The relevance of Pólya's random-walk problem for the single-species reaction-diffusion system

International Centre of Theoretical Physics, Condensed Matter Section, P.O.Box 586, 34100 Trieste, Italy Theoretische Polymerphysik - Rheinstr. 12, D-79104 Freiburg, Germany

Received: 6 September 1996
Accepted: 8 April 1997

Abstract

The diffusion-limited reactions and in dimension d > 2 are reconsidered from the point of view of the random-walk theory. It is pointed out that Pólya's theorem on the returning probability of a random walker to the origin, which would imply a probability less than one for the meeting of two typical particles, would predict the possibility of a state in which the reaction seems to have spontaneously ceased, in contradiction with the very well known asymptotic for the particles population of these reactions. In fact, a given condition is presented, in which the relative particle number N(t)/N(0) decays to a non-vanishing constant. The condition is that the initial distribution of particles in the d-dimensional space has a dimension γ, such that .