We investigate superfluid flow around an airfoil accelerated to a finite velocity from rest. Using simulations of the Gross-Pitaevskii equation we find striking similarities to viscous flows: from production of starting vortices to convergence of airfoil circulation onto a quantized version of the Kutta-Joukowski circulation. We predict the number of quantized vortices nucleated by a given foil via a phenomenological argument. We further find stall-like behavior governed by airfoil speed, not angle of attack, as in classical flows. Finally we analyze the lift and drag acting on the airfoil.
We provide a model for thinking about ADMR in the small magnetic field limit and discuss what recent measurements of ADMR near the pseudogap critical point of Nd-LSCO are actually able to deduce about the Fermi surface. We further include the quasiparticle residue in these calculations and use our model to understand why its inclusion doesn't qualitatively alter the predicted ADMR for the spin density wave reconstructed Fermi surface. We attempt to propose a Fermi surface where the inclusion of the quasiparticle residue will alter the qualitative form of ADMR.
We investigate the development of superfluid flow around an airfoil accelerated to a finite velocity from rest. Using both simulations of the Gross-Pitaevskii equation and analytical calculations we find striking similarities to viscous flows: from the production of starting vortices to the convergence of the airfoil circulation onto a quantized version of the classical Kutta-Joukowski circulation. Using a phenomenological argument we predict the number of vortices nucleated by a given foil and find good agreement with numerics. Finally we analyze the lift and drag acting on the airfoil.
Non-Equilibrium Statistical Mechanics (NESM) Journal Club - January 7, 2018
I’ll explore particle vortex duality, i.e. the idea that the 3D XY-Model and Abelian-Higgs Models are dual, with vortices in one model identified with particles in the other model. I’ll review the models and give some heuristic theoretical evidence for believing they describe the same physics, culminating in a revealing comparison of their phase diagrams. Next I’ll discuss how we can simulate a model in the same universality class as the Abelian-Higgs Model to verify that it undergoes an inverted XY-Model type transition. Finally I’ll conclude with applications of particle-vortex duality to the FQHE.
Vortex Νucleation in Superfluids
ChuSOARS - November 21, 2017
I will introduce superfluids and the vortex excitations they contain. I will also discuss how to produce these excitations via nucleation from moving objects.
Poisson Geometry with Applications to the Hamiltonian Formulation of Inviscid Fluid Mechanics
Chicago Mathematics REU - August 15, 2015
We explore particle vortex duality, i.e. the idea that the 3D XY-Model and Abelian-Higgs Models are dual, with vortices in one model identified with particles in the other model. First we review the models and give some heuristic theoretical evidence for believing they describe the same physics. This culminates in a table to allow easy comparison between their phases. Next we discuss how we can use numerics to find the critical exponents of the models and check that they are the same. The Ising model is used to provide a natural introduction to numerical simulations, before moving on to simulating a model in the same universality class as the Abelian-Higgs Model. Finally we conclude with a demonstration that the Abelian-Higgs Model undergoes an inverted XY-Model type transition.
We motivate Weyl’s law by demonstrating the relevance of the distribution of eigenvalues of the Laplacian to the ultraviolet catastrophe of physics. We then introduce Riemannian geometry, define the Laplacian on a general Riemannian manifold, and find geometric analogs of various concepts in real space, along the way using the two-sphere as a clarifying example. Finally, we use analogy with real space and the results we have built up to prove Weyl’s law, making sure at each step to demonstrate the physical significance of any major ideas.
We introduce Hamiltonian systems and derive an important stability result, along with giving some physical motivation. We then move onto the generalization of these systems found in symplectic geometry. Next we consider symplectic geometry’s natural generalization, Poisson geometry. After giving some definitions we present the motivating example of the torqueless Euler equations. These motivate us to consider the abstract structure of Poisson geometry, but we first need to introduce some concepts from multi- vector calculus. Finally we arrive at some important results connecting symplectic geometry and Poisson geometry, including the Darboux-Weinstein theorem.
Weakly nonlinear oscillations exhibit a wide range of phenomena not seen in simple linear oscillations. This paper considers weakly nonlinear oscillations with an analytic forcing term, and attempts to understand various quantitative solution methods for this problem. We give a quantitative demonstration of the failure of regular perturbation theory, and use this failure to motivate investigation into two-timing and averaging theory. We see that both two-timing and averaging theory give initially identical approximations of a solution, in fact giving excellent approximations even when the nonlinear effects cease to be weak. Finally, we see that the results given by two-timing and averaging theory are physically valid only when the exact solution is bounded.