A model of ballistic aggregation and fragmentation

Nikolay V. Brilliantov, Anna S. Bodrova, P. L. Krapivsky

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34 Citations (Scopus)


A simple model of ballistic aggregation and fragmentation is proposed. The model is characterized by two energy thresholds, Eagg and E frag, which demarcate different types of impacts: if the kinetic energy of the relative motion of a colliding pair is smaller than E agg 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 function and derive Smoluchowski-like equations for concentrations of particles of different mass. We analyze these equations analytically, solve them numerically and perform Monte Carlo simulations. When aggregation and fragmentation energy thresholds do not depend on the masses of the colliding particles, the model becomes analytically tractable. In this case we show the emergence of the two types of behavior: the regime of unlimited cluster growth arises when fragmentation is (relatively) weak and the relaxation towards a steady state occurs when fragmentation prevails. In a model with mass-dependent Eagg and Efrag the evolution with a crossover from one of the regimes to another has been detected.

Original languageEnglish
Article numberP06011
JournalJournal of Statistical Mechanics: Theory and Experiment
Issue number6
Publication statusPublished - 2009
Externally publishedYes


  • Boltzmann equation
  • Granular matter
  • Irreversible aggregation phenomena (theory)
  • Kinetic theory of gases and liquids


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