ONE-DIMENSIONAL BALLISTIC AGGREGATION - RIGOROUS LONG-TIME ESTIMATES

Aggregation of mass by perfectly inelastic collisions in a one-dimensi onal gas of point particles is studied. The dynamics is governed by la ws of mass and momentum conservation. The motion between collisions is free. An exact probabilistic description of the state of the aggregat ing gas is presented. For an initial configuration of equidistant part icles on the line with Maxwellian velocity distribution, the following results are obtained in the long-time limit. The probability for find ing empty intervals of length growing faster than t2/3 vanishes. The m ass 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 est imates are in accordance with numerical simulations predicting t-1 dec ay for the number density of initial masses and a slower t-2/3 decay f or the density of aggregates resulting from a large number of collisio ns (with masses approximately t2/3). Our proofs rely on a link between the considered aggregation dynamics and Brownian motion in the presen ce of absorbing barriers.