On the gravitational instability of a set of random walkers

Departamento de Matemática Aplicada, ETSII, Universidad de Salamanca 37700 Béjar, Spain

Corresponding author: acedo_physics@hotmail.com

Received: 17 September 2005
Accepted: 5 January 2006

Abstract

We consider the time-dependent distribution, , of identical point-like random walkers which interact via a Newtonian potential in a d-dimensional unbounded Euclidean space. This distribution is usually studied by means of the Smoluchowski-Poisson equation in which the self-interaction of the Brownian particles is incorporated as a mean-field dragging force. In this letter we find the exact solution of this equation in one dimension for any normalizable and symmetric initial condition. This solution is found by exploiting the similarity of the equation for , the mass inside a hypersphere of radius r, and Burgers's equation. We prove that the one-dimensional system is stable under arbitrary symmetric perturbations and always converges to the well-known stationary state. An exact scaling solution representing monotonous evaporation in two dimensions, , is also studied and we show that it only exists for , where N_c is the number of particles corresponding to the unstable stationary state.