Scaling limits of Smoluchowski particles

TL;DR

This paper establishes a law of large numbers and a central limit theorem for the empirical density in a Marcus-Lushnikov model, linking it to the Smoluchowski equation and describing fluctuations via an Ornstein-Uhlenbeck process.

Contribution

It provides rigorous probabilistic results connecting particle system limits to the Smoluchowski equation and characterizes the fluctuations around the limit.

Findings

Empirical density converges to the Smoluchowski equation solution.

Fluctuations are described by an Ornstein-Uhlenbeck process.

Proves a law of large numbers and a functional central limit theorem.

Abstract

We prove a law of large numbers and a functional central limit theorem for the empirical density of a Marcus-Lushnikov model. The limiting density turns out to be the solution of a Smoluchowski equation, and the fluctuations around this limit are shown to be described by an Ornstein-Uhlenbeck process with drift term given by the linearization of the Smoluchowski operator.