Spatial particle processes with coagulation: Gibbs-measure approach, gelation and Smoluchowski equation
Authors
- Andreis, Luisa
- König, Wolfgang
ORCID: 0000-0002-7673-4364 - Langhammer, Heide
ORCID: 0009-0005-0353-0343 - Patterson, Robert I. A.
ORCID: 0000-0002-3583-2857
2020 Mathematics Subject Classification
- 82C22 60J25 60F10 60G55 60K35 35Q70
Keywords
- Spatial coagulation process, spatial Marcus--Lushnikov process, empirical measures of particles, coagulation trajectories, monodispersed initial condition, Gibbsian representation, gelation phase transition, Smoluchowski equation, large deviations
DOI
Abstract
We study a spatial Markovian particle system with pairwise coagulation, a spatial version of the Marcus--Lushnikov process: according to a coagulation kernel K, particle pairs merge into a single particle, and their masses are united. We introduce a statistical-mechanics approach to the study of this process. We derive an explicit formula for the empirical process of the particle configuration at a given fixed time T in terms of a reference Poisson point process, whose points are trajectories that coagulate into one particle by time T. The non-coagulation between any two of them induces an exponential pair-interaction, which turns the description into a many-body system with a Gibbsian pair-interaction. Based on this, we first give a large-deviation principle for the joint distribution of the particle histories (conditioning on an upper bound for particle sizes), in the limit as the number N of initial atoms diverges and the kernel scales as 1/N K. We characterise the minimiser(s) of the rate function, we give criteria for its uniqueness and prove a law of large numbers (unconditioned). Furthermore, we use the unique minimiser to construct a solution of the Smoluchowski equation and give a criterion for the occurrence of a gelation phase transition. endabstract
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