Analysis of random curve and vortex filament topology

Xiong, Anda (2022). Analysis of random curve and vortex filament topology. University of Birmingham. Ph.D.

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Abstract

This thesis covers a range of topics through the analysis of random curves and vortex filaments in various contexts. We find that phase vortices in random wave model, which is a type of model for quantum and wave chaos, are closed random walks whose length distribution is of universal scaling relation. We analyze the phase vortex in two-plus-one dimensional random waves and find that they can be classified into three types based on length: small loops whose behaviour is model dependent, large loops that are closed random walks, and vortex lines that are long enough to penetrate through the space before getting closed.

Since vortex loops are closed random walks, we investigate the knot probability of them by generating a very large amount of sampling of equilateral and non-equilateral closed random polygons and determining their knot type by the Alexander polynomial. Then with statistical analysis we confirm a general probability equation for knots in random polygons. We also investigate a series of problems regarding to the knot probability of random polygons.

The vortex in three-plus-one-dimensional space-time is two-dimensional surface and we call them vortex worldsheet. We find that while the topology of the vortex worldsheet is relativistically invariant, the topology of the vortex in the timeslice of observers in different reference frame changes with the Lorentz boost. We discuss constructions we used for the scenario and the superluminal region on the vortex world sheet, which is where the speed of phase vortex exceeds the light speed.

Type of Work:

Thesis (Doctorates > Ph.D.)

Award Type:

Doctorates > Ph.D.

Supervisor(s):