Irreversible Markov-chains for particle systems and spin models: Mixing and dynamical scaling

Abstract : This thesis studies the irreversible Markov chain in the spin systems and particle systems, theoretically explains their dynamical specialties, proposes an improvement to the Monte Carlo methods with respect to the systematic properties. The first two chapters review the probability theory, Markov chain and Monte Carlo method. The irreversible Markov chain, with the “lifting” scheme and factorized Metropolis filter, increases the mixing speed at a higher scale in many models. The third chapter studies the hard sphere model. From the exact result obtained from the one-dimensional model in the continuous limit, the “event-chain” algorithm is related to the coupon-collector problem, in the evaluation of mixing time. A sequential “event-chain” algorithm is proposed to accelerate the mixing process, which is also valid in more general cases of higher dimensions. For more general Metropolis algorithms with “lifting”, their crossover with the “event-chain” algorithm is discussed. The fourth chapter presents the dynamics of the irreversible Markov-chain for continuous spin models using Metropolis filter, in the presence of topological excitations. The local nature of the Markov-chain dynamics leads to a slow vortex mode and a fast spin-wave mode in the XY model. The equilibrium correlation varies from z~2 at the critical temperature to z~0 at the low temperature limit, and the respective influence on three-dimensional Heisenberg model is also described. The fifth chapter, based on the knowledge of the previous two chapters, proposes an optimization of Metropolis filter for general particle models, by introducing an auxiliary field. Simulations on one dimensional Lennard-Jones chain exhibit an obvious acceleration as the spin-wave mode. Further studies verify a super-diffusive behavior of the “event-chain” algorithm, which may explain the high speed of the dynamics.
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Ze Lei. Irreversible Markov-chains for particle systems and spin models: Mixing and dynamical scaling. Statistical Mechanics [cond-mat.stat-mech]. PSL Research University; Ecole Normale Superieure de Paris - ENS Paris, 2018. English. ⟨tel-02063289⟩

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