Clustering in a stochastic model of one-dimensional gas

TL;DR

This paper analyzes the clustering dynamics of a one-dimensional gas model where particles stick together upon collision, providing a limit theorem for the number of clusters over time as the initial particle count grows.

Contribution

It introduces a functional limit theorem for the number of clusters in a stochastic one-dimensional gas model, based on a novel localization property of the aggregation process.

Findings

Established a limit theorem for cluster count over time

Discovered a localization property of the aggregation process

Characterized asymptotic behavior of clustering in the model

Abstract

We give a quantitative analysis of clustering in a stochastic model of one-dimensional gas. At time zero, the gas consists of $n$ identical particles that are randomly distributed on the real line and have zero initial speeds. Particles begin to move under the forces of mutual attraction. When particles collide, they stick together forming a new particle, called cluster, whose mass and speed are defined by the laws of conservation. We are interested in the asymptotic behavior of $K_n(t)$ as $n\to \infty$, where $K_n(t)$ denotes the number of clusters at time $t$ in the system with $n$ initial particles. Our main result is a functional limit theorem for $K_n(t)$. Its proof is based on the discovered localization property of the aggregation process, which states that the behavior of each particle is essentially defined by the motion of neighbor particles.

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