Smoluchowski coagulation equation

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The Smoluchowski coagulation equation is an integrodifferential equation introduced by Marian Smoluchowski in a seminal 1916 publication, describing the evolution of the number density of particles of size x at a time t. In the continuous case the equation is

\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.

If dy is interpreted as a discrete measure then the discrete form of the equation is recovered:

\frac{\partial n(x_i,t)}{\partial t}=\frac{1}{2}\sum^N_{i\neq j} K(x_i-x_j,x_j)n(x_i-x_j,t)n(x_j,t) - \sum^{\infty}_0K(x_i,x_j)n(x_i,t)n(x_j,t).

The operator, K, is known as the coagulation kernel and describes the rate at which particles of size x coagulate with particles of size y. Analytic solutions to the equation exist when the kernel takes one of three simple forms:

K = 1,\quad K = x + y, \quad K = xy,

known as the constant, additive, and 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 collisions in a dilute gas-phase system,

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.

Generally the coagulation equations which result from such physically realistic kernels are intractable, and as such, it is necessary to appeal to 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 and sectional methods. In the multi-variate case however, when two or more properties (such as size, shape, composition etc.) are introduced, the efficiency of deterministic methods suffers and stochastic particle (Monte-Carlo) methods are an attractive alternative.

[edit] See also

[edit] Literature

  • M. Smoluchowski: "Drei Vorträge über Diffusion, Brownsche Molekularbewegung und Koagulation von Kolloidteilchen", Phys Z, 17 (1916) 557-571 and 585-599.
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