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Out-of-equilibrium dynamics in classical field theories and Ising spin models

Abstract : This thesis is made up of two independent parts. In the first chapter, we introduce a novel numerical method to integrate partial differential equations representing the Hamiltonian dynamics of field theories. It is a multi-symplectic integrator that locally conserves the stress-energy tensor with an excellent precision over very long periods. Its major advantage is that it is extremely simple (it is basically a centered box scheme) while remaining locally well defined. We put it to the test in the case of the non-linear wave equation (with quartic potential) in one spatial dimension, and we explain how to implement it in higher dimensions. A formal geometric presentation of the multi-symplectic structure is also given as well as a technical trick allowing to solve the degeneracy problem that potentially accompanies the multi-symplectic structure. In the second chapter, we address the issue of the influence of a finite cooling rate while performing a quench across a second order phase transition. We extend the Kibble-Zurek mechanism to describe in a more faithfully way the out-of-equilibrium regime of the dynamics before crossing the transition. We describe the time and cooling rate dependence of the typical growing size of the geometric objects, before and when reaching the critical point. These theoretical predictions are demonstrated through a numerical study of the emblematic kinetic ferromagnetic Ising model on the square lattice. A description of the geometric properties of the domains present in the system in the course of the annealing and when reaching the transition is also given.
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Submitted on : Thursday, December 14, 2017 - 9:35:07 AM
Last modification on : Friday, May 29, 2020 - 4:01:55 PM


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  • HAL Id : tel-01663588, version 1


Hugo Ricateau. Out-of-equilibrium dynamics in classical field theories and Ising spin models. Physics [physics]. Université Pierre et Marie Curie - Paris VI, 2017. English. ⟨NNT : 2017PA066189⟩. ⟨tel-01663588⟩



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