A centrifuge
Channel: Nils Berglund
Date: 2022-05-19

A while ago, a commenter asked me if I could illustrate the principle of a centrifuge. So here is a first video doing that. It is of course a gross simplification of how a real centrifuge works, but it gives the general idea. 
The simulation features 436 particles, which are of two different types: 70% of the particles are shown in red in the first part of the video, the other 30% are shown in blue. The red particles are smaller, but 5 times as heavy as the blue ones. The idea is that red particles could represent a solid substance that is diluted in water, and that one wants to separate from the water by using the centrifuge.
The centrifuge consists here of 6 bins. The particles interact via a Lennard-Jones potential and are subject to gravity and to a thermostat. The speed of rotation of the centrifuge, indicated in radians per second, is first increased, and then decreased again. Due to the forces from the walls of the bins, the particles are pressed against each other, which results in the heavier particles partially separating from the lighter ones.
One thing which is not quite realistic here is that the particles do not really move outwards, as one would expect because of the centrifugal force (which is, of course, an inertial force that appears with respect to a rotating frame of reference). This may be due to the thermostat, which tries to keep the particles at a relatively low speed, pressing them strongly against the rotating walls and making it harder for them to move. A forthcoming simulation will show what happens without the thermostat, and without gravity.
The simulation has two parts, showing the same evolution with two different color codes: 
Particle type: 0:00
Direction of velocity: 2:46
In the first part, the particles' color indicates their type.In the second part, colors indicate the direction of the particles' velocity: white particles move to the right, blue ones upward, purple ones to the left, and red ones move downward. In addition, the luminosity depends on the particle's kinetic energy, so that very slow particles appear black. 
The temperature is controlled by a thermostat, implemented here with the "Nosé-Hoover-Langevin" algorithm introduced by Ben Leimkuhler,  Emad Noorizadeh and Florian Theil, see reference below. The idea of the algorithm is to couple the momenta of the system to a single random process, which fluctuates around a temperature-dependent mean value. Lower temperatures lead to lower mean values. The temperature shown on the top right is in fact the temperature multiplied by Boltzmann's constant. To save on computation time, particles are placed into a "hash grid", each cell of which contains between 3 and 10 particles. Then only the influence of other particles in the same or neighboring cells is taken into account for each particle. 
The Lennard-Jones potential is strongly repulsive at short distance, and mildly attracting at long distance. It is widely used as a simple yet realistic model for the motion of electrically neutral molecules. The force results from the repulsion between electrons due to Pauli's exclusion principle, while the attractive part is a more subtle effect appearing in a multipole expansion. For more details, see https://en.wikipedia.org/wiki/Lennard-Jones_potential

Render time: 51 minutes 
Color scheme: Part 1 - Turbo, by Anton Mikhailov
https://gist.github.com/mikhailov-work/6a308c20e494d9e0ccc29036b28faa7a
Part 2 - Twilight by Bastian Bechtold
https://github.com/bastibe/twilight

Music: "Starcrasher" by TrackTribe@TrackTribe

Reference: Leimkuhler, B., Noorizadeh, E. & Theil, F. A Gentle Stochastic Thermostat for Molecular Dynamics. J Stat Phys 135, 261–277 (2009). https://doi.org/10.1007/s10955-009-9734-0

http://www.maths.warwick.ac.uk/~theil/HL12-3-2009.pdf

Current version of the C code used to make these animations: 
https://github.com/nilsberglund-orleans/YouTube-simulations
https://www.idpoisson.fr/berglund/software.html
Some outreach articles on mathematics:
https://images.math.cnrs.fr/_Berglund-Nils-1343_.html
(in French, some with a Spanish translation)