We investigate the long time behaviour of the one-dimensional ballistic aggregation model that represents a sticky gas of N particles with random initial positions and velocities, moving deterministically, and forming aggregates when they collide. We obtain a closed formula for the stationary measure of the system which allows us to analyze some remarkable features of the final `fan' state. In particular, we identify universal properties which are independent of the initial position and velocity distributions of the particles. We study cluster distributions and derive exact results for extreme value statistics (because of correlations these distributions do not belong to the Gumbel-Frechet-Weibull universality classes). We also derive the energy distribution in the final state. This model generates dynamically many different scales and can be viewed as one of the simplest exactly solvable model of N-body dissipative dynamics.
We study far-from-equilibrium dynamics in models of freely cooling granular gas and ballistically aggregating compact clusters. For both the cases, from event driven molecular dynamics simulations we have presented detailed results on structure and dynamics in space dimensions $d=1$ and $2$. Via appropriate analyses it has been confirmed that the ballistic aggregation mechanism applies in $d=1$ granular gases as well. Aging phenomena for this mechanism, in both the dimensions, have been studied via the two-time density autocorrelation function. This quantity is demonstrated to exhibit scaling …
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.…
We study the one-dimensional ballistic aggregation process in the continuum limit for one-sided Brownian initial velocity (i.e. particles merge when they collide and move freely between collisions, and in the continuum limit the initial velocity on the right side is a Brownian motion that starts from the origin $x=0$). We consider the cases where the left side is either at rest or empty at $t=0$. We derive explicit expressions for the velocity distribution and the mean density and current profiles built by this out-of-equilibrium system. We find that on the right side the mean density remains …
In this paper, we analyze the scaling properties of a model that has as limiting cases the diffusion-limited aggregation (DLA) and the ballistic aggregation (BA) models. This model allows us to control the radial and angular scaling of the patterns, as well as, their gap distributions. The particles added to the cluster can follow either ballistic trajectories, with probability $P_{ba}$, or random ones, with probability $P_{rw}=1-P_{ba}$. The patterns were characterized through several quantities, including those related to the radial and angular scaling. The fractal dimension as a function of…
A simple model of ballistic aggregation and fragmentation is proposed. The model is characterized by two energy thresholds, Eagg and Efrag, which demarcate different types of impacts: If the kinetic energy of the relative motion of a colliding pair is smaller than Eagg or larger than Efrag, particles respectively merge or break; otherwise they rebound. We assume that particles are formed from monomers which cannot split any further and that in a collision-induced fragmentation the larger particle splits into two fragments. We start from the Boltzmann equation for the mass-velocity distribution…
In this work, the transition between diffusion-limited and ballistic aggregation models was revisited using a model in which biased random walks simulate the particle trajectories. The bias is controlled by a parameter $λ$, which assumes the value $λ=0$ (1) for ballistic (diffusion-limited) aggregation model. Patterns growing from a single seed were considered. In order to simulate large clusters, a new efficient algorithm was developed. For $λ\ne 0$, the patterns are fractal on the small length scales, but homogeneous on the large ones. We evaluated the mean density of particles $\barρ$ in th…
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 …
We propose two lattice models in one dimension, with stochastically hopping particles which aggregate on contact. The hops are guided by "velocity rates" which themselves evolve according to the rules of ballistic aggregation as in a sticky gas in continuum. Our lattice models have both velocity and density fields and an appropriate real time evolution, such that they can be compared directly with event driven molecular dynamics (MD) results for the sticky gas. We demonstrate numerically that the long time and large distance behavior of the lattice models are identical to that of the MD, and s…
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 $n\sim t^{-ξ}$ with $ξ=2d/(d+3)$ in the reaction-controlled limit in $d$ dimensions, in contrast with well-known mean-field prediction $ξ=2d/(d…
We study scaling properties of stochastic aggregation processes in one dimension. Numerical simulations for both diffusive and ballistic transport show that the mass distribution is characterized by two independent nontrivial exponents corresponding to the survival probability of particles and monomers. The overall behavior agrees qualitatively with the mean-field theory. This theory also provides a useful approximation for the decay exponents, as well as the limiting mass distribution.
The kinetic energy of a freely cooling granular gas decreases as a power law $t^{-θ}$ at large times $t$. Two theoretical conjectures exist for the exponent $θ$. One based on ballistic aggregation of compact spherical aggregates predicts $θ= 2d/(d+2)$ in $d$ dimensions. The other based on Burgers equation describing anisotropic, extended clusters predicts $θ=d/2$ when $2\le d \le 4$. We do extensive simulations in three dimensions to find that while $θ$ is as predicted by ballistic aggregation, the cluster statistics and velocity distribution differ from it. Thus, the freely cooling granular g…
We study the dynamic scaling properties of an aggregation model in which particles obey both diffusive and driven ballistic dynamics. The diffusion constant and the velocity of a cluster of size $s$ follow $D(s) \sim s^γ$ and $v(s) \sim s^δ$, respectively. We determine the dynamic exponent and the phase diagram for the asymptotic aggregation behavior in one dimension in the presence of mixed dynamics. The asymptotic dynamics is dominated by the process that has the largest dynamic exponent with a crossover that is located at $δ= γ- 1$. The cluster size distributions scale similarly in all case…
In this paper, we study a two-dimensional model for the growth of molecular clusters in superfluid helium at low temperature. In the model, particles of diameter a follow random ballistic moves of length δ= a-256a. Upon attachment on the cluster surface, particles allow one-step relaxation to the nearest twofold coordinated site. Average coordination numbers of particles show that in the presence of relaxation the screening for incoming particles modifies the microscopic structure of the cluster. These results are in qualitative agreement with experimental aggregation of methane in He droplets…
In this work we study numerically the effects of the angle of deposition of particles in the growth process of a thin-film generated by aggregation of particles added at random. The particles are aggregated in a random position of an initially flat surface and with a given angle distribution. This process gives rise to a rough interface after some time of deposition. We performed Monte Carlo simulations and, by changing the angle of deposition, we observed a transition from the random deposition (RD) universality class to the Kardar-Parisi-Zhang (KPZ) universality class. We measured the usual …
Aggregation phenomena are ubiquitous in nature, encompassing out-of-equilibrium processes of fractal pattern formation, important in many areas of science and technology. Despite their simplicity, foundational models such as diffusion-limited aggregation (DLA) or ballistic aggregation (BA), have contributed to reveal the most basic mechanisms that give origin to fractal structures. Hitherto, it has been commonly accepted that, in the absence of long-range particle-cluster interactions, the trajectories of aggregating particles, carrying the entropic information of the growing medium, are the m…
We investigate the aggregation kinetics of sedimenting particles theoretically and numerically, using the advection-diffusion equation. Agglomeration, caused by both transport mechanisms (diffusion and advection), is important for small particles, like primary ash or soot particles in atmosphere, and large particles of equal or close size, where the advection mechanism is weak. For small Peclet numbers, which quantify the relative importance of diffusion and advection, we obtain the aggregation rates, as an expansion in Peclet numbers. For large Peclet numbers we use purely ballistic aggregati…
We simulated models of diffusion and aggregation in long pores of small widths in order to represent the basic mechanisms of coke deposition in catalysts' pores. Coke precursors are represented by particles injected at the pore entrance. Knudsen diffusion, which is usually expected inside the pores, is modeled by ballistic motion of those particles. The regime of molecular diffusion is also analyzed via models of lattice random walks biased along the pores. The aggregation at the surface or near previously aggregated particles was modeled by different probabilistic rules, accounting for the po…
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 with…
We study surface and bulk properties of porous films produced by a model in which particles incide perpendicularly to a substrate, interact with deposited neighbors in its trajectory, and aggregate laterally with probability of order $a$ at each position. The model generalizes ballistic-like models by allowing attachment to particles below the outer surface. For small values of $a$, a crossover from uncorrelated deposition (UD) to correlated growth is observed. Simulations are performed in 1+1 and 2+1 dimensions. Extrapolation of effective exponents and comparison of roughness distributions co…
The idealized general model of aggregate growth is considered on the basis of the simple additive rules that correspond to one-step aggregation process. The two idealized cases were analytically investigated and simulated by Monte Carlo method in the Desktop Grid distributed computing environment to analyze "pile-up" and "wall" cluster distributions in different aggregation scenarios. Several aspects of aggregation kinetics (change of scaling, change of size distribution type, and appearance of scale-free size distribution) driven by "zero cluster size" boundary condition were determined by an…
Ballistic transport and resonance phenomena are elucidated in the one-dimensional $α$-Fermi-Pasta-Ulam-Tsingou (FPUT) model using an approach of computing thermal response functions. The existence of periodic oscillations in spatially sinusoidal temperature profiles seen in previous studies is confirmed. However, the results obtained using response functions enable a more complete understanding. In particular, it is shown that resonance involves beats between normal modes which tend to reinforce in a one-dimensional chain. Anharmonic scattering acts to destroy phase coherence across the statis…
A scaling theory is developed for diffusion-limited cluster aggregation in a porous medium, where the primary particles and clusters stick irreversibly to the walls of the pore space as well as to each other. Three scaling regimes are predicted, connected by smooth crossovers. The first regime is at low primary particle concentrations where the primary particles stick individually to the walls. The second regime is at intermediate concentrations where clusters grow to a certain size, smaller than the pore size, then stick individually to the walls. The third regime is at high concentrations wh…
We present results of computational modeling of the formation of uniform spherical silver particles prepared by rapid mixing of ascorbic acid and silver-amine complex solutions in the absence of a dispersing agent. Using an accelerated integration scheme to speed up the calculation of particle size distributions in the latter stages, we find that the recently reported experimental results -- some of which are summarized here -- can be modeled effectively by the two-stage formation mechanism used previously to model the preparation of uniform gold spheres. We treat both the equilibrium concentr…
