I'm studying a diffusion limited aggregation phenomenon. The $N$ particles diffuse in a box and when there is a contact they stick with a probability $p$, and let's say to simplify $p=1$. Meaning that there are reactions : $A+A\rightarrow AA,AA+A\rightarrow A^3 $ and so on.

I assume that the discrete mathematical equation that describes that is the master equation : $ \frac{dN_n}{dt}=-k_{on}(N_n,{\{N_i\}_{i\in \mathbb{N} }})+...$ with $N_n$ the numbre of agregated with size $n$. The rates depend on the diffusion coefficients of the aggregates meaning on there size, and everything is pretty complicated. And I'd like to have some insight with no heavy numerics.

So I thought a continuous approach would simplify my problem. I wondered if a potential like a Cahn-Hilliard : $F=\int -\rho^2+\alpha\rho^4+\epsilon (\nabla \rho)^2 d\mathbf{r}$ could be used... But I'm not so sure.

Would you know any method that could simplify the calculations ? Either in a discrete or continuous way ? Or would you know any books dealing with that question or at least that kind of issue ?


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