I am a chemical engineer working on an algorithm for the simulation of catalytic processes in a particle based simulation framework and unfortunately, I am struggling with a math problem and I'm not sure if my solution is correct.
I have a 2D surface ($n_{dimensions} = 2$) with periodic boundaries and a surface Area $A$. On this surface, I have random walking (brownian) particles of species $a$ and $b$ with diameter $d_a$ and $d_b$. The movement of those particles on the surface is defined by diffusion coefficients $Diff_a$ and $Diff_b$. The simulation is using a timestep $t$. All units are SI units. In the end, I want to calculate a probability $P_{ab}$ for two random walkers $W_a(1)$ and $W_b(1)$ to meet on the surface (without respect to their actual positions on the surface). Furthermore, I would like to compute an expectation value $E_{ab}$ for the number of collisions between $N_a$ particles of species $a$ and $N_b$ particles of species $b$. Collisions of particles with the same species are of no interest. Also, I only want to consider binary collisions.
My approach so far:
- I calculate the Root Mean Square Displacement on a 2D surface for both particles $RMS_x = 2 \times n_{dimensions} \times D_x \times t$
- Then, I calculate the Mean RMS: $\langle \Delta x^2 \rangle = \frac{RMS_a + RMS_b}{2}$
- Next, I calculate the Probability $P_{ab}$ as follows: $P_{ab} = 1- \exp \left( - \frac{\pi (d_a + d_b)}{4 \times A} \times \langle \Delta x^2 \rangle \right)$
- Finally, i calculate the expectation value $E_{ab} = N_a \times N_b \times P_{ab}$
I should add/admit here, that I used AI to get to the equations above, as my math skills are not good enough, to derive them myself. Hence I also fear, that the equations above may be absolutely wrong. Any help and feedback on this would be greatly appreciated.