Lifting the lid: heating a Lennard-Jones fluid in a container with movable cover
Channel: Nils Berglund
Date: 2022-03-14

Similarly to the video https://youtu.be/PSKYRQ5zdP0 this simulation shows the behavior of 661 particles interacting via a Lennard-Jones potential, subjected to gravity, and coupled to a thermostat with gradually rising temperature. The difference is that here, the top of the container, instead of being fixed, is a mobile lid modeled as a rigid object having 1000 times the mass of the particles. The pressure indicated at the top left is the pressure experienced by the lid, averaged over a time window to decrease fluctuations. The pressure appears to remain roughly constant, so that if the particles were forming an ideal gas (which is not the case except at low density), the volume (proportional to the height of the lid) should vary like the temperature.
There are two parts in this simulation, showing the same evolution with two different visualizations:
Kinetic energy: 0:00
Number of neighbors: 1:31
In the first part, colors depend on the particles' kinetic energy, red particles are moving faster than blue ones. In the second part, the colors depend on the number of other particles within a given distance, to which they are connected by bonds. Particles however interact also when no bonds are shown between them. 
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: 12 minutes
Color scheme: Turbo, by Anton Mikhailov
https://gist.github.com/mikhailov-work/6a308c20e494d9e0ccc29036b28faa7a

Music: "Running Out" by Patrick Patrikios@patrikiospatrick

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)