*Europhys. Lett.*,

**73**(5), pp. 698-704 (2006)

DOI: 10.1209/epl/i2005-10450-2

## On the gravitational instability of a set of random walkers

**L. Acedo**

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

acedo_physics@hotmail.com

received 17 September 2005; accepted in final form 5 January 2006

published online 18 January 2006

** Abstract **

We consider the time-dependent distribution,
*c*(**r**,*t*), 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 *u*(*r*,*t*), 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,
*c*(*r*,*t*)=*C*(*r*^{2}/*t*)/*t*, is also
studied and we show that it only exists for *N* < *N*_{c}, where *N*_{c} is the number of
particles corresponding to the unstable stationary state.

**PACS**

05.40.Jc - Brownian motion.

04.40.-b - Self-gravitating systems; continuous media and classical fields in curved spacetime.

**©**

*EDP Sciences 2006*