ONE-DIMENSIONAL BALLISTIC AGGREGATION - RIGOROUS LONG-TIME ESTIMATES

Aggregation of mass by perfectly inelastic collisions in a one-dimensional gas of point particles is studied. The dynamics is governed by laws of mass and momentum conservation. The motion between collisions is free. An exact probabilistic description of the state of the aggregating gas is presented. For an initial configuration of equidistant particles on the line with Maxwellian velocity distribution, the following results are obtained in the long-time limit. The probability for finding empty intervals of length growing faster than t2/3 vanishes. The mass spectrum can range from the initial mass up to mass of order t2/3, Aggregates with masses growing faster than t2/3 cannot occur. Our estimates are in accordance with numerical simulations predicting t-1 decay for the number density of initial masses and a slower t-2/3 decay for the density of aggregates resulting from a large number of collisions (with masses approximately t2/3). Our proofs rely on a link between the considered aggregation dynamics and Brownian motion in the presence of absorbing barriers.