# Stochastic simulation Gillespie algorithm for areas instead of volumes?

I am trying to find resources on the Gillespie stochastic simulation algorithm for my system which happens on a surface. The original algorithm was developed for a reactor of volume $V$, but my system is a flat surface of area $A$. My questions are as follows:

1. Are there any publications that do stochastic simulation on surfaces?
2. Is extending the SSA from a volume to a surface trivial? For example, consider the following second order reaction: $$R_1+R_2 \rightarrow P_1$$ and suppose that the rate constant is $k$. The stochastic and deterministic constants ($k^{stoc}$ and $k^{det}$) for a reactor of volume $V$ are related as follows $$k^{det}=\frac{N_aV}{2}k^{stoch}$$ where $N_a$ is Avogadro's number. Can I simply replace $V$ with the surface area $A$ to get $$k^{det}_{area}=\frac{N_aA}{2}k^{stoch}_{area}$$

Please note that I am not looking to discretize space for diffusion. Just like the SSA does not discretize volumes, I should like to summarize the surface by one number (i.e, the surface area).

• I am not familiar with this method, does it take transport into account or only mean concentration? – Ian Jul 22 '18 at 19:04
• It does not take transport into account, just random interaction of molecules in a fixed volume. So the concentration serves as a way to measure the probabilities of interaction – Distopia Jul 22 '18 at 19:06
• If there is no transport then mathematically the situation is exactly identical and you can treat surface area as analogous to volume. The only problem is that the approximation of neglecting where the substances are located in space is dramatically less accurate in surface chemistry vs. bulk chemistry. – Ian Jul 22 '18 at 21:02
• Just to make sure we are on the same page, by transport you are referring to some kind of drift right? Because there is still some diffusive processes which makes the system well mixed. Also do you have any references I can cite? – Distopia Jul 22 '18 at 23:18
• I just mean anything that creates spatial inhomogeneity in the concentration. If the concentration is spatially homogeneous then there's no place for the geometry to enter. – Ian Jul 22 '18 at 23:37