Exit Time from the perspective of random dynamical systems and its application in stochastic resonance

随机动力系统视角下的退出时间及其在随机共振中的应用

• 批准号: 2752048
• 金额: --
• 依托单位: Imperial College London
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2022
• 资助国家: 英国
• 起止时间: 2022 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-2752048
• 关键词: Exit; Time; perspective; random; dynamical

The exit time problem, or first passage time problem, typically involves determining the time it takes for a stochastic process (such as Brownian motion or Lévy motion) to leave a given domain. More specifically, it deals with calculating the probability distribution of the time it takes for a diffusion process to first reach the boundary of a bounded domain, starting from an initial point within that domain. We consider deterministic quantities in this problem because they can provide us with the dynamical information of random systems, that is, how the system evolves over time, and quantifying the dynamic behavior of random systems helps people understand how randomness affects these systems. People mainly consider the mean exit time and escape probability in the exit time problem. Mean exit time quantifies how long a random system will stay in a region, while escape probability describes the likelihood of the system transitioning from one region to another. The exit time problem has a wide range of applications in many fields, such as the time of a fund going bankrupt, the time it takes for a Brownian dust particle to escape from a region, the time for a protein molecule to gain sufficient energy to change shape and activate a biological process, and the time when the population of a species in ecology falls below a critical threshold, etc. One important and intriguing application of the exit time problem is stochastic resonance, the simplest example being the addition of a periodic force to a bistable system in addition to stochastic forces. In such a scenario, there exists an optimal noise intensity that, in conjunction with the periodic force, allows the system to escape from one metastable state to another with near-perfect precision within one period, meaning the mean exit time is half the period. This noise intensity is also when the signal-to-noise ratio of the system's output is maximized. This phenomenon, where noise enhances the signal under the influence of periodic forcing, is known as stochastic resonance. Stochastic resonance has widespread applications in engineering, most notably in the detection of weak signals, that is, extracting useful signals from a strong noise background or detecting extremely faint signals. By considering the exit time to find the optimal noise intensity and thus enhance the output noise, we grasp a significant value of studying the exit time problem.

退出时间问题或首次通过时间问题通常涉及确定随机过程（例如布朗运动或莱维运动）离开给定域所需的时间。更具体地说，它涉及计算扩散过程从该域内的初始点开始首先到达有界域的边界所需时间的概率分布。我们在这个问题中考虑确定性量，因为它们可以为我们提供随机系统的动态信息，即系统如何随时间演化，而量化随机系统的动态行为有助于人们理解随机性如何影响这些系统。人们在退出时间问题中主要考虑平均退出时间和逃生概率。平均退出时间量化了随机系统将在一个区域停留的时间，而逃逸概率描述了系统从一个区域转移到另一个区域的可能性。退出时间问题在许多领域都有广泛的应用，例如基金破产的时间、布朗尘埃粒子逃离某个区域所需的时间、蛋白质分子获得足够能量改变形状并激活生物过程的时间、生态中物种数量低于临界阈值的时间等。退出时间问题的一个重要且有趣的应用是随机共振，最简单的例子是 除了随机力之外，还向双稳态系统添加周期性力。在这种情况下，存在最佳噪声强度，与周期性力相结合，允许系统在一个周期内以近乎完美的精度从一种亚稳态逃逸到另一种亚稳态，这意味着平均退出时间是周期的一半。该噪声强度也是系统输出的信噪比最大化时的噪声强度。这种噪声在周期性强迫的影响下增强信号的现象被称为随机共振。随机共振在工程中有着广泛的应用，尤其是在微弱信号的检测方面，即从强噪声背景中提取有用信号或检测极微弱的信号。通过考虑退出时间来找到最佳噪声强度，从而增强输出噪声，我们掌握了研究退出时间问题的重要价值。