Is the Boltzmann Equation Reversible? A Large Deviation Perspective on the Irreversibility Paradox
- PDF / 534,599 Bytes
- 36 Pages / 439.37 x 666.142 pts Page_size
- 30 Downloads / 163 Views
Is the Boltzmann Equation Reversible? A Large Deviation Perspective on the Irreversibility Paradox Freddy Bouchet1 Received: 24 February 2020 / Accepted: 12 June 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract We consider the kinetic theory of dilute gases in the Boltzmann–Grad limit. We propose a new perspective based on a large deviation estimate for the probability of the empirical distribution dynamics. Assuming Boltzmann molecular chaos hypothesis (Stosszahlansatz), we derive a large deviation rate function, or action, that describes the stochastic process for the empirical distribution. The quasipotential for this action is the negative of the entropy when the conservation laws are verified, as should be expected. While the Boltzmann equation appears as the most probable evolution, corresponding to a law of large numbers, the action describes a genuine reversible stochastic process for the empirical distribution, in agreement with the microscopic reversibility. As a consequence, this large deviation perspective gives the expected meaning to the Boltzmann equation and explains its irreversibility as the natural consequence of limiting the physical description to the most probable evolution. More interestingly, it also quantifies the probability of any dynamical evolution departing from solutions of the Boltzmann equation. This picture is fully compatible with the heuristic classical view of irreversibility, but makes it much more precise in various ways. We also explain that this large deviation action provides a natural gradient structure for the Boltzmann equation. Keywords Boltzmann equation · Kinetic theory · Large deviation theory · Macroscopic fluctuation theory · Dilute gases · Gradient flows
1 Introduction The Boltzmann equation [7] (see [8,9] for an english translation) is a cornerstone of statistical physics. It describes dilute gas dynamics at a macroscopic level, and has been the starting point for the kinetic theory of many other physical phenomena: the derivation of hydrodynamic equations [11,36], the kinetic theory of self gravitating systems [2], the relativistic Boltzmann equation, lattice Boltzmann algorithms for fluid mechanics [20], nuclear physics, and so on. While the underlying microscopic Hamiltonian dynamics is time reversible, Boltzmann’s
Communicated by Herbert Spohn.
B 1
Freddy Bouchet [email protected] Univ Lyon, Ens de Lyon, Univ Claude Bernard, CNRS, Laboratoire de Physique, Lyon, France
123
F. Bouchet
equation increases the entropy, as proven by Boltzmann’s H-theorem. This irreversibility paradox has played a crucial role in the early development of statistical physics and led to long controversies, for instance between Boltzmann and Zermelo, that involved many of the leading physicists and mathematicians of the late nineteen century (see [8,9] for a collection of basic papers from the second half of the nineteenth century on the subject). This apparent paradox and the fact that the irreversible evolution of macroscopic laws i
Data Loading...
