# Techniques for the analysis of interacting particle systems with a finite number of particles, which do not resort to limiting arguments?

I am interested in pointers to (keywords/authors) recent research on the analysis of interacting particle systems with a finite number of particles which do not resort to limiting arguments converting the problem into the analaysis of an approximating PDE system. If it is not clear what I mean, see below.

Consider a system of finite number of particles, on a continuous (thus, not a lattice) background, like so:

Assume that there are $N_b$ black particles, and $N_r$ red particles.Say that the dynamics of the black and red particles are governed by the following ODEs ($\mathbf{b}_i$ is a vector representing the 2D position of the $i$th black particle, and $\mathbf{r}_i$ is similarly the position of the $i$th red particle):

$$\frac{\textrm{d}\mathbf{b}_i}{\textrm{d}t} = \sum_{j = 0}^{N_b-1} f_1(\mathbf{b}_i, \mathbf{b}_j) + \sum_{j = 0}^{N_r-1} g_1(\mathbf{b}_i, \mathbf{r}_j)$$

$$\frac{\textrm{d}\mathbf{r}_i}{\textrm{d}t} = \sum_{j = 0}^{N_r-1} f_2(\mathbf{r}_i, \mathbf{r}_j) + \sum_{j = 0}^{N_b-1} g_2(\mathbf{r}_i, \mathbf{b}_j)$$

where $f,g$ are vector functions which determine how the particles interact with each other.

It is of interest to predict some qualitative features of the system without resorting to simulation. For instance:

1) are there any interesting steady states (e.g. where the particles separate out based on colour, as shown in the cartoon above, but remain close to each other)?

2) more importantly, are there any interesting "non-steady" states, not necessarily analogous to periodic orbits (e.g. red particles form a clump, and the red clump chases black particles, keeping them for aggregating)

I am interested in recent research regarding the analysis of such problems, which do not resort to limiting arguments converting the ODE system into an approximate PDE system (basically, particle positions are replaced by approximating particle densities), and then analyzing the PDE system instead.

Converting the problem into a PDE system is helpful because instead of talking about the dynamics of each finite particle (which, looked at in isolation is likely not of much interest), one begins to talk about the dynamics of clumps of particles. However, interpretation of the PDE representation becomes problematic if there were not many particles to begin with, although there is the counter-argument of "so what? it's good enough". So, perhaps another way to ask my question is: what other ideas have people had to similarly reduce the complexity of the problem?

Do you know of any papers and authors of interest? Also, are there any topics in related fields (particularly, combinatorics or probability) which may be of interest?

• Have you looked at N-body problem literature ? The field is gravitational field and all particles are treated the same (no colors..but mass may vary). The general consensus seems to be that beyond special cases of 3 body problem.. N>3 these problems are very hard and not much can be said without taking the limit and using either a PDE (kinetic models or fluid models) or some mean field approach. Commented Jan 28, 2016 at 23:40
• For example, you might look at the Wikipedia article n-body choreography and references given there. Commented Jan 29, 2016 at 0:19
• @RobertIsrael Following your advice led me to: gminton.org/choreo.html --- try it out! Pretty neat. Commented Jan 29, 2016 at 1:21