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  "body": "\n## Example\n\nLet $n(x, t) \\ge 0$ be the number density of clusters of mass $x \u003e 0$ at time $t \\ge 0$.\nLet the coagulation kernel $K(x, y) = K(y, x) \\ge 0$ be the rate at which clusters of masses $x$ and $y$ merge.\nThe continuous Smoluchowski coagulation equation is\n$$\\frac{\\partial n(x, t)}{\\partial t} = \\frac12 \\int_0^x K(x - y, y) \\, n(x - y, t) \\, n(y, t) \\, dy - n(x, t) \\int_0^\\infty K(x, y) \\, n(y, t) \\, dy.$$\nThe first term counts the clusters of mass $x$ formed from two smaller ones, with the factor $\\frac12$ because each pair is counted twice.\nThe second term counts the clusters of mass $x$ lost by merging with any other cluster.\nIt is a nonlinear [[integro-differential equation|YRKW2KX2CW]]: a time derivative of $n$ and integrals of $n$ over mass.\n\nCheck for the constant kernel $K = k \u003e 0$: let $N(t) = \\int_0^\\infty n(x, t) \\, dx$ be the total number of clusters.\nIntegrating over $x$ and exchanging the order of integration in the first term gives $N'(t) = \\frac{k}{2} N^2 - k N^2 = -\\frac{k}{2} N^2$.\nHence $N(t) = \\frac{N(0)}{1 + k N(0) t / 2}$, so the number of clusters decreases like $\\frac{2}{kt}$.\n\n## Remarks\n\nFor bounded kernels the total mass $\\int_0^\\infty x \\, n(x, t) \\, dx$ is conserved.\nFor the product kernel $K(x, y) = xy$ the mass is conserved only up to a finite time, after which mass escapes into an infinitely large cluster; this is called gelation.\nThe discrete version replaces the integrals by sums over integer cluster sizes.\n\n## Explanations\n\n### L1 human\n\nSmall particles in a smoke or a polymer solution collide and stick together, forming bigger clusters.\nThe equation describes how the distribution of cluster sizes changes in time.\nA cluster of a given size is created when two smaller ones meet, which gives an integral over the ways to split it.\nIt is destroyed when it meets any other cluster.\n",
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