Nils Berglund | Growing quasicrystals from pentagons: Orientation and number of neighbors @NilsBerglund | Uploaded July 2024 | Updated October 2024, 1 second ago.
This video shows the same simulation as the video youtu.be/5133mYMFbeM , but with a different color code. Instead of showing clusters and their kinetic energy, the colors depend on the pentagons' orientation and their number of neighbors.
I have been preparing this simulation for quite some time. It is the first success in growing something resembling a quasicrystal, by dynamically aggregating rigid pentagons. The result is not yet perfect, because in a few cases pentagons do overlap, but this may still be improved.
Whenever two pentagons merge, they form a rigid cluster, which is henceforth considered as a single solid. This polygonal solid evolves according to the laws of Newtonian mechanics, as a result of the forces exerted on it from other polygonal solids. To help pentagons coagulate, a torque has been added between pentagons that are close that helps them match their orientation.
Particularly when the clusters become large, the merging sometimes results in an imperfect alignment of the parts. Therefore, a "repair" function has been added, that examines each cluster at regular time intervals. It computes the ideal relative placement of adjacent pentagons, and pushes the pentagons in that direction in little steps. The cluster size has been limited to 200, because larger clusters turn out to result in numerical instabilities.
To compute the force and torque of polygon j on polygon i, the code computes the distance of each vertex of polygon j to the faces of polygon 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 polygons have been added, whenever a vertex of polygon j is not on a perpendicular to a face of polygon i. This is important, because otherwise polygons 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 polygons 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
Number of neighbors: 1:28
In the first part, the hexagons' color depends on their orientation modulo 72 degrees. In the second part, it depends on the number of other pentagons they are attached to.
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 en.wikipedia.org/wiki/Lennard-Jones_potential
Render time: 43 minutes 3 seconds
Compression: crf 23
Color scheme: Turbo, by Anton Mikhailov
gist.github.com/mikhailov-work/6a308c20e494d9e0ccc29036b28faa7a
Music: "Looping Ascent" by Jeremy Cummins@JCStriking
Reference: Leimkuhler, B., Noorizadeh, E. & Theil, F. A Gentle Stochastic Thermostat for Molecular Dynamics. J Stat Phys 135, 261–277 (2009). doi.org/10.1007/s10955-009-9734-0
maths.warwick.ac.uk/~theil/HL12-3-2009.pdf
Current version of the C code used to make these animations:
github.com/nilsberglund-orleans/YouTube-simulations
https://www.idpoisson.fr/berglund/software.html
Some outreach articles on mathematics:
https://images.math.cnrs.fr/auteurs/nils-berglund/
(in French, some with a Spanish translation)
#molecular_dynamics #polygon #quasicrystal
This video shows the same simulation as the video youtu.be/5133mYMFbeM , but with a different color code. Instead of showing clusters and their kinetic energy, the colors depend on the pentagons' orientation and their number of neighbors.
I have been preparing this simulation for quite some time. It is the first success in growing something resembling a quasicrystal, by dynamically aggregating rigid pentagons. The result is not yet perfect, because in a few cases pentagons do overlap, but this may still be improved.
Whenever two pentagons merge, they form a rigid cluster, which is henceforth considered as a single solid. This polygonal solid evolves according to the laws of Newtonian mechanics, as a result of the forces exerted on it from other polygonal solids. To help pentagons coagulate, a torque has been added between pentagons that are close that helps them match their orientation.
Particularly when the clusters become large, the merging sometimes results in an imperfect alignment of the parts. Therefore, a "repair" function has been added, that examines each cluster at regular time intervals. It computes the ideal relative placement of adjacent pentagons, and pushes the pentagons in that direction in little steps. The cluster size has been limited to 200, because larger clusters turn out to result in numerical instabilities.
To compute the force and torque of polygon j on polygon i, the code computes the distance of each vertex of polygon j to the faces of polygon 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 polygons have been added, whenever a vertex of polygon j is not on a perpendicular to a face of polygon i. This is important, because otherwise polygons 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 polygons 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
Number of neighbors: 1:28
In the first part, the hexagons' color depends on their orientation modulo 72 degrees. In the second part, it depends on the number of other pentagons they are attached to.
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 en.wikipedia.org/wiki/Lennard-Jones_potential
Render time: 43 minutes 3 seconds
Compression: crf 23
Color scheme: Turbo, by Anton Mikhailov
gist.github.com/mikhailov-work/6a308c20e494d9e0ccc29036b28faa7a
Music: "Looping Ascent" by Jeremy Cummins@JCStriking
Reference: Leimkuhler, B., Noorizadeh, E. & Theil, F. A Gentle Stochastic Thermostat for Molecular Dynamics. J Stat Phys 135, 261–277 (2009). doi.org/10.1007/s10955-009-9734-0
maths.warwick.ac.uk/~theil/HL12-3-2009.pdf
Current version of the C code used to make these animations:
github.com/nilsberglund-orleans/YouTube-simulations
https://www.idpoisson.fr/berglund/software.html
Some outreach articles on mathematics:
https://images.math.cnrs.fr/auteurs/nils-berglund/
(in French, some with a Spanish translation)
#molecular_dynamics #polygon #quasicrystal
