Convection on a sphere with asymmetric heating

Описание: This is a variant of the simulation    • Convection on a sphere heated from below   showing convection of particles on a sphere heated from "below". The difference is that the heated region is confined to a smaller sector of longitudes. This breaks the rotational symmetry of the system, causing a particular geometry of convection, with particles rising above the heated region.
The particles in this simulation are subjected to a gravitational field directed towards the south pole of the sphere. Part of them, at low latitudes, are coupled to a thermostat with a temperature increasing over time. The remaining particles are subjected to a viscous friction force, proportional to their speed. After a while, a circular motion of the particles is created, consisting in two counter-rotating swirls. This means that the rotation symmetry is spontaneously broken.
This simulation has two parts, showing the same evolution with two different visualizations:
3D view: 0:00
2D view: 1:36
In the 3D part, the observer moves around the sphere in an orbit at constant latitude, located below the equator. The 2D part shows an equirectangular projection (the x- and y-coordinates are proportional to longitude and latitude). The particle color indicates their kinetic energy.
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. 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 https://en.wikipedia.org/wiki/Lennard...

Render time: 3D part: 53 minutes 47 seconds
2D part: 37 minutes 39 seconds
Color scheme: Turbo, by Anton Mikhailov
https://gist.github.com/mikhailov-wor...

Music: "Ocean View Forest" by Patrick Patrikios‪@patrick-jordan-patrikios‬

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/_Berglund...
(in French, some with a Spanish translation)