Construction of point vortex equilibria via Brownian ratchets

Abstract

A theory capable of producing equilibrium configurations of point vortices in the plane, along with a numerical scheme to compute them, is described. The theory is formulated as a problem in linear algebra where one must find solutions to the matrix equation , where A is the (1/2) N ( N −1)× N non-normal configuration matrix obtained by requiring that all intervortical distances remain fixed, and are the N -vortex strengths. For existence of an equilibrium, A must have a non-trivial nullspace. We consider the singular values of A ; when this has one or more zero singular values, the nullspace of A is non-empty and an equilibrium exists for some choice of Γ . New equilibrium configurations are found numerically by randomly depositing N points in the plane, which generically gives rise to a configuration matrix A with empty nullspace. Using the sum of squares of the k smallest singular values of A as a ‘ratchet’, we ‘thermally fluctuate’ the configuration, allowing each point to execute a random walk in the plane, retaining only those configurations which reduce this quantity at the next step. The configuration is thus driven to one with nullspace ( A )= k >0. These converged states are not necessarily nearby their initial configurations, typically they are asymmetric, and often we can drive the same initial state to several different equilibria. A reverse-ratchet method is also described, which can produce initial conditions that would evolve to a specified equilibrium state. Once a converged final state is achieved, the full singular value decomposition of A is used to calculate an optimal basis set for the nullspace of A and thus all allowable Γ . The distribution of the singular values gives important information on the size of each equilibrium state (as measured by Frobenius norm), their distance from each other (spacing and density) and how far a randomly chosen system of N points in the plane is from the nearest equilibrium configuration with a specified rank, as well as its Shannon entropy.

Spectral gradient flow and equilibrium configurations of point vortices

“…As the system is linear in the particle strengths, the relative equilibrium problem can be reduced to a problem in linear algebra (Jamaloodeen & Newton 2006;Newton & Chamoun 2007, 2009. In order to set the stage for the gradient flow method detailed in this paper, we first give the necessary background.…”

Vortex Lattice Theory: A Particle Interaction Perspective

“…An actual convergence path in the plane of one of the vortices making up the equilibrium is shown in Figure 4.4. We note that, in practice, the method has worked well and has uncovered new classes of equilibrium patterns (see Newton and Chamoun (2007) for examples). Of course, the method doesn't always converge and can sometimes get stuck, for various reasons that need to be better understood.…”

“…The advantage of a Brownian motion-driven algorithm, over say a gradient method, is that all portions of the configuration landscape are sampled in an unbiased way. We describe what we call a Brownian "ratchet" scheme, used in Newton and Chamoun (2007) to uncover new classes of equilibria. The idea behind a Brownian ratchet is simple-it is a device that sits in a heat bath and rectifies the nonequilibrium thermal fluctuations to generate motion in a preferred direction.…”

Force-enhancing vortex equilibria for two parallel plates in uniform flow

“…This poses some difficulties in solving for an equilibrium by numerical means; thus we introduce a simple linear algebraic approach called the Brownian ratchet scheme. The numerical scheme was originally developed by Newton & Chamoun (2007) to obtain point-vortex equilibria in the plane. We show that the numerical method is successfully applicable to the present problem.…”