Subject: CCL: Re: Diffusion Equation from Kinetic Lattice Monte Carlo

From: [email-masked]
Date: Wed, 15 Sep 1999 19:41:25 -0700

Andrew Horsfield wrote:
> I have a question about kinetic Monte Carlo simulations on a lattice, and
> how they relate to traditional diffusion and reaction equations.
> ...
> Can anyone tell me where I have gone wrong? Is there a standard reference
> for this kind of approximation?

Andrew,

One standard reference, which I found extremely useful, is "Stochastic processes
in physics and chemistry", by N. G. van Kampen, Elsevier Science Publishers
B.V., 2n ed. 1992. He extensively treats the problem of how to derive
macroscopic rate equations from master equations, the method is a systematic
expansion being called "Omega-expansion". Higher-order terms in the expansion
can even be related to Langevin-like fluctuations.

In particular, a master equation is always a linear equation for the probability
density P, whereas the resulting equation for the concentrations C can of course
be non-linear.

In van Kampen's book, spatial dependence and diffusion is not explicitely
treated, but this is straightforward.

Harold Schranz wrote:
> ..If you are indeed describing bimolecular reactions on a lattice
> (or whatever) then your matrices L_ij and R_rn,sm will have to have
> a dependence on concentration as well.
I'm afraid this is not correct. See the above reference. The crucial point is
that the probability density in the master equation refers to the many-particle
picture, i.e. you are looking at P(m, n1, n2, ...), where m is the index of the
lattice site (or spatial cell), and n1, n2,... are the occupation numbers of
atoms of type 1, 2,..

It is also good to note that there are master equations which do have a rate
equation, the (formal) reason being that the Omega-expansion does not converge.
I could try to dig out more references if you're interested.

Wolfgang Huber
IBM Almaden Research Center