Hello Stack Exchange Mathematics,
I am curious if there is an analytical or efficient numerical solution for the collision of hard spheres in a rectangular unit cell with periodic boundary conditions.
Consider a system with two hard spheres. The first sphere is located at $\vec r_i$ with a velocity $\vec v_i$ and a radius $\sigma_i$. The second sphere is located at $\vec r_j$ with a velocity $\vec v_j$ and a radius $\sigma_j$. Note $\vec v_{ij} = v_i - v_j$ and $\vec r_{ij} = r_i - r_j$.
The hard sphere collision condition is well known. If $\vec r_{ij} \cdot \vec v_{ij} < 0$ and $b_{ij}^{2} - v_{ij}^{2}(r_{ij}^2-\sigma^2) > 0$, the spheres will collide. The time of collision is: $$\frac{-b_{ij}-\sqrt{b_{ij}^{2}-v_{ij}^{2}(r_{ij}^{2}-\sigma^{2})}}{v_{ij}^{2}}$$
How is this condition modified if the spheres are enclosed in a rectangular unit cell $L_x \times L_y \times L_z$ with periodic boundary conditions in the $x$, $y$, and $z$ directions? What is the most efficient way to determine if and when the spheres will collide?
Best, Eric
Take a look at https://cs.stackexchange.com/q/49070/755 for a 2D version of this kind of problem, with a single point (sphere of radius 0), in an integer grid. There is an efficient solution to that one involving modular arithmetic. You might be able to adapt the techniques there to your problem. I don't imagine the generalization to 3D will be difficult to handle, but I haven't thought about the consequences of continuity (non-integral values) here.