After the falling triangles in the video    • Falling triangles   , and the falling squares in the video    • Falling squares   , here come the falling pentagons. In case you wonder how many other regular polygons I am going to try, I plan to stop at hexagons, and then add some coagulation reactions, as well as other shapes.
One observation one can make here is that despite not being hexagons, the pentagons favor a hexagonal closed-packing lattice.
To compute the force and torque of pentagon j on pentagon i, the code computes the distance of each vertex of pentagon j to the faces of pentagon i. If this distance is smaller than a threshold, the force increases linearly with a large spring constant. In addition, radial forces between the vertices of the pentagons have been added, whenever a vertex of pentagon j is not on a perpendicular to a face of pentagon i. This is important, because otherwise pentagons can approach each other from the vertices, and when one vertex moves sideways, it is suddenly strongly accelerated, causing numerical instability. A weak Lennard-Jones interaction between pentagons has been added, as it seems to increase numerical stability.
The temperature is controlled by a thermostat with constant temperature. There is a constant gravitational force directed downward.
This simulation has two parts, showing the evolution with two different color gradients:
Orientation: 0:00
Kinetic energy: 1:27
In the first part, the particles' color depends on their orientation modulo 72 degrees. In the second part, it depends on their kinetic energy, averaged over a sliding time window.
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...

Render time: 49 minutes 6 seconds
Compression: crf 23
Color scheme: Part 1 - Twilight by Bastian Bechtold
https://github.com/bastibe/twilight
Part 2 - Turbo, by Anton Mikhailov
https://gist.github.com/mikhailov-wor...

Music: "City By Nght" by ELPHNT‪@ELPHNT‬

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/s10955-009-97...
http://www.maths.warwick.ac.uk/~theil...

Current version of the C code used to make these animations:
https://github.com/nilsberglund-orlea...
https://www.idpoisson.fr/berglund/sof...
Some outreach articles on mathematics:
https://images.math.cnrs.fr/auteurs/n...
(in French, some with a Spanish translation)

#molecular_dynamics #polygon