Island myriads in periodic potentials

What is it about?

Potentials are the playground over which the movement of particles occurs and how their dynamic is mathematically modeled, analogous to a ball flowing over a skate bowl. We have discovered a phenomenon, here called island myriads, where infinite periodic trajectories appear in place of previous chaotic ones, for potentials that are periodic in space. The studied potentials have a periodic structure where they tile the plane completely without gaps. This tiling aspect, and its connection to translational and rotational symmetries, is shown to be the cause of the periodic orbits that form a complex fractal-like structure that is the myriad when seen in phase space. The phase space in its turn, is the mathematical space where all possible movements for any velocity and position of the particle can be visuallized. In this space, the stable trajectories found appear as 'islands' in a fractal mosaic, and despite being stable, they emerge when particles approach unstable equilibrium points of the potential surface, which is rather counterintuitive and never previously identified.

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Photo by Faruk Kaymak on Unsplash

Photo by Faruk Kaymak on Unsplash

Why is it important?

The structure found, the island myriad, was never previously identified and beyond its curious aesthetical elegance, it can provide new ways to confine particles or create specific symmetric movements in systems modeled as potentials with tiling symmetry. Some examples would be cold atoms in periodic lattices, atoms diffusing over crystal (metallic) surfaces, electrons guided by ExB waves or electrons in semiconductors.

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