Smoluchowski coagulation equation

In statistical physics, the Smoluchowski coagulation equation is a population balance equation introduced by Marian Smoluchowski in a seminal 1916 publication,[1] describing the time evolution of the number density of particles as they coagulate (in this context "clumping together") to size x at time t. Simultaneous coagulation (or aggregation) is encountered in processes involving polymerization,[2] coalescence of aerosols,[3] emulsication,[4] flocculation.

In the case when the sizes of the coagulated particles are continuous variables, the equation involves an integral: If dy is interpreted as a discrete measure, i.e. when particles join in discrete sizes, then the discrete form of the equation is a summation: There exists a unique solution for a chosen kernel function.

[6] The operator, K, is known as the coagulation kernel and describes the rate at which particles of size

it could be mathematically proven that the solution of Smoluchowski coagulation equations have asymptotically the dynamic scaling property.

[8] This self-similar behaviour is closely related to scale invariance which can be a characteristic feature of a phase transition.

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, Some coagulation kernels account for a specific fractal dimension of the clusters, as in the diffusion-limited aggregation: or Reaction-limited aggregation: where

is the exponent of the product kernel, usually considered a fitting parameter.

are the radius and fall speed of the cloud particles usually expressed using power law.

Generally the coagulation equations that result from such physically realistic kernels are not solvable, and as such, it is necessary to appeal to numerical methods.

are introduced, one has to seek special approximation methods that suffer less from curse of dimensionality.

Approximation based on Gaussian radial basis functions has been successfully applied to the coagulation equation in more than one dimension.

[16][17] When the accuracy of the solution is not of primary importance, stochastic particle (Monte Carlo) methods are an attractive alternative.

The accuracy of this transformation can be adjusted such that just those coagulation events are considered while keeping the number of simulation entries constant.

[18] In addition to aggregation, particles may also grow in size by condensation, deposition or by accretion.

Hassan and Hassan recently proposed a condensation-driven aggregation (CDA) model in which aggregating particles keep growing continuously between merging upon collision.

This reaction scheme can be described by the following generalized Smoluchowski equation Considering that a particle of size

i.e. One can solve the generalized Smoluchowski equation for constant kernel to give which exhibits dynamic scaling.

is always a conserved quantity which is responsible for fixing all the exponents of the dynamic scaling.