Smoluchowski Coagulation Equation
Encyclopedia
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
If dy is interpreted as a discrete measure then the discrete form of the equation is recovered:
Marian Smoluchowski
Marian Smoluchowski was an ethnic Polish scientist in the Austro-Hungarian Empire. He was a pioneer of statistical physics and an avid mountaineer.-Life:...
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
If dy is interpreted as a discrete measure then the discrete form of the equation is recovered:
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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:
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,
Generally the coagulation equations that 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) methodsMonte Carlo methodMonte Carlo methods are a class of computational algorithms that rely on repeated random sampling to compute their results. Monte Carlo methods are often used in computer simulations of physical and mathematical systems...
are an attractive alternative.
See also
- Einstein–Smoluchowski relationEinstein relation (kinetic theory)In physics the Einstein relation is a previously unexpected connection revealed independently by Albert Einstein in 1905 and by Marian Smoluchowski in their papers on Brownian motion...
- Marian SmoluchowskiMarian SmoluchowskiMarian Smoluchowski was an ethnic Polish scientist in the Austro-Hungarian Empire. He was a pioneer of statistical physics and an avid mountaineer.-Life:...
- Smoluchowski factorSmoluchowski factorThe Smoluchowski factor, also known as von Smoluchowski's f-factor is related to inter-particle interactions. It is named after Marian Smoluchowski.-See also:* Flocculation* Smoluchowski coagulation equation* Einstein–Smoluchowski relation...
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