic energy from molecular, particle, and ion sources, has not yet been reported to generate electricity. (...) This study provides insights into the development of self-charging technologies to harvest energy from ambient heat, and the power output is comparable to several environmental energy harvesting techniques such as ZnO nanogenerator, liquid and gas flow-induced electricity generation across carbon nanotube thin films and graphene, although this remains a challenge to the second law of thermodynamics..."

http://link.springer.com/article/10.1007%2Fs10701-014-9781-5 Experimental Test of a Thermodynamic Paradox, D. P. Sheehan et al, Foundations of Physics, March 2014, Volume 44, Issue 3, pp 235-247: "...there arise between the vane faces permanent pressure and temperature differences, either of which can be harnessed to perform work, in apparent conflict with the second law of thermodynamics. Here we report on the first experimental realization of this paradox, involving the dissociation of low-pressure hydrogen gas on high-temperature refractory metals (tungsten and rhenium) under blackbody cavity conditions. The results, corroborated by other laboratory studies and supported by theory, confirm the paradoxical temperature difference and point to physics beyond the traditional understanding of the second law."

  • 1
    $\begingroup$ Whoah, whoah, whoah. Thermodynamics is not dead, and perpetual motion is definitely impossible. Have either of these results been tested by other, independent groups? $\endgroup$
    – HDE 226868
    Nov 22, 2014 at 20:07
  • $\begingroup$ They have not been NOTICED, let alone "tested by other, independent groups". $\endgroup$ Nov 22, 2014 at 20:24
  • $\begingroup$ Other groups such as . . . ? $\endgroup$
    – HDE 226868
    Nov 22, 2014 at 20:28

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