We describe collective oscillatory behavior in the kinetics of irreversible coagulation with a constant input of monomers and removal of large clusters. For a broad class of collision rates, this system reaches a nonequilibrium stationary state at large times and the cluster size distribution tends to a universal form characterized by a constant flux of mass through the space of cluster sizes. Universality, in this context, means that the stationary state becomes independent of the cutoff as the cutoff grows. This universality is lost, however, if the aggregation rate between large and small clusters increases sufficiently steeply as a function of cluster sizes. We identify a transition to a regime in which the stationary state vanishes as the cutoff grows. This nonuniversal stationary state becomes unstable as the cutoff is increased. It undergoes a Hopf bifurcation after which the stationary state is replaced by persistent and periodic collective oscillations. These oscillations, which bear some similarities to relaxation oscillations in excitable media, carry pulses of mass through the space of cluster sizes such that the average mass flux through any cluster size remains constant. Universality is partially restored in the sense that the scaling of the period and amplitude of oscillation is inherited from the dynamical scaling exponents of the universal regime.

It is proposed to revisit the inverse problem associated with Smoluchowski's coagulation equation. The objective is to reconstruct the functional form of the collision kernel from observations of the time evolution of the cluster size distribution. A regularised least squares method originally proposed by Wright and Ramkrishna (1992) based on the assumption of self-similarity is implemented and tested on numerical data generated for a range of different collision kernels. This method expands the collision kernel as a sum of orthogonal polynomials and works best when the kernel can be expressed exactly in terms of these polynomials. It is shown that plotting an "L-curve" can provide an a-priori understanding of the optimal value of the regularisation parameter and the reliability of the inversion procedure. For kernels which are not exactly expressible in terms of the orthogonal polynomials it is found empirically that the performance of the method can be enhanced by choosing a more complex regularisation function.

If the rates, K(x, y), at which particles of size x coalesce with particles of size y is known, then the mean-field evolution of the particle-size distribution of an ensemble of irreversibly coalescing particles is described by the Smoluchowski equation. We study the corresponding inverse problem which aims to determine the coalescence rates, K(x, y) from measurements of the particle size distribution. We assume that K(x, y) is a homogeneous function of its arguments, a case which occurs commonly in practice. The problem of determining, K(x, y), a function to two variables, then reduces to a simpler problem of determining a function of a single variable plus two exponents, µ and ν, which characterise the scaling properties of K(x, y). The price of this simplification is that the resulting least squares problem is nonlinear in the exponents µ and ν. We demonstrate the effectiveness of the method on a selection of coalescence problems arising in polymer physics, cloud science and astrophysics. The applications include examples in which the particle size distribution is stationary owing to the presence of sources and sinks of particles and examples in which the particle size distribution is undergoing self-similar relaxation in time.

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