Condensates in driven aggregation processes

We investigate aggregation driven by mass injection. In this stochastic process, mass is added with constant rate r and clusters merge at a constant total rate 1, so that both the total number of clusters and the total mass steadily grow with time. Analytic results are presented for the three classic aggregation rates K_i,j between clusters of size i and j. When K_i,j=const, the cluster size distribution decays exponentially. When K_i,j∝i+j or K_i,j∝i×j, there are two phases: (i) a condensate phase with a condensate containing a finite fraction of the mass in the system as well as finite clusters and (ii) a cluster phase with finite clusters only. For K_i,j∝i+j, the cluster size distribution, c_k, has a power-law tail, c_k∼k^−γ in either phase. The exponent is a nonmonotonic function of the injection rate γ=r∕(r−1) in the condensate phase r<2 and γ=r in the cluster phase r>2.