A Rayleigh-Taylor instability (short version)
Channel: Nils Berglund
Date: 24/08/2022

A Rayleigh-Taylor instability occurs when there are two layers of fluid, subject to gravity, and the upper layer has a larger density than the lower one. Here the effect is reproduced with 5663 particles interacting with a Lennard-Jones potential. Half of the particles are blue, and their mass is 10 times as large as the red particles. This is a #short version of the longer simulation https://youtu.be/n5MAsOkAENU
The initial state is a regular arrangement, with the heavy blue particles on top of the lighter red ones. At the beginning, the particles behave more like a crystal than like a fluid, and have difficulty changing position. During the simulation, however, the temperature is increased. This results in the particles becoming more fluid. The boundary between the blue and red particles becomes unstable, and after a while, most heavy particles manage to move below the light ones. Interestingly, large clusters of particles of the same type remain throughout the process. 
The entropies shown at the right are computed as follows. For each type of particle, let p be the proportion of particles in the upper half. Then the entropy is defined as -p*log(p) - (1-p)*log(1-p). This quantity is minimal, with value 0, when p is 0 or 1, and maximal when p = 1/2, where it has value log(2). The figures indicate the entropy divided by log(2), which has 1 as theoretical maximum. 
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. 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: 31 minutes 54 seconds
Color scheme: Turbo, by Anton Mikhailov
https://gist.github.com/mikhailov-work/6a308c20e494d9e0ccc29036b28faa7a

Music: Jenny's Theme by Audionautix is licensed under a Creative Commons Attribution 4.0 licence. https://creativecommons.org/licenses/by/4.0/
Artist: http://audionautix.com/

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)