918 research outputs found
The random gas of hard spheres
The inconsistency between the time-reversible Liouville equation and
time-irreversible Boltzmann equation has been pointed out long ago 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 nonincreasing function
of time. Unlike the conventional hard sphere model, the proposed random gas
model 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.
In the dilute gas approximation, the Enskog equation simplifies to the
Boltzmann equation, while the Enskog-Euler and Enskog-Navier-Stokes equations
become the conventional Euler and Navier-Stokes equations.Comment: 49 pages, 1 figure (improved the part with conventional hard spheres,
also improved the formulation of the random jump process
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