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Spatial particle processes with coagulation: Gibbs-measure approach, gelation and Smoluchowski equation

Luisa Andreis, Wolfgang König, Heide Langhammer, Robert I. A. Patterson

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 $\frac 1N K$. 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.

Spatial particle processes with coagulation: Gibbs-measure approach, gelation and Smoluchowski equation

Abstract

We study a spatial Markovian particle system with pairwise coagulation, a spatial version of the Marcus--Lushnikov process: according to a coagulation kernel , 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 in terms of a reference Poisson point process, whose points are trajectories that coagulate into one particle by time . 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 of initial atoms diverges and the kernel scales as . 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.
Paper Structure (28 sections, 35 theorems, 210 equations, 1 figure)

This paper contains 28 sections, 35 theorems, 210 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Summary of our results
  3. The spatial Marcus--Lushnikov process
  4. Decomposition into coagulation trajectories
  5. Results
  6. First main result: identification of the distribution
  7. Second main result: a large-deviations principle
  8. Third main result: criteria for convergence and gelation
  9. Fourth main result: the Smoluchowski equation
  10. Background discussion
  11. Comments on the main results
  12. Literature survey
  13. Comparison to inhomogeneous Erdős--Rényi graph
  14. Proof of Theorem \ref{['thm-distMi']}: distribution of the configuration
  15. Preparations
  16. ...and 13 more sections

Key Result

Theorem 2.1

Fix $\mu\in{\mathcal{M} }_1({\mathcal{S} })$ and $T>0$ and $N\in\mathbb{N}$ and a measurable bounded function $f\colon{\mathcal{M} }(\Gamma^{{{({1}})}}_T)\to[0,\infty)$. Then, for any $b\in(0,\infty)$, where $Y_N=\sum_i\delta_{\Xi_i}\sim\mathrm{Poi}_{N M^{{{{({T}})}}}_{b\mu,N}}$ is a PPP with intensity measure $N M^{{{{({T}})}}}_{b\mu,N}$.

Figures (1)

  • Figure 1: An illustration of the decomposition of $(\Xi_t)_{t\in [0,T]}$ into four subprocesses $(\xi^{{{({i}})}}_t)_{t\in[0,T]}$, $i=1, \ldots, 4$, that are distinguished by their colour. The process $(\Xi_t)_{t\in [0,T]}$ is started from $15$ atoms at time 0 and ends up with $4$ particles. However, each subprocess $(\xi^{{{({i}})}}_t)_{t\in[0,T]}$, $i=1, \ldots, 4$, ends in a single-particle configuration by definition. At time $t$, there are $6$ particles with masses $3, 2, 1, 2, 5$ and $2$.

Theorems & Definitions (72)

  • Remark 1.1: Special choices
  • Remark 1.2: Inhomogeneous Erdős--Rényi graph
  • Theorem 2.1: Poissonian description of the empirical measure
  • Lemma 2.2: Convergence of $N^{|k|-1}\mathbb{P}_k^{{{{({N}})}}}(\Xi\in\cdot)|_{\Gamma_T^{{{{({1}})}}}}$
  • Theorem 2.3: LDP for ${\mathcal{V} }_{N}^{{{{({T}})}}}$
  • Corollary 2.4: Accumulation points
  • Lemma 2.5: Continuity of $\nu\mapsto \rho(\nu)$
  • Corollary 2.6: LDP for $(\frac{1}{N}\Xi_t)_{t\in[0,T]}$
  • Remark 2.7: Convexity of $I_\mu^{{{{({T}})}}}$ by nonnegative definiteness of $K$
  • Theorem 2.8: Criteria for non-gelation and for gelation
  • ...and 62 more