Non-equilibrium statistical theory and information geometry

自然界和实验室中的许多系统远离平衡，并表现出显著的波动，使处于或接近平衡状态的小波动和短记忆时间的关键假设无效。充分了解概率分布函数（pdf），特别是时间相关的pdf，对于理解这些随机系统的动力学至关重要。一旦计算出来，时间相关的pdf提供了一个关键的信息，这在任何只使用固定pdf的研究中是完全缺失的。例如，可以通过为概率分配适当的度量来获得对系统中信息变化的关键洞察。从理论和实践考虑，人们对概率度量的兴趣确实越来越大，根据感兴趣的问题提出了不同的度量。从理论上讲，给概率分配一个适当的度量使我们能够在数学上量化不同pdf之间的差异，在随机过程和几何之间提供一个美丽的概念联系。在实际层面上，它可以用于优化各种期望的结果。通过将Fisher度量扩展到与时间相关的问题，我们最近引入了一种与系统无关的方法，通过量化与pdf的时间演化相关的信息变化来理解信息几何。本项目旨在研究非平衡随机系统中的信息几何。具体来说，我们将研究布朗棘轮，它指的是一种新现象，即各向异性系统中的非平衡波动可以在经典系统中引起净机械力和运动。我们将对与噪声诱导输运相关的信息几何以及棘轮理论在纳米系统中的应用感兴趣。由于纳米系统（分子马达）的能量水平仅比波动热能大几倍，因此随机性在其运行中起主要作用。数学上，我们将计算时间相关的pdf和信息长度，以理解与布朗棘轮操作相关的信息几何。然后，我们将工作扩展到量子布朗棘轮，包括量子效应在势和其他随机系统。该项目是跨学科的，是应用统计学、概率和连续介质力学（目前EPSRC的战略领域）的桥梁，并补充了英国的其他研究。

Many systems in nature and laboratories are far from equilibrium and exhibit significant fluctuations, invalidating the key assumptions of small fluctuations and short memory time in or near equilibrium. A full knowledge of Probability Distribution Functions (PDFs), especially time-dependent PDFs, is crucial for understanding the dynamics of these stochastic systems. Once computed, time-dependent PDFs provide a key information that is completely missing in any studies using only stationary PDFs. For instance, a key insight into the information change in the system can be obtained by assigning an appropriate metric to probability. There has indeed been increasing interest in a probability metric from theoretical and practical considerations, with different metrics proposed depending on the question of interest. Theoretically, the assignment of an appropriate metric to probability enables us to mathematically quantify the difference among different PDFs, providing a beautiful conceptual link between a stochastic process and geometry. At a practical level, it can be utilized for optimising various desired outcomes. By extending the Fisher metric to time-dependent problems, we recently introduced a system-independent way of understanding information geometry by quantifying information change associated with time-evolution of PDFs. This project aims to investigate information geometry in non-equilibrium stochastic systems. Specifically, we will investigate the Brownian Ratchet which refers to a novel phenomenon that non-equilibrium fluctuations in an anisotropic system can induce a net mechanical force and motion in classical systems. We will be interested in information geometry associated with noise-induced transport and the application of the ratchet theory to nano systems. Since the energy level in nano systems (molecular motors) is only a few times greater than fluctuating thermal energy, stochasticity plays a primary role in their operation. Mathematically, we will compute time-dependent PDFs and the information length to understand information geometry associated with the operation of Brownian ratchet. We will then extend the work to quantum Brownian Ratchet by including the quantum effect in the potential and also to other stochastic systems.This project is interdisciplinary and bridges Applied Statistics and Probability and Continuum Mechanics (current EPSRC strategic areas) and complements other research in the UK.