In statistical physics, the Smoluchowski coagulation equation is a population balance equation introduced by Marian Smoluchowski in a seminal 1916 publication, 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, coalescence of aerosols, emulsication, flocculation.
Equation The distribution of particle size changes in time according to the interrelation of all particles of the system. Therefore, the Smoluchowski coagulation equation is an integrodifferential equation of the particle-size distribution. In the case when the sizes of the coagulated particles are continuous variables, the equation involves an integral:
∂ n ( x , t ) ∂ t = 1 2 ∫ 0 x K ( x − y , y ) n ( x − y , t ) n ( y , t ) d y − ∫ 0 ∞ K ( x , y ) n ( x , t ) n ( y , t ) d y . {\displaystyle {\frac {\partial n(x,t)}{\partial t}}={\frac {1}{2}}\int _{0}^{x}K(x-y,y)n(x-y,t)n(y,t)\,dy-\int _{0}^{\infty }K(x,y)n(x,t)n(y,t)\,dy.}
If dy is interpreted as a counting measure, i.e. when particles join in discrete sizes, then the discrete form of the equation is a summation:
∂ n ( x i , t ) ∂ t = 1 2 ∑ j = 1 i − 1 K ( x i − x j , x j ) n ( x i − x j , t ) n ( x j , t ) − ∑ j = 1 ∞ K ( x i , x j ) n ( x i , t ) n ( x j , t ) . {\displaystyle {\frac {\partial n(x_{i},t)}{\partial t}}={\frac {1}{2}}\sum _{j=1}^{i-1}K(x_{i}-x_{j},x_{j})n(x_{i}-x_{j},t)n(x_{j},t)-\sum _{j=1}^{\infty }K(x_{i},x_{j})n(x_{i},t)n(x_{j},t).}
There exists a unique solution for a chosen kernel function.
Coagulation kernel The operator, K, is known as the coagulation kernel and describes the rate at which particles of size x 1 {\displaystyle x_{1}} coagulate with particles of size x 2 {\displaystyle x_{2}} . Analytic solutions to the equation exist when the kernel takes one of three simple forms:
K = 1 , K = x 1 + x 2 , K = x 1 x 2 , {\displaystyle K=1,\quad K=x_{1}+x_{2},\quad K=x_{1}x_{2},}
known as the constant, additive, and multiplicative kernels respectively. For the case K = 1 {\displaystyle K=1} it could be mathematically proven that the solution of Smoluchowski coagulation equations have asymptotically the dynamic scaling property. 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,
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