In kinetic theory, one often comes across interacting particle systems with a collisional flavour. I'll currently prefer to think about them as systems of ODEs (or SDEs, Jump Processes, $\ldots$), though of course it's of considerable interest to consider them at the PDE level as well.

The two standard examples of which I'm aware are:

**Boltzmann**: At the particle level, one has a system of $N$ hard spheres (each with position and velocity), following the dynamics $\dot{x_i} = v_i, \dot{v_i} = 0$. When two of the spheres collide, the velocities of both spheres jump according to the law of specular reflection.**Kac**: At the particle level, one has a system of $N$ particles (each with*only*a velocity). Any two particles collide randomly at some rate which depends only on the velocities of the two particles, and at such an event, their velocities are changed in a correlated way, e.g. $$(v^i, v^j ) \mapsto ( v^i \cos \theta + v^j \sin \theta, -v^i \sin \theta + v^j \cos \theta )$$ with $\theta \sim \mathcal{U} [ 0, 2 \pi ] $.

In each of these examples, the system's collisions each involve two particles at a time.

What I'm curious about is whether there are interesting examples of collisional systems in a similar vein to the previous two, but in which the collisions involve **more than 2 particles at once**.

Strictly speaking, I know that Boltzmann-type systems do, in principle, accommodate $k$-particle collisions. However, these are in some sense lower-order terms which wash out in the usual $N \to \infty$ limit. I am more interested in cases where the higher-order interactions are instead a key ingredient of the story.

Some other remarks:

For the purposes of this question, I'm interested in collisional models, rather than full-blown mean-field-type systems, in which each particle is actively interacting with all other particles. e.g. McKean-Vlasov diffusions would not be a satisfactory answer. I'm imagining a model with $k$-particle collisions, with $k$ fixed as $N$ grows.

I'm aware that it's possible to construct systems of this form artificially (e.g. one could easily imagine a Kac-type model in which the velocities interacted $k$-at-a-time with some $k > 2$). I highlight that I'm interested in examples of such systems which arise naturally, ideally through scientific modelling, or possibly in algorithmic applications.

- I would be okay with an example which is somewhat artificial, provided it makes some other interesting point, e.g. if the system is physically meaningless, but illustrates some mathematical / analytical point, then this would be satisfying to me.