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If the rates, $K(x,y)$, at which particles of size $x$ coalesce withparticles of size $y$ is known, then the mean-field evolution of theparticle-size distribution of an ensemble of irreversibly coalescing particlesis described by the Smoluchowski equation. We study the corresponding inverseproblem which aims to determine the coalescence rates, $K(x,y)$ frommeasurements of the particle size distribution. We assume that $K(x,y)$ is ahomogeneous function of its arguments, a case which occurs commonly inpractice. The problem of determining, $K(x,y)$, a function to two variables,then reduces to a simpler problem of determining a function of a singlevariable plus two exponents, $\mu$ and $\nu$, which characterise the scalingproperties of $K(x,y)$. The price of this simplification is that the resultingleast squares problem is nonlinear in the exponents $\mu$ and $\nu$. Wedemonstrate the effectiveness of the method on a selection of coalescenceproblems arising in polymer physics, cloud science and astrophysics. Theapplications include examples in which the particle size distribution isstationary owing to the presence of sources and sinks of particles and examplesin which the particle size distribution is undergoing self-similar relaxationin time.

A nonlinear least squares method for the inverse droplet coagulation problem