Long-term Effects of Small Perturbations and Other Multiscale Asymptotic Problems

小扰动和其他多尺度渐近问题的长期影响

• 批准号: 1411866
• 负责人: Mark Freidlin
• 金额: $ 21万
• 依托单位: University of Maryland, College Park
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-08-15 至 2017-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1411866&HistoricalAwards=false
• 关键词: Long; term; Effects; Small; Perturbations

Long term effects of small perturbations, both deterministic and stochastic are of great interest in real life and for various intellectual enterprises from history, sociology, and economics to biology, physics, and engineering. When one wants to understand the time evolution of a complex system, a simplified model of the system, that can be analyzed, is usually a way forward. In this case one chooses a relatively small number of main 'factors' or variables which are guiding the evolution of the system while neglecting other factors that are relatively insignificant. On a short time interval, these small factors are not essential. However, on long time scales, the factors, which were considered as negligible, can become important and even critical for determining the system's behavior. The investigator and his colleagues are developing new models and methods for studying such problems. 'Problem' examples include climate change, biological evolution, and phase transitions in physical or economic systems as well as the appearance of 'stable' oscillations, equilibriums, and other patterns caused by small perturbations.The investigator and his colleagues study deterministic and stochastic perturbations of various dynamical systems as well as semi-flows corresponding to the evolution of PDEs. A general approach to, at first glance, different problems is suggested. Metastability and stochastic resonance, which are manifestation of the large deviation theory, as well as the averaging principle and its modifications can be considered from this point of view. This general approach is based on the study of limiting slow evolution as a motion on the cone of invariant measures of the non-perturbed system. The approach underlines the importance of joint consideration of deterministic and stochastic perturbations. Perturbations of the generalized Landau-Lifshitz equation for multiple spins, incompressible flows in 3D having a conservation law, reaction-advection-diffusion equations in narrow channels arising in models of molecular motors can be studied using this approach. Various boundary problems for second order elliptic equations with a small parameter can be considered as well. Finally, the PI will also study the small mass asymptotics for the Langevin equation.

在真实的生活中，以及从历史、社会学、经济学到生物学、物理学和工程学的各种智力活动中，小扰动的长期影响，无论是确定性的还是随机的，都引起了极大的兴趣。当人们想要了解一个复杂系统的时间演化时，一个可以分析的简化系统模型通常是一个前进的方向。在这种情况下，人们选择了相对较少的主要“因素”或变量，这些因素或变量引导着系统的演变，而忽略了其他相对不重要的因素。在短时间间隔内，这些小因素并不重要。然而，在长时间尺度上，这些被认为可以忽略不计的因素可能变得重要，甚至是决定系统行为的关键因素。研究人员和他的同事正在开发研究这些问题的新模型和方法。“问题”的例子包括气候变化，生物进化，物理或经济系统的相变，以及由小扰动引起的“稳定”振荡，平衡和其他模式的出现。研究员和他的同事研究各种动力系统的确定性和随机扰动，以及与偏微分方程演化相对应的半流。一个一般的方法，乍一看，不同的问题提出。作为大偏差理论的表现形式的亚稳和随机共振，以及平均原理及其修正，都可以从这个角度来考虑。这个一般的方法是基于研究的限制慢演化作为一个运动的锥不变的措施的非扰动系统。该方法强调了联合考虑确定性和随机扰动的重要性。利用这种方法可以研究多自旋的广义Landau-Lifshitz方程、具有守恒律的三维不可压缩流、分子马达模型中窄通道中的反应-对流-扩散方程的扰动。还可以考虑带有小参数的二阶椭圆方程的各种边界问题。最后，PI还将研究朗之万方程的小质量渐近性。