Decay kinetics of ballistic annihilation

Controlling Uncertainty of Empirical First-Passage Times in the Small-Sample Regime

We derive general bounds on the probability that the empirical first-passage time τ[over ¯]_{n}≡∑_{i=1}^{n}τ_{i}/n of a reversible ergodic Markov process inferred from a sample of n independent realizations deviates from the true mean first-passage time by more than any given amount in either direction. We construct nonasymptotic confidence intervals that hold in the elusive small-sample regime and thus fill the gap between asymptotic methods and the Bayesian approach that is known to be sensitive to prior belief and tends to underestimate uncertainty in the small-sample setting. We prove sharp bounds on extreme first-passage times that control uncertainty even in cases where the mean alone does not sufficiently characterize the statistics. Our concentration-of-measure-based results allow for model-free error control and reliable error estimation in kinetic inference, and are thus important for the analysis of experimental and simulation data in the presence of limited sampling.

Dimension dependence of clustering dynamics in models of ballistic aggregation and freely cooling granular gas

Via event-driven molecular dynamics simulations we study kinetics of clustering in assemblies of inelastic particles in various space dimensions. We consider two models, viz., the ballistic aggregation model (BAM) and the freely cooling granular gas model (GGM), for each of which we quantify the time dependence of kinetic energy and average mass of clusters (that form due to inelastic collisions). These quantities, for both the models, exhibit power-law behavior, at least in the long time limit. For the BAM, corresponding exponents exhibit strong dimension dependence and follow a hyperscaling relation. In addition, in the high packing fraction limit the behavior of these quantities become consistent with a scaling theory that predicts an inverse relation between energy and mass. On the other hand, in the case of the GGM we do not find any evidence for such a picture. In this case, even though the energy decay, irrespective of packing fraction, matches quantitatively with that for the high packing fraction picture of the BAM, it is inversely proportional to the growth of mass only in one dimension, and the growth appears to be rather insensitive to the choice of the dimension, unlike the BAM.

Ballistic annihilation with superimposed diffusion in one dimension

We consider a one-dimensional system with particles having either positive or negative velocity, and these particles annihilate on contact. Diffusion is superimposed on the ballistic motion of the particle. The annihilation may represent a reaction in which the two particles yield an inert species. This model has been the subject of previous work, in which it was shown that the particle concentration decays faster than either the purely ballistic or the purely diffusive case. We report on previously unnoticed behavior for large times when only one of the two species remains, and we also unravel the underlying fractal structure present in the system. We also consider in detail the case in which the initial concentration of right-going particles is 1/2+ɛ, with ɛ≠0. It is shown that remarkably rich behavior arises, in which two crossover times are observed as ɛ→0.

Molecular dynamics simulations of ballistic annihilation

Using event-driven molecular dynamics we study one- and two-dimensional ballistic annihilation. We estimate exponents xi and gamma, which describe the long-time decay of the number of particles [n(t) approximately t-xi] and of their typical velocity [v(t) approximately t-gamma]. To a good accuracy our results confirm the scaling relation xi+gamma=1. In the two-dimensional case our results are in good agreement with those obtained from Boltzmann kinetic theory.

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.

Probabilistic ballistic annihilation with continuous velocity distributions

We investigate the problem of ballistically controlled reactions where particles either annihilate upon collision with probability p, or undergo an elastic shock with probability 1-p. Restricting to homogeneous systems, we provide in the scaling regime that emerges in the long time limit, analytical expressions for the exponents describing the time decay of the density and the root-mean-square velocity, as continuous functions of the probability p and of a parameter related to the dissipation of energy. We work at the level of molecular chaos (nonlinear Boltzmann equation), and using a systematic Sonine polynomials expansion of the velocity distribution, we obtain in arbitrary dimension the first non-Gaussian correction and the corresponding expressions for the decay exponents. We implement Monte Carlo simulations in two dimensions, which are in excellent agreement with our analytical predictions. For p<1, numerical simulations lead to the conjecture that unlike for pure annihilation (p=1), the velocity distribution becomes universal, i.e., does not depend on the initial conditions.

Correlations in ballistic processes

We investigate a class of reaction processes in which particles move ballistically and react upon colliding. We show that correlations between velocities of colliding particles play a crucial role in the long time behavior. In the reaction-controlled limit when particles undergo mostly elastic collisions and therefore are always near equilibrium, the correlations are accounted analytically. For ballistic aggregation, for instance, the density decays as n approximately t(-xi) with xi=2d/(d+3) in the reaction-controlled limit in d dimensions, in contrast with the well-known mean-field prediction xi=2d/(d+2).

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.