Scaling properties of one-dimensional cluster–cluster aggregation with Lévy diffusion

Abstract

We present a study of the scaling properties of cluster–cluster aggregation with a source ofmonomers in the stationary state when the spatial transport of particles occurs by Lévyflights. We show that the transition from mean-field statistics to fluctuation-dominatedstatistics which, for the more commonly considered case of diffusive transport,occurs as the spatial dimension of the system is tuned through two from above,can be mimicked even in one dimension by varying the characteristic exponent,β, of the Lévy jump length distribution. We also show that the two-pointmass correlation function, responsible for the flux of mass in the stationarystate, is strongly universal: its scaling exponent is given by the mean-fieldvalue independent of the spatial dimension and independent of the value ofβ. Finally, we study numerically the two-point spatial correlation function whichcharacterizes the structure of the depletion zone around heavy particles in thediffusion-limited regime. We find that this correlation function vanishes with a non-trivialfractional power of the separation between particles as this separation goes to zero. Weprovide a scaling argument for the value of this exponent which is in reasonable agreementwith the numerical measurements.