Like the simulation [ Ссылка ] this video shows the motion of particles coupled to a thermostat, and interacting with an anisotropic Lennard-Jones type potential with pentagonal symmetry. Here the lattice is visualized by drawing line segments between particles that are closer than a given distance. Attentive viewers will spot the pentagram appearing at some point.
For this potential, two isolated particles have 10 stable configurations, located on the vertices of two regular pentagons of different radius, rotated by 36 degrees with respect to each other. The exact form of the potential in polar coordinates is
V(r,phi) = (req/r)^12 - A(phi)*(req/r)^6,
where req is 5 times the radius of the particles, and
A(phi) = 1 + 2*cos(pi/10) + [1-2*cos(pi/10)]*cos(5*phi).
While there isn't an actual quasicrystal formed, as I was initially hoping, there are still several interesting features. The pentagonal symmetry causes a lot of instability and "frustration" in the arrangement of particles, making them rearrange quite a lot, and producing interesting oscillations, including shear waves.
The color of the particles represents the number of neighbors it is connected to by a link. Blue particles have more neighbors than cyan, yellow, orange and red ones. Temperature is controlled by a thermostat, given here by 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 temperature shown on the top right is in fact the temperature multiplied by Boltzmann's constant. 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 [ Ссылка ]

Render time: 34 minutes
Color scheme: Turbo, by Anton Mikhailov
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Music: Summer Sidewalk by Audionautix is licensed under a Creative Commons Attribution 4.0 licence. [ Ссылка ]
Artist: [ Ссылка ]

Reference: Leimkuhler, B., Noorizadeh, E. & Theil, F. A Gentle Stochastic Thermostat for Molecular Dynamics. J Stat Phys 135, 261–277 (2009). [ Ссылка ]
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Current version of the C code used to make these animations:
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Some outreach articles on mathematics:
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(in French, some with a Spanish translation)

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