Nonlocal interactions in coagulating particle systems
  We consider a three dimensional system consisting of a large number of small spherical particles, which move due to gravity or with laminar shear and which merge when they cross. A size ratio criterion may be applied to restrict merging to similar sized particles (locality of interactions) or particles dissimilar in size (nonlocality). We perform direct numerical simulations (DNS) of this particle system and study the resulting mass spectra. In mean field approximation, these systems can be described by the Smoluchowski coagulation equation (SCE). DNS of the particle system with locality enforced show the scaling solutions or Kolmogorov-Zakharov spectra for the SCE, signifying a constant mass flux. DNS without a size ratio criterion show -4/3 scaling for large particles in a system with gravity, signifying a constant flux in number of particles, which we also find analytically by assuming nonlocality of interactions in the SCE. For laminar shear, this nonlocality is only marginal, and our DNS show that a correction to the scaling solution is required. 
T.H.M. Stein and S.V. Nazarenko