Aggregation of neutral particles
Channel: Nils Berglund
Date: 19/03/2024

Several recent simulations, such as https://youtu.be/rWUYoEb8wPE , showed charged particles forming clusters. This simulation involved neutral particles instead, that only interact with a Lennard-Jones potential. It turns out that one can obtain clusters of particles as well, but the parameters have to be tuned more carefully. I also had to decouple some particles from the thermostat: particles that have two "partners" or more in a cluster no longer interact with the thermostat, but are subject to a weak viscous friction instead. Otherwise, the system tended to slow down after a while, probably because the particles inside clusters soak up a lot of energy in the form of tiny vibrations. This can be seen in the second part of the simulation, showing the particles' kinetic energy.
This simulation features particles that interact in two different ways. Particles that belong to different "molecules" interact via a Lennard-Jones potential. In addition, particles that come within a certain distance can "react", thereby becoming members of a same "molecule". These reactions are highlighted by a quickly fading white disc. Particles within a molecule no longer interact via an LJ potential, but instead via a strong harmonic force, that tends to maintain their distance constant.  
The video has two parts, showing the same simulation with two different color schemes:
Clusters: 0:00
Kinetic energy: 1:41
In the first part, particles in the same cluster are given the same color, which is updated when several clusters merge. In the second part, the particles' color hue depends on their kinetic energy.
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 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 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: 23 minutes 16 seconds
Compression: crf 23
Color scheme: Part 1 - HSL/Jet
Part 2 - Turbo, by Anton Mikhailov
https://gist.github.com/mikhailov-work/6a308c20e494d9e0ccc29036b28faa7a

Music: Freddy's Menagerie by Kevin MacLeod is licensed under a Creative Commons Attribution 4.0 licence. https://creativecommons.org/licenses/by/4.0/
Source: http://incompetech.com/music/royalty-free/index.html?isrc=USUAN1100579
Artist: http://incompetech.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)

#molecular_dynamics #coagulation