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(Ir)reversibility and Entropy

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Nonlinear Partial Differential Equations

Part of the book series: Abel Symposia ((ABEL,volume 7))

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Abstract

In the 1860’s emerges a revolutionary idea: many properties of the world around us can be explained by combining the atomistic hypothesis with the statistical theory. Some of the great scientific conquests from this time are the Boltzmann equation, which triggers one of the first qualitative studies of a complicated nonlinear partial differential equation; the notion of statistical entropy, which would later be fundamental in other areas of physics and mathematics, including information theory; and the notion of macroscopic irreversibility emerging from microscopically reversible laws. Thus the basic rules of statistical physics were set until Boltzmann’s irreversibility paradigm was shaken by Landau’s discovery of the Landau damping effect, about 80 years later, which opened the idea that equilibration is compatible with preservation of information, and led to a number of problems concerning the statistical theory of matter.

La cosa più meravigliosa è la felicità del momento

L. Ferré

Reprinted, with kind permission, from: Duplantier, B. (ed.): Time—Poincaré Seminar 2010. Progress in Mathematical Physics, vol. 63, pp. 19–79. © Springer Basel AG 2012.

Translated by Lester Senechal.

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Notes

  1. 1.

    Even if this formula accurately reflects Boltzmann’s thoughts, it was Planck who first wrote it in this particular form, around 1900.

  2. 2.

    The monography [27] was incomplete at the time of Carleman’s passing away, and was completed by Carleson.

  3. 3.

    This program culminates in a recent manuscript by Mischler and Mouhot.

  4. 4.

    According to a personal communication by Mouhot, there are clues that Sturm’s measure may be too singular to do the job.

  5. 5.

    In real life, I think it likely that the validity of the Boltzmann equation is longer, because of slight non-Newtonian randomness, like quantum perturbations, which “renew” the equation; but this does not invalidate the reasoning.

  6. 6.

    Maxwell’s Demon has been the object of many discussions, in particular by Smoluchowski, Szilard, Gabor, Brillouin, Landauer and Bradbury; it has also inspired novelists like Pynchon. A recent paper by Binder and Danchin suggests to look for such concepts in the heart of living mechanisms.

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    Cédric Villani

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  1. of Science and Technology, Dept. of Mathematics, Norwegian University, Trondheim, 7491, Norway

    Helge Holden

  2. Centre of Mathematics for Applications, University of Oslo, Oslo, 0316, Norway

    Kenneth H. Karlsen

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Villani, C. (2012). (Ir)reversibility and Entropy. In: Holden, H., Karlsen, K. (eds) Nonlinear Partial Differential Equations. Abel Symposia, vol 7. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-25361-4_16

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