• Corpus ID: 119612054

Kac Interaction Clusters: A Bilinear Coagulation Equation and Phase Transition

  title={Kac Interaction Clusters: A Bilinear Coagulation Equation and Phase Transition},
  author={Daniel Heydecker and Robert I. A. Patterson},
  journal={arXiv: Probability},
We consider the interaction clusters for Kac's model of a gas with quadratic interaction rates, and show that they behave as coagulating particles with a bilinear coagulation kernel. In the large particle number limit the distribution of the interaction cluster sizes is shown to follow an equation of Smoluchowski type. Using a coupling to random graphs, we analyse the limiting equation, showing well-posedness, and a closed form for the time of the gelation phase transition $t_\mathrm{g}$ when a… 


Existence of Gelling Solutions for Coagulation-Fragmentation Equations
Abstract:We study the Smoluchowski coagulation-fragmentation equation, which is an infinite set of non-linear ordinary differential equations describing the evolution of a mono-disperse system of
Coagulation in finite systems
Kinetic theory of cluster dynamics
A Model for Coagulation with Mating
We consider in this work a model for aggregation, where the coalescing particles initially have a certain number of potential links (called arms) which are used to perform coagulations. There are two
A consistency estimate for Kac's model of elastic collisions in a dilute gas
An explicit estimate is derived for Kac's mean-field model of colliding hard spheres, which compares, in a Wasserstein distance, the empirical velocity distributions for two versions of the model
The Boltzmann–Grad limit of a hard sphere system: analysis of the correlation error
It is shown that, provided k < \varepsilon ^{-\alpha }$$k<ε-α, such an error flows to zero with the average density, namely j different particles behave as dictated by the Boltzmann equation even when j diverges as a negative power of £ε, providing an information on the size of chaos.
Kinetics of polymer gelation
The kinetics of polymer growth and especially gelation are discussed for a system of reacting f‐functional monomeric units. The gelation models of Flory and Stockmayer are examined, and their
Polymers and random graphs
We establish a precise connection between gelation of polymers in Lushnikov's model and the emergence of the giant component in random graph theory. This is achieved by defining a modified version of