## Chaos

Program `Chaos` simulates 11 particles initially moving with the same velocity such that the net force on each particle is zero.
The particles interact through the Lennard-Jones potential. Perturb the system slightly and see what happens. The velocities
can also be reversed to see to what extent the system is reversible.

The goal of this simulation is to explore how a small perturbation changes a special initial state to one that we would expect to see at thermal equilibrium. The idea of chaos or sensitivity to initial conditions helps us understand irreversibility and why a system of particles reaches equilibrium.

**Problem: Irreversibility**

Program `Chaos`
simulates a system of
$N=11$ particles with a special initial condition.

- Perturb the velocity of particle 6 and discuss the qualitative results about chaos and irreversibility.
- Stop the simulation at a time $t$ after the perturbation and reverse all the velocities. Confirm that if $t$ is sufficiently short, the particles will return approximately to their initial state. What is the maximum value of $t$ that allows the particles to return to their initial positions if $t$ is replaced by $-t$ (all velocities reversed)?

## Resource

Problem 1.8 in*Statistical and Thermal Physics: With Computer Applications*, 2nd ed., Harvey Gould and Jan Tobochnik, Princeton University Press (2021).