Frameworks for Fluctuation Analysis in Semi-Markov Systems: Large Deviations, Renewal Theory and Reinforcement Learning

In recent decades, rigorous frameworks for studying non-equilibrium phenomena, rooted in the theory of stochastic processes and their large deviations have spawned several successful techniques for investigating Markovian (or memoryless) processes. Adaption of these frameworks for fluctuation analysis in non-Markov systems has become an active area of research, with recent works showing analytical and computational results for some types of memory-dependent models. The present thesis reflects this growing interest and is concerned with the analysis of fluctuations in semi-Markov systems, a class of non-Markov processes where memory is encoded via non-exponential waiting-time distributions. We present contributions towards two complementary frameworks for large deviations in semi-Markov systems, aiming to obtain the scaled cumulant generating function (SCGF); a quantity capturing non-equilibrium fluctuations. The first is a computational framework using advanced neural reinforcement learning, which takes inspiration from recent machine learning trends. The method extends the Markovian method by Rose et al. [New J. Phys. 23 013013 (2021)] to semi-Markov systems and effective SCGF computations are demonstrated for current fluctuations in single and many-body systems. The second contribution uses an analytical framework for calculating the SCGF in certain semi-Markov systems where waiting-time distributions are renewed after a transition occurs. The method combines renewal theory and the pole analysis notably utilized by Poland and Scheraga [J. Chem. Phys. 45 1456–1463 (1966)] for studying DNA denaturation. The method is then applied for computing fluctuations of a special type of ratchet current induced via asymmetry in the shapes of waiting-time distributions having the same mean.