Particle number vs time in collisional N-body problem

Question

I have created a crude N-body simulator which allows N bodies of equal masse​s to interact gravitationally in 2 dimensions.

Each body is modeled as a circle with a radius as a function of its mass, in such a way that all bodies have the same density.

When two bodies collide, i.e their radii overlap, they stick together inelastically (but linear momentum is conserved).

I have initialized the simulation with N=300 particles, and initial positions and velocties randomized (all positions bounded to a certain rectangular window, all velocties of the same modulus).

As the simulation progresses, particles move about, collide and form larger particles, and after some time the system appears to reach a stable state in which the number of particles is very few, usually between 2-5 (the most common case is a planet-sun system)

I have plotted the number of bodies vs. time, and the graph shows something like an exponential decrease (maybe subexponential, I haven't done any regression analysis yet) until the system reaches a steady state in which particle number is constant:

Is there a means of predicting an exponential decrease on theoretical grounds, for example by assuming that for large numbers of particles the distribution of velocities is approximately random, and then using probability?

Does this question have anything to do with scattering theory?

Answer 1

Collision theory is probably more what you're looking for. The rate constant for a bimolecular gas reaction (instantaneously analogous to your situation) is a falling exponential, thus similarly giving a falling exponential for the number of reactant particles.

Of course, the key here is instantaneously analogous, since in your situation, the particle (body) size increases with time. This should make your resultant graph somewhat sub-exponential. Have you verified this?

(For the record, scattering theory is more to do with waves, or wave-matter interaction, AFAIK.)

N.B. Ignoring the gravitational attraction would seem to be reasonable, at least while the density and rate of collisions are high, just like ignoring the van der Waals force is reasonable for the gaseous model.

Answer 2

For a first approximation, let's define $n(t)$ as the number of bodies at time $t$. When there are a lot of bodies, the rate of collisions will be proportional to the average distance between the bodies (also called the mean free path), which does come out of scattering theory:

$$\frac{dn}{dt} = -kn^{1/m},$$

where $m$ is the number of dimensions of space--two in your case--and $k$ is some positive real number (this number will involve things like the volume of space the bodies occupy, the size of the bodies, and the average speed of the bodies). We can solve this to find an expression for $n(t)$:

$$\frac{dn}{n^{1/m}} = -k\,dt$$ $$\int_{n_0}^{n_t}\frac{dn}{n^{1/m}} = \int_0^t -k\,dt$$ $$\left.\frac{1}{\frac{1}{m}+1}n^{\frac{1}{m}+1}\right|_{n_0}^{n_t} = -kt$$ $$\left.\frac{m}{m+1}n^{\frac{m+1}{m}}\right|_{n_0}^{n_t} = -kt$$ $$\frac{m}{m+1}n(t)^{\frac{m+1}{m}} - \frac{m}{m+1}n_0^{\frac{m+1}{m}} = -kt$$ $$n(t) = \left(n_0^{\frac{m+1}{m}} - \frac{m+1}{m}kt\right)^{\frac{m}{m+1}}$$

In the final equation, $n_0$ is the initial number of bodies.

As an example, here's a plot of the number of particles with with $m = 2$, $k = 0.01$, and $n_0 = 200$:

You can see a plot of this function and experiment with the parameters here: https://www.desmos.com/calculator/fdz39m6wog

Now, the above calculation did not include gravity, nor does it hold up with a small number of particles. With gravity, you would expect a larger number of collisions in the beginning of the simulation (due to gravity) and much fewer as time runs on (due to collisions becoming rare due to stable orbits). This will make the graph more curved and exponential-looking than what's shown.