Nils Berglund | Growth of a mono-quasicrystal from pentagons @NilsBerglund | Uploaded July 2024 | Updated October 2024, 1 minute ago.
This simulation of merging pentagons uses a somewhat different rule than previous simulations on this channel. At the beginning, one pentagon is singled out, and other pentagons are only allowed to merge with the cluster containing this pentagon. As a result, there is only one large cluster growing, and defects due to merging clusters are reduced.
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 to 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.
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 playback speeds:
Time lapse: 0:00
Normal speed: 1:37
In the first part, the speed has been multiplied by a factor 3.
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: 1 hour 16 minutes
Compression: crf 23
Color scheme: Viridis by Nathaniel J. Smith, Stefan van der Walt and Eric Firing
github.com/BIDS/colormap
Music: "URL Melt" by the Unicorn Heads@UnicornHeads
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 simulation of merging pentagons uses a somewhat different rule than previous simulations on this channel. At the beginning, one pentagon is singled out, and other pentagons are only allowed to merge with the cluster containing this pentagon. As a result, there is only one large cluster growing, and defects due to merging clusters are reduced.
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 to 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.
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 playback speeds:
Time lapse: 0:00
Normal speed: 1:37
In the first part, the speed has been multiplied by a factor 3.
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: 1 hour 16 minutes
Compression: crf 23
Color scheme: Viridis by Nathaniel J. Smith, Stefan van der Walt and Eric Firing
github.com/BIDS/colormap
Music: "URL Melt" by the Unicorn Heads@UnicornHeads
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
![[Flash warning] Two planar waves crossing a gradient index lens](https://i.ytimg.com/vi/aYvsBgyJgIE/hqdefault.jpg "[Flash warning] Two planar waves crossing a gradient index lens
The first part of this video shows some flashing due to the formation of standing waves.
This is a variant of the simulation https://youtu.be/CdpTq3dSrXA , with a different wave form. Though it looks like several parallel beams, the incoming wave is actually obtained by superimposing two planar waves, one of them angled upwards, and the other one downwards. This results in two spots forming in the focal plane of the lens.
The videos are inspired by Huygens Optics recent short https://youtube.com/shorts/VGd3Ajnp6e0 showing the principle of a gradient index lens.
Lenses focus incoming rays of light by delaying them more near the center of the lens than at its periphery. This is often done with a material of constant index of refraction, by making the lens thicker near the center, as shown for instance in the simulation https://youtu.be/rrJJBh9ubUE . However, one can also build lenses of constant thickness, by making the index of refraction of their material depend on the location in the lens. In this simulation, the index decreases like sqrt(n0² - a*r²), where r is the distance to the axis of symmetry. This results in the incoming planar waves being focused at two points in the (estimated) focal plane, marked by a vertical line. The plot to the right shows a time-averaged value of the field along that plane.
This video has two parts, showing the same evolution with two different color gradients:
Wave height: 0:00
Averaged wave energy: 1:08
In the first part, the color hue depends on the height of the wave. In the second part, it depends on the energy of the wave, averaged over a sliding time window.
There are periodic boundary conditions between the top and bottom boundaries, absorbing boundary conditions on the right boundary, and a time-periodic signal is imposed on the left boundary. The display at the right shows the signal along the focal plane, which is indicated by a vertical line.
Render time: 32 minutes 40 seconds
Compression: crf 23
Color scheme: Part 1 - Twilight by Bastian Bechtold
https://github.com/bastibe/twilight
Part 2 - Inferno by Nathaniel J. Smith and Stefan van der Walt
https://github.com/BIDS/colormap
Music: Surrender by Asher Fulero@AsherFulero
See also https://images.math.cnrs.fr/Des-ondes-dans-mon-billard-partie-I.html for more explanations (in French) on a few previous simulations of wave equations.
The simulation solves the wave equation by discretization. The algorithm is adapted from the paper https://hplgit.github.io/fdm-book/doc/pub/wave/pdf/wave-4print.pdf
C code: https://github.com/nilsberglund-orleans/YouTube-simulations
https://www.idpoisson.fr/berglund/software.html
Many thanks to Marco Mancini and Julian Kauth for helping me to accelerate my code!
#wave #lens #gradient_index")
