Algorithms and applications of the Monte Carlo method : Two-dimensional melting and perfect sampling

Abstract : This thesis deals with the Monte Carlo method and some of its applications to statistical physics. The first part concerns the study of the melting transition in two dimensions. The nature of this transition is an old problem of statistical physics, and especially for the fundamental model of hard disks. A Monte Carlo algorithm, called ''event-chain'', is developed for this model and is used to study the melting transition. The results show that the transition follows a two-step scenario with a hexatic phase between the liquid and the solid. The solid-hexatic transition is continuous of the Kosterlitz-Thouless type, and the hexatic-liquid transition is discontinuous. These results confirm the existence of the hexatic phase, and pose a new theoretical basis for experiments on two-dimensional melting transitions. The second part concerns perfect sampling algorithms using the ''coupling-from-the-past'' approach. This paradigm of the Markov-chain Monte Carlo method allows to sample configurations from the exact desired distribution, and this suppresses the long-standing problem of accessing the mixing time of a Markov chain. This method is however difficult to apply to physical systems such as spin glasses at low temperature or hard spheres at high density. Perfect-sampling algorithms are studied for these systems. The results show that the limitation of this method is related to transitions toward chaos of Markov chains. These dynamical transitions are not caused by thermodynamical changes.
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Submitted on : Tuesday, November 1, 2011 - 2:32:59 AM
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Etienne Bernard. Algorithms and applications of the Monte Carlo method : Two-dimensional melting and perfect sampling. Data Analysis, Statistics and Probability [physics.data-an]. Université Pierre et Marie Curie - Paris VI, 2011. English. ⟨tel-00637330⟩

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