Abstract
The inconsistency between the time-reversible Liouville equation and time-irreversible Boltzmann equation has been pointed out by Loschmidt. To avoid Loschmidt’s objection, here we propose a new dynamical system to model the motion of atoms of gas, with their interactions triggered by a random point process. Despite being random, this model can approximate the collision dynamics of rigid spheres via adjustable parameters. We compute the exact statistical steady state of the system, and determine the form of its marginal distributions for a large number of spheres. We find that the Kullback–Leibler entropy (a generalization of the conventional Boltzmann entropy) of the full system of random gas spheres is a non-increasing function of time. Unlike the conventional hard sphere model, the proposed random gas system results in a variant of the Enskog equation, which is known to be a more accurate model of dense gas than the Boltzmann equation. We examine the hydrodynamic limit of the derived Enskog equation for spheres of constant mass density, and find that the corresponding Enskog–Euler and Enskog–Navier–Stokes equations acquire additional effects in both the advective and viscous terms.
Highlights
It is known that the atoms in an electrostatically neutral monatomic gas interact via the Lennard–Jones potential [1]
To avoid Loschmidt’s objection [14] and derive the Boltzmann equation in a consistent fashion, we propose a new random process to model the underlying hard sphere dynamics, where the changes in the velocities of the spheres still obey the mechanics of the rigid sphere collision, but the
We examine the hydrodynamic limit of the Enskog equation for spheres of constant mass density, as it appears to be physically plausible for atoms of the noble gases [23]
Summary
It is known that the atoms in an electrostatically neutral monatomic gas interact via the Lennard–Jones potential [1]. To avoid Loschmidt’s objection [14] and derive the Boltzmann equation in a consistent fashion, we propose a new random process to model the underlying hard sphere dynamics, where the changes in the velocities of the spheres still obey the mechanics of the rigid sphere collision, but the “collisions” themselves are triggered by a point process This random dynamical system possesses the infinitesimal generator, so that the corresponding forward Kolmogorov equation for its probability density is readily available via the integration by parts. We examine the hydrodynamic limit of the Enskog equation for spheres of constant mass density, as it appears to be physically plausible for atoms of the noble gases [23] In this limit, we find that the resulting Enskog–Euler and Enskog–Navier–Stokes equations acquire additional non-vanishing terms, which are not present in the conventional gas dynamics equations originating from the Boltzmann equation [7,8,9,10,13]. These additional effects disappear in the dilute gas approximation, which yields the usual Boltzmann, Euler, and Navier–Stokes equations, respectively
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