Smoluchowski's coagulation equation describes the evolution in time of a system of atmospheric particles, coagulating upon binary collision. In this talk I will present a generalization of the classical Smoluchowski's coagulation equation, i.e. the coagulation equation with a source term of small clusters. The source term drives the system out-of-equilibrium, leading to a rich range of different possible long-time behaviours, including anomalous self-similarity.

We assume that the coagulation kernel is non gelling, homogeneous, with homogeneity \( γ < 1 \), and behaves like \( x^{(γ+λ)} y^{−λ} \) when \( y\ll x \) with \( γ+2λ \ge 1 \). We argue that, when \( γ+2λ > 1 \) the long-time behaviour is self-similar and that the scaling of the self-similar solutions depends on the sign of \( γ + λ \) and on whether \( γ = −1 \) or \( γ < −1 \) or \( −1 < γ < 1 \). We present also some conjectures supporting the self-similar ansatz for the critical case \( γ + 2λ = 1 \), \( γ \le −1 \).