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| In [[statistical physics]], the '''Smoluchowski coagulation equation''' is an [[integrodifferential equation]] introduced by [[Marian Smoluchowski]] in a seminal 1916 publication,<ref>{{cite journal|last=Smoluchowski|first=Marian|title=Drei Vorträge über Diffusion, Brownsche Molekularbewegung und Koagulation von Kolloidteilchen|journal=Physik. Z.|year=1916|volume=17|pages=557–571, 585–599|bibcode=1916ZPhy...17..557S}}</ref> describing the [[time evolution]] of the [[number density]] of particles as they coagulate (in this context "clumping together") to size ''x'' at time ''t''.
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| ==Equation==
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| In the case when the sizes of the coagulated particles are [[continuous variable]]s, the equation involves an [[integral]]:
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| : <math>\frac{\partial n(x,t)}{\partial t}=\frac{1}{2}\int^x_0K(x-y,y)n(x-y,t)n(y,t)\,dy - \int^\infty_0K(x,y)n(x,t)n(y,t)\,dy.</math>
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| If ''dy'' is interpreted as a discrete [[List of integration and measure theory topics#Intuitive foundations|measure]], i.e. when particles join in [[discrete variable|discrete]] sizes, then the discrete form of the equation is a [[summation]]:
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| : <math>\frac{\partial n(x_i,t)}{\partial t}=\frac{1}{2}\sum^{i-1}_{j=1}
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| K(x_i-x_j,x_j)n(x_i-x_j,t)n(x_j,t) - \sum^\infty_{j=1}K(x_i,x_j)n(x_i,t)n(x_j,t).</math>
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| ==Coagulation kernel==
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| The [[linear operator|operator]], ''K'', is known as the coagulation [[integral kernel|kernel]] and describes the rate at which particles of size ''x'' coagulate with particles of size ''y''. [[Closed-form expression|Analytic solutions]] to the equation exist when the kernel takes one of three simple forms:
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| : <math>K = 1,\quad K = x + y, \quad K = xy,</math>
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| known as the [[constant (mathematics)|constant]], [[Additive function|additive]], and [[Multiplicative function|multiplicative]] kernels respectively. However, in most practical applications the kernel takes on a significantly more complex form, for example the free-molecular kernel which describes [[collision]]s in a dilute [[gas]]-[[Phase (matter)|phase]] system,
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| : <math>K = \sqrt{\frac{\pi k_b T}{2}}\left(\frac{1}{m(x)}+\frac{1}{m(y)}\right)^{1/2}\left(d(x)+d(y)\right)^2.</math>
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| Generally the coagulation equations that result from such physically realistic kernels are not solvable, and as such, it is necessary to appeal to [[Numerical analysis|numerical methods]]. There exist well-established [[deterministic]] methods that can be used when there is only one particle property (''x'') of interest, the two principal ones being the [[Method of moments (probability theory)|method of moments]] and [[sectional method]]s. In the [[Multivariable calculus|multi-variate]] case however, when two or more properties (such as size, shape, composition etc.) are introduced, special approximation methods that suffer less from [[curse of dimensionality]] has to be applied. For instance, approximation based on Gaussian [[radial basis functions]] has been successfully applied to the two-dimensional coagulation equation.<ref>Predicting multidimensional distributive properties of hyperbranched polymer resulting from AB2 polymerization with substitution, cyclization and shielding, I. Kryven, P.D. Iedema, Polymer 54 (14), 3472–3484</ref>
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| When accuracy of the solution is not of primary importance, [[Monte Carlo method|stochastic particle (Monte-Carlo) methods]] are an attractive alternative.
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| ==See also== | |
| *[[Einstein relation (kinetic theory)|Einstein–Smoluchowski relation]]
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| *[[Smoluchowski factor]]
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| == References ==
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| {{Reflist}}
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| {{DEFAULTSORT:Smoluchowski Coagulation Equation}}
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| [[Category:Differential equations]]
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| [[Category:Polish physicists]]
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