Université Paris 6Pierre et Marie Curie Université Paris 7Denis Diderot CNRS U.M.R. 7599 Probabilités et Modèles Aléatoires''

### Clustering in a self-gravitating one-dimensional gas at low temperature

Auteur(s):

Code(s) de Classification MSC:

• 70F10 $n$-body problem
• 60G50 Sums of independent random variables

Résumé: We study a system of gravitationally interacting sticky particles. At the initial time, we have $n$ particles, each with mass $1/n$ and momentum 0, independently spread on $[0,1]$ according to the uniform law. Due to the confining of the system, all particles merge into a single cluster after a finite time. We give the asymptotic laws of the time of the last collision and of the time of the $k$-th collision, when $n\to\infty$. We prove also that clusters of size $k$ appear at time $\sim n^{-1/2(k-1)}$. We then investigate the system at a fixed time $t<1$. We show that the biggest cluster has size of order $\log n$, whereas a typical cluster is of finite size.

Mots Clés: sticky particles ; gravitational interacting ; uniform law ; Brownian bridge

Date: 2001-06-13

Prépublication numéro: PMA-668