# Examples of particle systems with higher-order collisions

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.
• are you interested in classical statistical mechanics only? or are you willing to consider quantum field theory models as well? Dec 22, 2019 at 0:45
• if you do not consider these (QM/QFT/algerbaic models etc ..) as "mean-field-type systems" would you count as examples the Feynmann diagrams or integrable systems via Yang-Baxter and R-matrix type methods? Imo, these seem to include higher order contributions (although not always in the explicilt sense ..). Dec 22, 2019 at 0:49
• @KonstantinosKanakoglou I'm not super familiar with QFT, and might struggle to comment on {Feynman, Yang-Baxter, etc.} without doing a little bit of extra reading, but I'm certainly willing to consider these other families of models.
– πr8
Dec 22, 2019 at 1:26
• You might be interested in the work of Ampatzoglou and Pavlovic and references therein on the derivation of a Boltzmann-type equation which takes incorporates three-particle collisions. Dec 22, 2019 at 17:12
• @MattRosenzweig Thank you! Yes, that's very much the sort of thing I'm looking for. The various references to colloids are also quite useful.
– πr8
Dec 22, 2019 at 18:58

I understand that the OP's original focus is classical statistical mechanics. However, i think that the question is of interest from a more general viewpoint including the dynamical systems/integrability and/or the quantum statistical mechanics point of view.
In this sense, i am not sure if this is the kind of answer you are hoping for, but if you are

interested in cases where the higher-order interactions are instead a key ingredient of the story

then you should look into the field of diffractive scatering models. That is, many-body models, non-integrable in the Bethe ansatz sense, not necessarily preserving the total number of particles and with the two-particle scattering matrices violating the Yang-Baxter equation:

In non-diffractive scattering, many-body collisions factorize in a sequence of two-body collisions; these are generally elastic collisions, the matrices satisfy the Yang-Baxter eq and we have complete integrability in the Bethe ansatz sense. These generalize -in a sense- the Boltzmann eq models you are mentioning. They preserve the number of particles and furthermore the set of outgoing momenta is the same with the the set of the incoming momenta. (some particles exchange velocities during these collisions). Binary collisions dominate the behaviour.
You can have a look at factorized scattering: sect 3.3 at Quantum integrable systems or sect. 4.3 at Elements of Classical and Quantum integrable systems

In diffractive scattering, all the above are violated. Only momentum and total energy conservation survive. Non-factorizable higher order (that is many particle) interactions dominate and in some cases they seem to be necessary in order to comply with experimental evidence of cooling down. If we consider a model with 3-particle collisions, this means that instead of elastic collisions we can have "chemical reactions" like

• Particle asscociation: $$A+B+C\rightarrow AB+C$$,
• rearrangement: $$AB+C\rightarrow AC+B$$,
• dissociation: $$AB+C\rightarrow A+B+C$$
• elastic collisions: $$AB+C\rightarrow AB+C$$

See for example Diffractive scattering of three particles in one dimension: a simple result for weak violations of the Yang--Baxter equation, where a proposal for the formulation of a kinetic theory with three-particle collisions is suggested. (see especially the conclusions section of the paper).
(I think this should be studied in conjuction with A rigorous derivation of a ternary Boltzmann equation for a classical system of particles proposed in the comments by user Matt Rosenzweig).

• Thank you, this is great! I will take a look at the references you provide. I appreciate the comparison with chemical reactions, as this is an area with which I have some prior familiarity.
– πr8
Dec 23, 2019 at 13:31