Ballistic aggregation: a solvable model of irreversible many particles dynamics

Abstract

The adhesive dynamics of a one-dimensional aggregating gas of point particles is rigorously described. The infinite hierarchy of kinetic equations for the distributions of clusters of nearest neighbours is shown to be equivalent to a system of two coupled equations for a large class of initial conditions. The solution to these nonlinear equations is found by a direct construction of the relevant probability distributions in the limit of a continuous initial mass distribution. We show that those limiting distributions are identical to those of the statistics of shocks in the Burgers turbulence. The analysis relies on a mapping on a Brownian motion problem with parabolic constraints.

Scaling theory and exactly solved models in the kinetics of irreversible aggregation

“…This model was first considered by Carnevale et al [15] and later investigated further by Jiang and Leyvraz [45] using numerical simulations together with qualitative scaling arguments. We present these first, and later discuss shortly an elegant exact solution [75,29], which confirms the results anticipated in [45] in a very satisfactory way.…”

“…In the model considered in [75,28,29,30], the particle positions are fixed (they occupy a lattice, which we take to have unit spacing), so that everything is reduced to averaging over the initial momenta. This is in turn considerably simplified by the assumption that the initial momenta are independent Gaussian random variables.…”

Dynamics of strongly localized Lyapunov vectors in many-hard-disk systems

“…However we use the name brick accumulation model for the one-dimensional version of the clock model, because in a simple image of brick accumulations, it is easily to visualize the configuration of brick heights which is essential for the analytical approach to the hopping rate discussed in the next section V, although this image is applicable only for the one-dimensional case. The image of brick accumulations also helps us to easily recognize the similarity of this model to ballistic aggregation models, whose scaling properties have been studied analytically and numerically [42,43]. In the brick model described by the brick height K j (n), the site number j corresponds to the particle index, and the number n of dropped bricks corresponds to the collision number n t for the quasi-one-dimensional hard-disk system.…”

Stochastic aggregation: scaling properties

“…We consider the momentum conserving case, also known as "ballistic aggregation" or "sticky gas" [27][28][29][30][31][32][33]. The initial velocities are assigned according to the dis-tribution P 0 (v).…”