A simple model for catalysis
Channel: Nils Berglund
Date: 30/01/2023

This simple model for catalysis can be summarized by the equation "2A + C → B + C". It means that when two A molecules (shown in red) encounter each other, they can form a B molecule (in blue), but only if a molecule of catalyst (C, in green) is present. After such a reaction has occurred, the catalyst molecule remains free to facilitate other reactions. 
In this simulation, there are initially 3025 molecules of type A, and 629 molecules of catalyst. The molecules interact via a Lennard-Jones potential. Reactions between two A molecules occur with a given probability per time step when they are closer than a fixed distance, and a C molecule is present. The total momentum is conserved during a reaction. At the beginning of the simulations, the catalyst is only present in the left half of the displayed region (which has periodic boundary conditions, meaning in particular that the left boundary touches the right boundary). Therefore, reactions only occur in the left half, until the catalyst has had a chance to diffuse to the right side.
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.
This simulation has two parts, showing the same evolution at two different speeds:
Time lapse: 0:00
Slow motion: 1:05
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: "75 & Lower" by DJ Williams@djwmusic

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)