Falling sticks @NilsBerglund

Nils Berglund | Falling sticks @NilsBerglund | Uploaded June 2024 | Updated October 2024, 2 hours ago.
This is a preparatory simulation, for a new way to simulate interacting polygonal shapes. The particles in this simulation are very thin rectangles. They interact via a harmonic potential as soon as they come close to each other, and their dynamics is determined from the total force and torque. A weak Lennard-Jones interaction has been added for stability, but it may not be necessary.
If the motion of the sticks does not seem very realistic, this can probably be improved by tweaking the parameters (force constants of the harmonic interaction, moment of inertia, friction for the rotational dynamics). 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: 1:08
In the first part, the particles' color depends on their orientation. 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 en.wikipedia.org/wiki/Lennard-Jones_potential

Render time: 52 minutes 46 seconds
Compression: crf 23
Color scheme: Part 1 - HSL/Jet
Part 2 - Turbo, by Anton Mikhailov
gist.github.com/mikhailov-work/6a308c20e494d9e0ccc29036b28faa7a

Music: "Come and Get It!" by Dan Lebowitz@lebo_tone

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 #ions #quasicrystal

$Cherenkov radiation of a slightly faster particle This is a variant of the video https://youtu.be/z_KZnXUNyus , showing Cherenkov radiation of a slightly faster particle, in a medium with an index of refraction that increases slightly faster. Update: I finally found the origin of this singularity, which is, sadly, a mistake in the way I initialized the refractive index of the medium. Instead of decreasing linearly while remaining positive, the wave speed in the medium decreases linearly, then reaches zero, and then increases linearly again. The singularity happens where the wave speed vanishes. A corrected version will be released in a couple of days. Cherenkov radiation is produced when a fast moving particle enters a medium in which the speed of light is smaller than the speed of the particle. This does not contradict special relativity, because only the speed of light in a vacuum cannot be reached by a massive particle, while the speed of light in a medium is in general smaller than in a vacuum. In this simulation, a particle moves at constant speed from left to right, emitting pulses at regular time intervals. The speed of light decreases linearly, reaches zero and increases linearly again. The ratio of the indices between the left to the right boundary of the simulation rectangle is 2.78. When the speed of light in the medium becomes smaller than the speed of a particle, a shock wave appears, which is the origin of Cherenkov radiation. This video has two parts, showing the same evolution with two different color gradients: Wave height: 0:00 Averaged wave energy: 1:12 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 absorbing boundary conditions on the borders of the simulated rectangle. The display at the bottom shows the signal along a horizontal line, slightly below the path of the particle. Render time: 32 minutes 52 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: Target Fuse by French Fuse@frenchfuse 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 #Cherenkov$

$Waves escaping a ring of obstacles: Hex grid This is a simulation of waves originating from a point source crossing a set of circular obstacles placed on concentric rings. There are 25 rings having 60 obstacles each. The obstacles of each ring are shifted by 3 degrees with respect to the previous ring, forming a polar analogue of a hexagonal grid. This video has two parts, showing the same evolution with two different color gradients: Averaged wave energy: 0:00 Wave height: 1:11 In the first part, the color hue depends on the energy of the wave, averaged over a sliding time window. In the second part, it depends on the height of the wave. The contrast has been enhanced by a shading procedure, similar to the one I have used on videos of reaction-diffusion equations. The process is to compute the normal vector to a surface in 3D that would be obtained by using the third dimension to represent the field, and then to make the luminosity depend on the angle between the normal vector and a fixed direction. The color in the central region has been brightened to white, because the waves tend to make large-amplitude oscillations there, which would not be very pleasant to watch. There are absorbing boundary conditions on the borders of the simulated rectangle. The display at the right shows a time-averaged version of the signal near the right boundary of the simulated rectangular area. More precisely, it shows the square root of an average of squares of the respective field value (wave height or energy). Render time: 26 minutes 18 seconds Compression: crf 23 Color scheme: Part 1 - Inferno by Nathaniel J. Smith and Stefan van der Walt Part 2 - Viridis by Nathaniel J. Smith, Stefan van der Walt and Eric Firing https://github.com/BIDS/colormap Music: Amber by VYEN@JohannesWeberTV See also https://images.math.cnrs.fr/des-ondes-dans-mon-billard-partie-i/ 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 #diffraction$

Trying to model tides with a shallow water equation

$Cherenkov radiation, with corrected refractive index$

Exciting resonant modes in a circle, with a single source and sixteen secondary cavities

All those moments will be lost in time: Longer simulation of DNA replication

DNA-like molecules in higher temperature and flying ice cubes

Rattling the sieves in a particle sorter avoids clogging