Subject: Diffusion Equation from Kinetic Lattice Monte Carlo

From: Andrew Horsfield <[email-masked]>
Date: Wed, 15 Sep 1999 15:18:41 +0100 (BST)

Hi,

I have a question about kinetic Monte Carlo simulations on a lattice, and
how they relate to traditional diffusion and reaction equations.

I understand Monte Carlo in the following way:

1) You define a set of allowed configurations which you can index

2) You evaluate by some means a set of rates for transitions between these
   configurations. Let the rate for a transition from configuration i to
   configuration j be W_{ji} (note the order of the indices).

3) You start from some initial configuration, and make a transition to a
   new configuration with a probability proportional to the rate. This
   process is repeated for a number of transitions.

I want to derive continuum equations from this algorithm. To do this I
begin by defining the probability that my system is in configuration i to
be P_i. I then ask how P_i varies with time. The answer (I think) is:

d P_i
----- = \sum_j L_{ij} P_j
 dt

where

L_{ij} = W_{ij}             i \= j
       = -\sum_k W_{ki}     i  = j

This all looks fine until one transforms this equation into a
corresponding equation for the rate of change of concentration (C):

d C(r)_n
-------- = \sum_{sm} R_{rn,sm} C(s)_m
   dt

where r and s are indices for the type of atom (C, O, H or whatever), n
and m are indices for spatial position (lattice site), and R_{rn,sm} is a
concentration independent rate matrix that can be derived from L_{ij}. The
feature that worries me about this is that it is *linear* in C (as it has
to be, given the starting point).

I believe that this cannot be correct if there are chemical reactions
taking place, since then we would expect the rate to depend on the
*product* of two concentrations (at least).

Can anyone tell me where I have gone wrong? Is there a standard reference
for this kind of approximation?

Cheers,

Andrew

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