# How is the number of successful collisions in a time-step distributed?

In a spatial homogeneous reaction system with two different molecules, how is the number of collisions which will lead to a reaction in a fixed time-step distributed?

My guess: Poisson distribution, as collisions are Bernoulli-experiments, usually have a low probability for success and the number of trials are huge.

• It's OK to use a Gaussian approximation so long as the time step is not too small. The Gaussian noise is a small perturbation on the dynamics of the reaction. But it can be important to consider in microtubes or cells of living things. Jan 18, 2018 at 22:07
• I want a very detailed description, so I need a discrete distribution. I also do not know how to calculate the variance, if I would use a Gaussian. A Poisson would solve that because I know the mean number of successful collisions. Is it reasonable to use a Poisson? Jan 19, 2018 at 16:29
• I think so. But in case the result does not agree with experiment or expectation, you still can allow the variance to be an adjustable parameter. The Poisson distribution gives you a guideline of how big the variance can be. Going beyond the Poisson theory would be difficult. Jan 19, 2018 at 16:35
• With "how big it can be" you do not mean an that Poisson variance is an upper limit but an estimation, right? Jan 21, 2018 at 14:23
• Right. Let's say the true std should be within an order of magnitude from the Poisson std ($\mathcal{O}(\sqrt{N})$). Jan 21, 2018 at 14:38