Large deviations in non-equilibrium dynamics: Markov models and thermodynamic uncertainty relations

Fu, Rueih-Sheng

doi:https://doi.org/10.21985/n2-3nj1-va60

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Large deviations in non-equilibrium dynamics: Markov models and thermodynamic uncertainty relations

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This manuscript describes and contextualizes the research I performed as a PhD student in Northwestern University. The first three chapters, on Markov chains, stochastic thermodynamics, and large deviation theory, describe three interrelated topics that serve as the background for subsequent research detailed in the next three chapters, on understanding the current reversal in a two-dimensional ratchet, extending thermodynamic uncertainty bounds to underdamped Langevin dynamics, and developing a stochastic thermodynamic model for a catenane motor. The combination of Markov chains, stochastic thermodynamics, and large deviation theory yields a potent framework for constructing and studying models of non-equilibrium systems as long as those systems satisfy relevant thermodynamic constraints. In particular, the assumption of Markovian—memoryless—dynamics implies that the system’s memory about its past state is insignificant. Where this assumption is valid, the dynamics simplify tremendously as a result, and the system’s dynamics may be represented as transitions between states on a graph. The intuitive picture is discrete in time and space,but can be generalized to continuous time and space with modest effort. Integrating stochastic thermodynamics into these Markov models allow us to interpret their physical behavior and calculate pertinent thermodynamic quantities, such as heat, work, and entropy. In equilibrium, conservation of probability between each pair of transitions in a Markov chain is known as detailed balance. By coupling these transitions to work expenditure, or by varying the energy of the underlying states according to a time-dependent protocol, we can break detailed balance and end up with an out-of-equilibrium system. Detailed balance is a remarkably strong constraint, and non-equilibrium systemsthat break detailed balance can exhibit rich dynamics as a result. The study of these rich non-equilibrium dynamics has, until the last few decades, largely been relegated to the linear response regime. The development of large deviation theory afforded a formal framework by which to make sense of the statistics of thermodynamic parameters in non-equilibrium systems, establishing a rigorous and considerably moregeneral system for studying non-equilibrium systems and models. After providing this background, I present three examples of research studies I undertook following this schema, exemplifying its utility and wide generality

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