Non-equilibrium statistical theory and information geometry

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.