Simulating a closed particle system

Question

Suppose we are given a two-dimensional plane. Also, we are given $n$ particles with "charges" $Q_1, Q_2, \dots, Q_n > 0$. Since all of them have positive charges, they repel each other. Each particle has a mass: $m_1, m_2, \dots, m_n$. Also, when a particle collides with a border, it bounces off in the same manner as a snooker ball would bounce.

Now, the plane and the particles comprise a closed system, and its total energy is

$$E_{\text{total}} = \sum_{i = 1}^n \frac{1}{2}m_i v_i^2 + \sum_{i = 1}^n \sum_{j = i + 1}^n \gamma \frac{Q_i Q_j}{r_{i,j}},$$ where $r_{i,j}$ is the distance between $Q_i$ and $Q_j$.

I am working on a computer simulation of such a system, yet I have a problem: the total energy grows, so we cannot speak about a closed system. My question is:

Does there exist a unique $\gamma$ such that the total energy remains constant in such a simulation?

Answer 1

You appear to want to use electrostatic interactions in a dynamic simulation. That's problematic. You don't get a truly correct description without something like delayed potentials (check a electrodynamics textbook). In reality, the system would of course also emit electromagnetic radiation, as the charges accelerate, thus it's actually dissipative.

With regard your current setup, without radiation this system would be conservative, so its energy cannot grow with time. If it's not a typo, you used masses instead of charges. The electrostatic potential energy should read:

$$ \sum_{\langle i,j\rangle} \gamma \frac{Q_i Q_j}{r_{i,j}}.$$

You might also want to recheck those summation limits.

As for $\gamma$, you only have to be careful to use the same in the calculation of both the force and energy. Other than that, it's only a scaling factor that measures whether the system is more strongly driven by inertia or electrical charge.