Enhanced Dissipation, Accelerating Langevin Dynamics, and Bose--Einstein Condensation

Enhanced dissipation arises in many situations of physical importance, ranging from micro fluids to oceanography, and is even commonly observed when cream is poured into coffee and it mixes quickly when stirred but very slowly if left alone. This is the phenomenon by which the combination of stirring and diffusion increases the rate of convergence to equilibrium.

This project plans to develop a theoretical understanding of enhanced dissipation, including quantifying this effect and producing criterion describing scenarios where enhanced dissipation occurs at the optimal rate. The methods developed will also be used to speed up sampling algorithms and are useful in scientific computation. In addition to enhanced dissipation, the project also involves the study of the formation of Bose—Einstein condensates in situations which are of interest in modern cosmology.

Students and post-docs working on this project will be exposed to a broad set of fundamental tools in partial differential equations, probability, and scientific computation, positioning them to contribute to the ever-changing scientific landscape.

Advection and diffusion are two fundamental phenomena that arise in a wide variety of applications ranging from micro-fluids to meteorology, and even cosmology. In many situations the interaction between advection and diffusion results in an increased rate of convergence to equilibrium -- a phenomenon known as “enhanced dissipation”. This project involves a quantitative study of enhanced dissipation, obtaining sharp bounds, determining criterion describing scenarios where it occurs at the optimal rate, and investigating its properties.

The methods developed can also be used to speed up certain Markov processes and may improve rates of convergence of commonly used Monte Carlo Markov Chain algorithms. In addition to enhanced dissipation, the project will also study the formation of Bose--Einstein condensates in high temperature plasmas. This arises in cosmological applications such as the study of the interaction between matter and radiation in the early universe, the radiation spectra for the accretion disk around black holes.

The project aims to classify mechanisms by which condensates form, prove convergence and stability of a numerical scheme, and study condensates in the three-dimensional versions.

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.