Spatial particle processes with coagulation: Gibbs-measure approach, gelation and Smoluchowski equation
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| author | Andreis, Luisa König, Wolfgang Langhammer, Heide Patterson, Robert I. A. |
| author_facet | Andreis, Luisa König, Wolfgang Langhammer, Heide Patterson, Robert I. A. |
| contents | 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 $\frac 1N K$. 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_06668 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Spatial particle processes with coagulation: Gibbs-measure approach, gelation and Smoluchowski equation Andreis, Luisa König, Wolfgang Langhammer, Heide Patterson, Robert I. A. Probability 82C22, 60J25, 60F10, 60G55, 60K35, 35Q70 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 $\frac 1N K$. 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. |
| title | Spatial particle processes with coagulation: Gibbs-measure approach, gelation and Smoluchowski equation |
| topic | Probability 82C22, 60J25, 60F10, 60G55, 60K35, 35Q70 |
| url | https://arxiv.org/abs/2401.06668 |