The chemical dissociation reaction B → A + A
Channel: Nils Berglund
Date: 03/02/2023

The chemical reactions shown so far on this channel were examples of synthesis, where different molecules can merge to become a new type of molecule, but the new molecules do never dissociate into their constituents. This simulation shows by contrast the dissociation reaction "B → A + A", where B molecules spontaneously disintegrate into two A molecules. The logical next step will be to simulate reversible reactions, in which molecules can both merge and dissociate.
In this simulation, there are initially 3079 molecules of type B, shown in blue. The molecules interact via a Lennard-Jones potential. At each time step, there is a small probability that a B molecule separates into two A molecules, each having half the mass of a B molecule. The total momentum is conserved during a reaction. To avoid unwanted interference with the thermostat, the code checks that no other molecule is too close when attempting to split a B molecule.
Reactions are highlighted by a small disc that quickly fades. To guide the eye, whenever three molecules of the same type are close enough, the triangular region between them is filled with the same color. In addition, a line is drawn between molecules when they are closer than a given distance (the reaction distance is smaller, however). The graph at the top right shows the number of molecules of each type as a function of time.
The theory of chemical kinetics predicts that the number of B molecules should decrease like an exponential, while the number of A molecules increases accordingly. This is however only true for very large numbers of molecules, as a consequence of the law of large numbers. 
This simulation has two parts, showing the same evolution at two different speeds:
Time lapse: 0:00
Slow motion: 1:01
In the first part, time has been accelerated by a factor 3. 
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: 1 hour 2 minutes
Color scheme: Turbo, by Anton Mikhailov
https://gist.github.com/mikhailov-work/6a308c20e494d9e0ccc29036b28faa7a

Music: "Golden" by Vibe Tracks

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)