And the same approach of just splitting in half in every dimension at each tree level can extend to arbitrary dimension, but usually something else like a kdtree is used instead.

The general algorithm used here (of computing attraction and repulsion forces between pairs of particles) is very similar to that used in simulations of many interesting phenomena in physics. Start with Smoothed Particle Hydrodynamics (https://en.wikipedia.org/wiki/Smoothed-particle_hydrodynamic...) and then check out Lagrangian Vortex Particle Methods and other N-Body problems (https://en.wikipedia.org/wiki/N-body_problem).

And the algorithms to solve these quickly is another deep area of research.

Quadtrees and octrees are themselves quite deep research areas. If the acceleration data structures interest you, I highly recommend Hanan Samet's book "Foundations of Multidimensional and Metric Data Structures". It's from 2006, but is basically the bible for the field.