Random Perturbations of Excited Deterministic Systems

Iowa State University, Ames IA

Investigators

Abstract

For over a century, randomness has been a core modeling tool used to describe phenomena such as, but not limited to, the value of stock prices over time, the motion of particles subject to thermal fluctuations (e.g. pollen grains or lipids suspended in water) and the time evolution of competing populations in biology. Additionally, randomness is a key component in many algorithms used to process and understand big data. Crucial to analyzing such models and algorithms is understanding precisely how the randomness and nonlinear effects, i.e. the excitation, combine to relax the system to equilibrium. The research supported by this award will seek to quantify rates of convergence to equilibrium in a number of models in statistical physics, turbulence and molecular dynamics, especially in those systems used in random sampling algorithms. The project includes collaborations with high school, undergraduate and graduate students, the results of which will be disseminated through publications and through presentations at domestic and international conferences. This award supports several projects at the interface of stochastic analysis and dynamics. The first part of this research focuses on extracting explicit rates of convergence to equilibrium for stochastic differential equations (SDEs) used in random sampling algorithms. Of particular importance is understanding the precise nature of convergence in high dimensions in the presence of singular coefficients in the equations. Such singularities arise naturally in the context of molecular dynamics (e.g. Lennard-Jones, Coloumb potential functions) as they are used to describe repulsive effects as particles approach one another. Although natural from a modeling and statistical standpoint, the singularities lead to challenges in both analyzing the dynamics itself as well as in analyzing the corresponding numerical simulation due to energy spikes generated by the singular interactions. The second part focuses on studying large-time properties of various statistical models in turbulence. In each of these models, there is also a source of energy often leading to interesting effects in equilibrium such as, for example, heavy-tailed invariant measures. Understanding how randomness, nonlinearities, dissipation as well as the excitation balance out to produce these interesting effects is of central importance. 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.