### Molecules' motion

Ideal gas animation (http://commons.wikimedia.org/wiki/File:Gaz_molecules.gif) |

$\frac{N_{i}}{N}=\frac{g_{i}exp(-E_{i}/kT)}{\sum_{j}g_{j}exp(-E_{j}/kT)}$

By using this formula we can think of all the different ways the molecules could be in (microstate) and we can work out the probability, which is in a particular arrangement. We can work out the distribution of the momentum (vector). $E=p^{2}/2m$. By putting this into the Maxwell distribution we can calculate the distribution of the velocity (vector) in a single direction.
$f_{v}(v_{i})=\sqrt{\frac{m}{2\pi kT}}exp(\frac{-mv_{i}^{2}}{2kT})$

Working out the distribution of speed which is now in three dimensions is easiest done using polar coordinates (like that of a world map) to switch from x,y,z to $v,\theta,\phi$ This requires a coordinate transformation.

$dv_{x}d_{y}d_{z}=v^{2}sin(\phi)dv d\theta d\phi$

This yields the probability distribution in three dimensions.

$f(v)=\sqrt{(\frac{m}{2 \pi kT})^{3}}4 \pi v^{2}exp(-\frac{mv^{2}}{2kT})$

The squared speed term says that the probability of having zero speed is zero and increases. The long tail comes from the exponential term. The assumptions made in modeling molecules like this are that:

- The molecule has no volume.
- All collisions are elastic (energy is conserved).
- The molecules move randomly and are distinguishable from each other.

### Moshing motion

Particle tracking was applied to people in mosh and circle pits in heavy metal concerts. The most remarkable finding was that the velocities of the people in the mosh pit matched a Maxwell-Boltzmann distribution. The velocity-velocity correlation was then found to decay away with a decay length of 0.39+-0.03 m (about shoulder width apart) which suggests that the people move distinctly from each other.This motion is then modelled using flocking based forces, repulsion forces, propulsion forces and some noise. This is found to model the mosh pit accurately.

They then compare three different time scales - flocking time scales, noise time and collision time-related to the mean free path. Random motion from collisions and noise over long time scales reproduce statistics of classical gases which is how the non-equilibrium system can show equilibrium characteristics. When flocking dominates then you can get vortex states which result in angular momentum. These are called circle pits. Interestingly, the majority of these observed rotate counter-clockwise (95%). This was not predicted by the model but may be due to handedness/footedness. This presents an interesting analogue to a classical gas with all the particles distinguishable not correlated with enough chaos to show Maxwell-Boltzmann distribution in the macroscopic world. I wonder if Boltzmann would have been into heavy metal moshing. Maybe it’s just the beard.

They also have this awesome javascript mosh pit simulator.

Silverberg, J. L., Bierbaum, M., Sethna, J. P., & Cohen, I. (2013). Collective Motion of Humans in Mosh and Circle Pits at Heavy Metal Concerts.

*Physical Review Letters*,*110*(22), 228701. doi:10.1103/PhysRevLett.110.228701