Bilinear Coagulation Equations

Abstract

We consider coagulation equations of Smoluchowski or Flory type where the total merge rate has a bilinear form π(y) • Aπ(x) for a vector of conserved quantities π, generalising the multiplicative kernel. For these kernels, a gelation transition occurs at a finite time t g ∈ (0, ∞), which can be given exactly in terms of an eigenvalue problem in finite dimensions. We prove a hydrodynamic limit for a stochastic coagulant, including a corresponding phase transition for the largest particle, and exploit a coupling to random graphs to extend analysis of the limiting process beyond the gelation time.

Cluster expansion for a dilute hard sphere gas dynamics

“…and we simply write F (H) (λ ) when the trajectory is an ε-cluster path λ . We stress the fact that, by definition of the ε-cluster paths λ j appearing in (22), the particles in λ j are not allowed to overlap initially…”

“…In (22) the integration variables are the initial particle configurations Z N (which as recalled in Remark II.2 fix the dynamics on [0,T ] and therefore the partition into cluster paths). Fubini's theorem enables us to consider now as the variables of interest the cluster paths λ j .…”

“…ε,T as in ( 27) with the modulus F (H) to take into account test functions H with complex values. With these notations, the partition function (22) can be rewritten as a partition function of a gas of cluster paths interacting by exclusion. Indeed a particle configuration Z N is partitioned into k cluster paths with cardinalities n 1 ,...,n k…”

“…We will make this analogy more precise in Section IV B when analysing the dynamical clustering process. We refer to [15,22,29,32,33] for previous works on this dynamical phase transition in the case of particle dynamics. Reaching longer time asymptotics requires therefore new ideas and techniques.…”

“…Later on in [32] it was shown that, in the Boltzmann-Grad limit, the cluster dynamics can be described by explicit formulas and that in this limit, the critical transition takes place in times of order one. Finally in [22,33], the cluster process was studied for Kac particle systems, obtaining more precise results in spatially homogeneous cases. Theorem IV.3 below derives rigorously the cluster process from hard spheres, giving in particular a proof to the formal statements in [32], at least for short times.…”