Branched Flow and Caustics in Two-Dimensional Random Potentials and Magnetic Fields

Abstract

Branched flow is a universal phenomenon of two-dimensional wave or particle flows which propagate through a weak random potential. Its origin is the formation of caustics, which are locations where the flow is focused by the cumulative effect of weak random forces acting along the flowpath. Branched flow has been observed on length scales spanning at least twelve orders of magnitude and in a variety of systems. For example, it has been studied in semiconductor microdevices, has been argued to be the mechanism underlying the formation of giant freak sea waves and has been predicted for the propagation of sound through the ocean on scales of several thousands of kilometers. A thorough understanding of the mechanism dictating how a random potential can cause such drastic effects such as branching is therefore important in many areas of physics, and is interesting to experimentalists and theoreticians alike.In this thesis, we contribute to the theory of branched flow in the following ways. First, we consider the statistics of caustics along particle trajectories in a random potential with an additional deterministic focusing mechanism, a constant magnetic field. By extending existing theories and with detailed numerical simulations we can study the interplay between random focusing by the disorder potential and deterministic focusing by the magnetic field.We then apply our theory to data from a magnetic focusing experiment in a semiconductor microstructure. We can reproduce the results of the experiments numerically and show them to be a result of random and deterministic focusing. Our results have important consequences for the conductance properties of semiconductor microstructures.In the second part of the thesis, we consider the statistics of branches transverse to the flow, since this, although not as directly analytically and numerically accessible, is a quantity which can be measured more easily in an experiment. For the first time, we obtain statistics of the number of branches valid for all distances from a source, analytically and numerically. Also for the first time, we analyze the effect of different correlation functions and find an analytic expression for the universal curve describing the number of branches, which is valid for a wide range of correlation functions and parameters of the random potential. KurzfassungBranched flow ("verästelter Fluss") ist ein universelles Phänomen zweidimensionaler Teilchen-und Wellenflüsse, die durch ein schwaches, korreliertes Zufallspotential propagieren. Ihm liegt die Entstehung von Kaustiken zugrunde, an denen die Flussdichte besonders hoch ist. Die Flussverästelung ist auf Längenskalen, die sich auf mehr als zwölf Größenordnungen erstrecken, und in einer Vielzahl verschiedener Systeme beobachtet worden. Sie wurde unter anderem in Halbleiter-Mikrostrukturen untersucht, als Ursache für Riesenwellen (Monsterwellen) beschrieben und für die Propagation von Schallwellen durch den Ozean auf Skalen von tausenden von Kilometern vorhergesagt. Eine genaues Verständnis des Mec...

Branched flows of flexural elastic waves in non-uniform cylindrical shells

“…0. See [29,31] for details. The first focusing event is, therefore, detected by finding the time and location when the ks À s curve folds over itself for the first time.…”

Branched flows of flexural waves in non-uniform elastic plates

“…For numerical ray simulations, we use a similar method as the one described in our earlier work 7 to detect focusing events. The first focusing event is detected by finding the time and location when the ks − s curve folds over itself 26,27 . This is done numerically by tracking the local slope of the curve and detecting locations when the slope become higher than π/2.…”