Maxwell and very hard particle models for probabilistic ballistic annihilation: hydrodynamic description

TL;DR

This paper develops hydrodynamic models for probabilistic ballistic annihilation using Maxwell and very hard particles models, providing analytical bounds, transport coefficients, and stability analysis to understand far-from-equilibrium dynamics.

Contribution

It introduces simplified kinetic models for probabilistic ballistic annihilation and derives hydrodynamic equations, bounds, and stability criteria for such non-conservative systems.

Findings

Analytical bounds for quantities in probabilistic ballistic annihilation.

Derived hydrodynamic equations and transport coefficients for simplified models.

Identified the role of dissipation in spatial inhomogeneity development.

Abstract

The hydrodynamic description of probabilistic ballistic annihilation, for which no conservation laws hold, is an intricate problem with hard sphere-like dynamics for which no exact solution exists. We consequently focus on simplified approaches, the Maxwell and very hard particles (VHP) models, which allows us to compute analytically upper and lower bounds for several quantities. The purpose is to test the possibility of describing such a far from equilibrium dynamics with simplified kinetic models. The motivation is also in turn to assess the relevance of some singular features appearing within the original model and the approximations invoked to study it. The scaling exponents are first obtained from the (simplified) Boltzmann equation, and are confronted against Monte Carlo simulation (DSMC technique). Then, the Chapman-Enskog method is used to obtain constitutive relations and transport coefficients. The corresponding Navier-Stokes equations for the hydrodynamic fields are derived for both Maxwell and VHP models. We finally perform a linear stability analysis around the homogeneous solution, which illustrates the importance of dissipation in the possible development of spatial inhomogeneities.