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 $K_{i,j}=\mathrm{const}$, the cluster size distribution decays exponentially. When $K_{i,j}\propto i+j$ or $K_{i,j}\propto i\times j$, 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 $K_{i,j}\propto i+j$, the cluster size distribution, $c_k$, has a power-law tail, $c_k\sim k^{-\gamma }$ in either phase. The exponent is a nonmonotonic function of the injection rate $\gamma =r∕(r-1)$ in the condensate phase $r<2$ and $\gamma =r$ in the cluster phase $r>2$.

Figure 1

(Color online) The condensate mass $m_{*}$ as a function of $r$ for the sum rate. Shown in a solid line is the theoretical prediction (20). Shown in solid lines with symbols are Monte Carlo simulations results for the size of the largest cluster in the system.

Figure 2

The exponent $\gamma$ versus $r$ for the sum rate.

Figure 3

The exponent $\gamma$ versus $r$ for the product rate.

Figure 4

(Color online) The condensate mass $m_{*}$ as a function of $r$ obtained from Monte Carlo simulations of the aggregation process (product rate). The results are from a single run with $M={10}^6$ particles. The inset displays the same behavior using a semilog scale. These simulation results represent an average over 100 independent realizations with a varying number of particles.