Deterministic patterns of random motion: fundamentals, electronics, lasers and the human cardiovascular system

随机波动在物理学、生物学和其他科学技术分支的许多过程中起着至关重要的作用。人们逐渐认识到，随机过程不仅会干扰系统，而且还会诱发全新的行为，这使得人们对噪声诱发现象的兴趣急剧增加。最近发现的例子包括：(i)随机共振，它体现在例如地球物理学，光学双稳性和神经动力学中)随机棘轮，它为生物细胞内的纳米级机械提供了基础；共振激活;和相干共振。波动理论对工业的重要性日益明显。现在人们普遍认识到，热波动对大多数纳米技术，特别是量子计算范式至关重要。非线性系统的波动动力学通常是丰富而复杂的。然而，在许多情况下，它可以分为两个部分，对应于波动的不同表现。一部分是系统亚稳态附近的扩散运动；另一种是大波动事件，系统在相空间中远离初始状态。大波动虽然不常见，但在广泛的过程中发挥着根本作用，从地震和相变成核到纳米线过程、DNA序列突变、电子设备故障以及人类心血管系统状态之间的转换。描述大波动的数学基础是严格的大偏差理论，适用于渐近零噪声极限。该理论预测，并通过实验证明，在大波动事件中，系统沿着定义良好的最可能（最佳）路径运动，而且波动以确定的方式产生特定的最优力。最优路径和力导致波动运动的确定性模式，大大简化了随机动力学的理解。大波动的严格（零噪声强度）理论的使用澄清了统计物理学的许多基本问题，并提出了几种控制波动的方法，例如，寻找能量最小的确定性力来诱导从一种状态到另一种状态的过渡，以及通过分析随机运动来根据数学模型推断（重建）系统动力学。然而，对于这些方法的实际应用，我们必须超越严格的（但渐近的）理论，并考虑波动强度的有限值。该项目正是针对这一目标：开发描述有限噪声强度下大波动所需的形式化和数值工具；并在此基础上，开发可供实验人员使用的新的应用方法和途径；然后用这些方法来表征人类心血管系统的状态。拟议的研究将在连续的阶段进行：理论背景的制定-发展数值方法-在模拟实验中测试方法-应用推理方法来揭示心血管功能障碍。数值方法与实验方法的结合和迭代将为激光物理、生物物理和电子学提供可靠的实用方法。所开发的控制方法在通信设备中具有潜在的应用，以增加对波动的稳定性和/或降低功耗。医学界将获得一项特别重要的好处，这是一种工具，可以改进对一系列病理的早期诊断，更好地评估治疗效果。该项目的心血管部分将与皇家兰开斯特医院的心脏病专家合作进行，他们已经是兰开斯特非线性集团的密切合作者。

Random fluctuations play a crucially important role in many processes in physics, biology and other branches of science and technology. The dawning understanding that random process can not only disturb a system, but also induce totally new kinds of behaviour, has led to sharply increased interest in noise-induced phenomena. Recently discovered examples include: (i) stochastic resonance, which manifests in e.g. geophysics, optical bistability and neural dynamics) stochastic ratchets, which provide the basis for nanoscale machinery within biological cells; resonant activation; and coherence resonance. The importance of fluctuation theory to industry is becoming apparent. It is now generally appreciated that thermal fluctuations are critical for most nanotechnologies, and especially to the paradigms of quantum computation.The fluctuational dynamics of a nonlinear system is usually rich and complex. In many cases, however, it can be divided into two parts, corresponding to different manifestations of the fluctuations. One part is diffusional motion in the vicinity of metastable states of system; the other is large f uctuation events, in which the system moves far away from the initial state in phase space. Large fluctuations, although infrequent, play a fundamental role in a wide range of processes, from earthquakes and nucleation at phase transitions to processes in nanowires, mutations in DNA sequences, failures of electronic devices, and transitions between states of the human cardiovascular systems. A mathematical basis to describe large fluctuations is the rigorous large deviations theory, applicable in the asymptotic zero-noise limit. The theory predicted, and it was shown experimentally, that during a large fluctuation event the system moves along a well-defined most probable (optimal) path, and moreover that fluctuations generate that specific optimal force in a deterministic way. The optimal path and force result in a deterministic pattern of fluctuational motion that significantly simplifies understanding of random dynamics. The use of the rigorous (zero noise intensity) theory of large fluctuations clarified many fundamental problems of statistical physics and also suggested several approaches to control of fluctuations, e.g. finding an energy-minimal deterministic force for inducing the transition from one state to another, and for the inference (reconstruction) of the system dynamics in terms of a mathematical model through the analysis of random motion.However, for real practical applications of these approaches, we have to go beyond the rigorous (but asymptotic) theory and take into account a finite value of fluctuational intensity. The project is directed to exactly this aim: to develop the formalism and numerical tools needed for the description of large fluctuations at finite noise intensity; and, on this basis, to develop new applied methods and approaches which are ready for use by experimentalists; then to use these methods for characterization of state of the human cardiovascular system.The proposed investigations will be developed in sequential stages: the formulation of the theoretical background - developing numerical methods - testing the methods in analogue experiments - the application of inference methods to reveal cardiovascular dysfunctions. The combination and iterations between numerical methods and experiments will provide robust practical methods which can be used in laser physics, biophysics, and electronics. The developed control methods have a potential application in communication devices for increasing stability to fluctuations and/or for decreasing power consumption. An especially important benefice will be received by the medical community in form of tools to improved early diagnosis of a range of pathologies and better assessment of the effect of treatment. The cardiovascular part of the project will be undertaken in collaboration with cardiologists at the Royal Lancaster Infirmary who are already close collaborators of the Lancaster Nonlinear Group.