Smoluchowski coagulation equation

Example

Let $n(x, t) \ge 0$ be the number density of clusters of mass $x > 0$ at time $t \ge 0$. Let the coagulation kernel $K(x, y) = K(y, x) \ge 0$ be the rate at which clusters of masses $x$ and $y$ merge. The continuous Smoluchowski coagulation equation is $$\frac{\partial n(x, t)}{\partial t} = \frac12 \int_0^x K(x - y, y) \, n(x - y, t) \, n(y, t) \, dy - n(x, t) \int_0^\infty K(x, y) \, n(y, t) \, dy.$$ The first term counts the clusters of mass $x$ formed from two smaller ones, with the factor $\frac12$ because each pair is counted twice. The second term counts the clusters of mass $x$ lost by merging with any other cluster. It is a nonlinear integro-differential equation: a time derivative of $n$ and integrals of $n$ over mass.

Check for the constant kernel $K = k > 0$: let $N(t) = \int_0^\infty n(x, t) \, dx$ be the total number of clusters. Integrating over $x$ and exchanging the order of integration in the first term gives $N'(t) = \frac{k}{2} N^2 - k N^2 = -\frac{k}{2} N^2$. Hence $N(t) = \frac{N(0)}{1 + k N(0) t / 2}$, so the number of clusters decreases like $\frac{2}{kt}$.