Derivations

Explicit formulae have been derived for the Smoluchowski equation in isotropic media for different geometries and for its logarithmically transformed version (concentration, time, space).

Law of mass action

The well established law of mass action give the rate of change of a concentration \(c_i\) of species \(i\):

\[\frac{\partial c_i}{\partial t} = \sum_l r_l S_{il}\]

\(t\) is time, \(c_i\) is the concentration of species \(i\), \(S_{il}\) is the net stoichiometric coefficient of species \(i\) in reaction \(l\) and \(r_l\) is the rate of reaction \(l\), which for a mass-action type of rate law is given by:

\[\begin{split}r_l = \begin{cases} \kappa_l\prod_k c_k^{R_{kl}} &\mbox{if } \sum_k R_{kl} > 0 \\ 0 &\mbox{otherwise} \end{cases}\end{split}\]

where \(\kappa_l\) is the rate constant, \(R_{kl}\) is the stoichiometric coefficient of species \(k\) on the reactant side.

If we introduce the logarithmically transormed concentration \(z\):

\[\begin{split}z_i &= \log(c_i)\end{split}\]

we have:

\[\begin{split}\frac{\partial z_i}{\partial t} &= \frac{\frac{\partial c_i}{\partial t}}{c_i}\end{split}\]

which can be expressed in \(z_i\):

\[\begin{split}\frac{\partial z_i}{\partial t} &= e^{-z_i} \sum_l r_l S_{il}\end{split}\]

where we may now express \(r_l\) as:

\[\begin{split}r_l = \begin{cases} \kappa_l e^{\sum_k R_{kl} z_k} &\mbox{if } \sum_k R_{kl} > 0 \\ 0 &\mbox{otherwise} \end{cases}\end{split}\]

Diffusion equation

Laplace operator

Jacobian elements

Boundary conditions

Finite difference scheme