Ballistic annihilation in a one-dimensional fluid

Effects of velocity relaxation on the anomalous kinetics of a one-dimensional A+A-->Ø reaction

A one-dimensional A+A-->slashed circle reaction with alternating dichotomic velocities is investigated to study the effects of velocity relaxation on the anomalous kinetics of this reaction. While we keep the magnitudes constant, the particle velocities are allowed to change from one moving direction to the other with a relaxation time, beta. The kinetics of the reaction is studied for various relaxation times. Although the anomalous slower reaction rate is anticipated for the reaction in one-dimension, compared to the classical rate law, the rate is found to speed up at intermediate times for intermediate values of beta . The time evolution for the spatial distribution of particles is also discussed to elucidate the effects of the velocity relaxation times on the kinetics of the reaction.

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

The hydrodynamic description of probabilistic ballistic annihilation, for which no conservation laws hold, is an intricate problem with hard spherelike dynamics for which no exact solution exists. We consequently focus on simplified approaches, the Maxwell and very-hard-particle (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 direct Monte Carlo simulations. 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.

Hydrodynamics of probabilistic ballistic annihilation

We consider a dilute gas of hard spheres in dimension d> or =2 that upon collision either annihilate with probability p or undergo an elastic scattering with probability 1-p . For such a system neither mass, momentum, nor kinetic energy is a conserved quantity. We establish the hydrodynamic equations from the Boltzmann equation description. Within the Chapman-Enskog scheme, we determine the transport coefficients up to Navier-Stokes order, and give the closed set of equations for the hydrodynamic fields chosen for the above coarse-grained description (density, momentum, and kinetic temperature). Linear stability analysis is performed, and the conditions of stability for the local fields are discussed.

Some exact results for Boltzmann's annihilation dynamics

The problem of ballistic annihilation for a spatially homogeneous system is revisited within Boltzmann's kinetic theory in two and three dimensions. Analytical results are derived for the time evolution of the particle density for some isotropic discrete bimodal velocity modulus distributions. According to the allowed values of the velocity modulus, different behaviors are obtained: power law decay with nonuniversal exponents depending continuously upon the ratio of the two velocities, or exponential decay. When one of the two velocities is equal to zero, the model describes the problem of ballistic annihilation in the presence of static traps. The analytical predictions are shown to be in agreement with the results of two-dimensional molecular dynamics simulations.

Dynamics of ballistic annihilation

The problem of ballistically controlled annihilation is revisited for general initial velocity distributions and an arbitrary dimension. An analytical derivation of the hierarchy equations obeyed by the reduced distributions is given, and a scaling analysis of the corresponding spatially homogeneous system is performed. This approach points to the relevance of the nonlinear Boltzmann equation for dimensions larger than 1 and provides expressions for the exponents describing the decay of the particle density n(t) proportional, variant t(-xi) and the root-mean-square velocity v proportional, variant t(-gamma) in terms of a parameter related to the dissipation of kinetic energy. The Boltzmann equation is then solved perturbatively within a systematic expansion in Sonine polynomials. Analytical expressions for the exponents xi and gamma are obtained in arbitrary dimension as a function of the parameter mu characterizing the small velocity behavior of the initial velocity distribution. Moreover, the leading non-Gaussian corrections to the scaled velocity distribution are computed. These expressions for the scaling exponents are in good agreement with the values reported in the literature for continuous velocity distributions in d=1. For the two-dimensional case, we implement Monte Carlo and molecular dynamics simulations that turn out to be in excellent agreement with the analytical predictions.

Kinetics and scaling in ballistic annihilation

We study the simplest irreversible ballistically controlled reaction, whereby particles having an initial continuous velocity distribution annihilate upon colliding. In the framework of the Boltzmann equation, expressions for the exponents characterizing the density and typical velocity decay are explicitly worked out in arbitrary dimension. These predictions are in excellent agreement with the complementary results of extensive Monte Carlo and molecular dynamics simulations. We finally discuss the definition of universality classes indexed by a continuous parameter for this far from equilibrium dynamics with no conservation laws.

Ballistic annihilation with continuous isotropic initial velocity distribution

Ballistic annihilation with continuous initial velocity distributions is investigated in the framework of the Boltzmann equation. The particle density and the rms velocity decay as c approximately t(-alpha) and velocity approximately t(-beta), with the exponents depending on the initial velocity distribution and the spatial dimension d. For instance, in one dimension for the uniform initial velocity distribution beta = 0.230 472ellipsis. In the opposite extreme d-->infinity, the dynamics is universal and beta-->(1-2(-1/2))d(-1). We also solve the Boltzmann equation for Maxwell particles and very hard particles in arbitrary spatial dimension. These solvable cases provide bounds for the decay exponents of the hard sphere gas.