Rare fluctuations in one dimensional hydrodynamics

Clark, Michael (2021). Rare fluctuations in one dimensional hydrodynamics. University of Birmingham. Ph.D.

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Abstract

In this thesis we study the particular rare fluctuation of emptiness formation probability in a one dimensional harmonically trapped system, and the associated last particle distribution. Inspired by the Tracy-Widom distribution we present the last particle distribution in the weakly interacting regime. We then construct a hydrodynamic formalism which allows us to extend the region of study of these phenomena to include dynamical properties. Using this new formalism we show two key results. First, how to link the Gaussian unitary ensemble of random matrix theory to the Kardar–Parisi–Zhang equation using a hydrodynamic approach. In doing so we gain understanding of why the Tracy-Widom distribution appears in the edge fluctuations of an equilibrium system of fermions in a harmonic trap as well as the surface fluctuations of the non equilibrium surface growth with specific initial conditions. Second, using the now available dynamics of our formalism we calculate time dependent emptiness configurations. Analysis of these configurations shows that a good description of emptiness configurations is that of the algebraic curve and its associated topology. We use this to analyse known emptiness configurations and find new ones.

Type of Work:

Thesis (Doctorates > Ph.D.)

Award Type:

Doctorates > Ph.D.

Supervisor(s):