Branched flow
Scattering phenomenon in wave dynamics
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Examples
Two-dimensional electron gas
Branched flow was first identified in experiments with a two-dimensional electron gas. Electrons flowing from a quantum point contact were scanned using a scanning probe microscope. Instead of usual diffraction patterns, the electrons flowed forming branching strands that persisted for several correlation lengths of the background potential.
Ocean dynamics
Focusing of random waves in the ocean can also lead to branched flow. The fluctuation in the depth of the ocean floor can be described as a random potential. A tsunami wave propagating in such medium will form branches which carry huge energy densities over long distances. This mechanism may also explain some statistical discrepancies in the occurrence of freak waves.
Light propagation
Given the wave nature of light, its propagation in random media can produce branched flow too. Experiments with laser beams in soap bubbles have shown this effect, which has also been proposed to control light focusing in a disordered medium.
Flexural waves in elastic plates
Flexural waves travelling in elastic plates also produce branched flows. Disorder, in this case, appears in the form of inhomogeneous flexural rigidity.
Other examples
Other examples where branched flow has been proposed to happen include microwave radiation of pulsars refracted by interstellar clouds, the Zeldovitch model for the large structure of the universe and electron-phonon interaction in metals.
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Dynamics: Kick and drift map
The dynamical mechanism that originates the branch formation can be understood by means of the kick and drift map, an area preserving map defined by: where n accounts for the discrete time, x and p are position and momentum respectively, and V is the potential. The equation for the momentum is called the "kick" stage, whereas the equation for the position is the "drift". Given an initial manifold in phase space, it can be iterated under the action of the kick and drift map. Typically, the manifold stretches and folds (although keeping its total area constant) forming cusps or caustics and stable regions. These regions of phases space with high concentration of trajectories are precisely the branches.
Scaling properties of branched flow in random potentials
When plane waves or parallel trajectories propagate through a weak random medium, several caustics can arise at more or less regularly ordered positions. Taking the direction perpendicular to the flow, the distance separating the caustics is determined by the correlation length of the potential d.
Another characteristic length is the distance L downstream where the first generation of caustics appear. Taking into account the energy of the trajectories E and the height of the potential ɛ<<E, it can be argued that the following relation holds