Is the Boltzmann equation reversible? A large deviation perspective on the irreversibility paradox

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, 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.

References in corpus (1)

• Fluctuation theory in the Boltzmann--Grad limit

Cited by in corpus (15)

• Macroscopic Stochastic Thermodynamics
• Multistability and rare spontaneous transitions in barotropic $\beta$-plane turbulence
• Landau theory for finite-time dynamical phase transitions
• Cosh gradient systems and tilting
• Dynamical large deviations for homogeneous systems with long range interactions and the Balescu--Guernsey--Lenard equation
• Large deviations for Kac-like walks
• Fluctuating kinetic theory and fluctuating hydrodynamics of aligning active particles: the dilute limit
• Path large deviations for the kinetic theory of weak turbulence
• Dynamical and structural properties of an absorbing phase transition: a case study from granular systems
• Dynamical large deviations for plasmas below the Debye length and the Landau equation
• Dissipation in Hydrodynamics from Micro- to Macroscale: Wisdom from Boltzmann and Stochastic Thermodynamics
• Dynamical large deviations for an inhomogeneous wave kinetic theory: linear wave scattering by a random medium
• On the long-term evolution of razor-thin galactic discs: Balescu-Lenard prediction and perspectives
• Typical = random
• Kinetic Theory of Chiral Active Disks: Odd Transport and Torque Density