The Geometry of Transport in Symplectic and Volume-Preserving Dynamics

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Anyone who has poured cream into hot coffee has observed the complex patterns that occur during the mixing of two fluids. It is perhaps surprising that the underlying process is not fully understood, especially when the fluid motion is laminar, i.e., either it is slow or the viscosity is high; then uniform, efficient mixing is hard to achieve. Such laminar processes are important to many applications including the development of micrometer scale bioreactors, effective mixing of polymer and granular materials, spreading of pollutants in the atmosphere, and even nutrient dispersal and spawning efficacy for life in the sea. Laminar flows can cause chaotic motion of advected particles, yielding rapid loss of accuracy for prediction of individual trajectories that turn out to be extremely sensitive to the miniscule changes in environment. The investigator and his colleagues study the design of efficient mixers by developing an understanding of the causes of this sensitivity, and by developing methods to globally optimize stirring protocols. The mathematics of these models is closely related to that, ubiquitous for conservative motion in physics. Techniques the investigator develops are used to predict the lifetime of particles in accelerators, obtain rates for elemental chemical reactions, calculate confinement times in plasma devices, understand the spectra of highly excited atomic systems, and predict asteroid and spacecraft trajectories. Graduate students will be trained through participation in this research project.Subsonic fluids are incompressible and the resulting flows are volume-preserving. Though chaotic motion in incompressible fluids is similar to that in Hamiltonian or symplectic systems, there are profound geometrical differences due to the lack of a canonical pairing between momenta and coordinates. The investigator studies how the geometry of such dynamics changes when it is "nearly" symplectic, leading to novel elliptic structures and to a discovery of the violation of the exponential quasi-stability of nearly-integrable systems implied by Nekhoroshev's theory. The studies include the development of techniques for understanding transport through destroyed invariant tori and the long-time correlations engendered by regular islands and accelerator modes. Mixing due to chaotic advection is caused by stretching and folding, which promotes homoclinic tangles and structure so fine that diffusivity can be effective even when small. The investigator and his students will model this by a finite sequence of stirring events that determine a mixing protocol. Optimization techniques under geometric and energy constraints are used to extremize Sobolev norms that give measures of mixing.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.