Random Perturbations of Excited Deterministic Systems

世纪以来，随机性一直是用于描述现象的核心建模工具，这些现象包括但不限于股票价格随时间的变化、受热波动影响的粒子（例如花粉粒或悬浮在水中的脂质）的运动以及生物学中竞争种群的时间演化。 此外，随机性是许多用于处理和理解大数据的算法的关键组成部分。 分析这种模型和算法的关键是精确地理解随机性和非线性效应（即激励）如何结合联合收割机以使系统松弛到平衡。 该奖项支持的研究将寻求量化统计物理学，湍流和分子动力学中许多模型的平衡收敛速率，特别是在随机抽样算法中使用的那些系统中。 该项目包括与高中生、本科生和研究生的合作，其结果将通过出版物和在国内和国际会议上的介绍加以传播。 该奖项支持随机分析和动力学接口的几个项目。 本研究的第一部分集中于随机抽样算法中使用的随机微分方程（SDEs）的显式收敛到平衡点的速度。 特别重要的是理解在方程中存在奇异系数时高维收敛的精确性质。 这种奇异性在分子动力学（例如Lennard-Jones，Coloumb势函数）中自然出现，因为它们被用来描述粒子相互接近时的排斥效应。 虽然从建模和统计的角度来看是自然的，但由于奇异相互作用产生的能量尖峰，奇异性导致了分析动力学本身以及分析相应的数值模拟的挑战。 第二部分着重研究湍流中各种统计模型的大时间特性。 在这些模型中的每一个中，也有一个能量源，经常导致平衡中的有趣效应，例如，重尾不变测度。 理解随机性、非线性、耗散以及激励如何平衡以产生这些有趣的效应是至关重要的。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.