Condensates in Driven Aggregation Processes

TL;DR

This paper studies how mass injection influences aggregation processes, revealing phase transitions and condensate formation depending on the aggregation kernel, with detailed analytic results for different kernel types.

Contribution

It provides new analytic insights into driven aggregation, characterizing phase behavior and condensate formation for various aggregation kernels.

Findings

Exponential decay of cluster size distribution for constant kernel.

Identification of condensate and cluster phases for kernels involving (i+j) and (ij).

Power-law tail with a non-monotonic exponent depending on injection rate.

Abstract

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} (ij), 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^{-gamma} in either phase. The exponent is a non-monotonic function of the injection rate: gamma=r/(r-1) in the condensate phase, r<2, and \gamma=r in the cluster phase, r>2.