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Classical and Quantum Out-Of-Equilibrium Dynamics Classical and quantum out-of-equilibrium dynamics. Formalism and applications. Camille Aron To cite this version: Camille Aron. Classical and quantum out-of-equilibrium dynamics. Formalism and applications.. Data Analysis, Statistics and Probability [physics.data-an]. Université Pierre et Marie Curie - Paris VI, 2010. English. tel-00538099 HAL Id: tel-00538099 https://tel.archives-ouvertes.fr/tel-00538099 Submitted on 21 Nov 2010 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. THESE` DE DOCTORAT DE L’UNIVERSITE´ PIERRE ET MARIE CURIE Specialit´ e´ : Physique theorique´ Ecole´ doctorale : ≪ La physique, de la particule au solide ≫ Realis´ ee´ au Laboratoire de Physique Theorique´ et Hautes Energies´ Present´ ee´ par M. Camille ARON Pour obtenir le grade de Docteur de l’Universite´ Pierre et Marie Curie Dynamique hors d’equilibre´ classique et quantique. Formalisme et applications. Soutenue le 20 septembre 2010 devant le jury compose´ de M. Denis BERNARD (President´ du jury) M. Federico CORBERI (Examinateur) me M Leticia CUGLIANDOLO (Directrice de these)` M. Gabriel KOTLIAR (Rapporteur) M. Marco PICCO (Invite)´ M. Fred´ eric´ Van WIJLAND (Rapporteur) A bracadabra. REMERCIEMENTS ES tous premiers vont bien naturellement a` ma directrice de these,` Leticia Cugliandolo. L Au cours de ces trois annees´ de these,` sa porte m’a toujours et´ e´ grande ouverte, ce qui m’a permis d’abuser un peu de son indefectible´ bonne humeur et de son vaste savoir. J’espere` conserver sa confiance et sa sympathie. Je remercie Marco Picco, mon co-directeur de these,` pour sa disponibilite´ et ses conseils toujours avises.´ Je tiens egalement´ a` remercier vivement Giulio Biroli avec qui j’ai eu l’occasion de collaborer sur des problemes` passion- ants. Je mesure la chance qui fut la mienne d’apprendre a` leurs cotˆ es.´ Je suis tres` reconnaissant a` Denis Bernard et Federico Corberi d’avoir accepte´ de faire partie du jury cette these,` et tout particulierement` a` Gabriel Koliar et Fred´ eric´ Van Wijland d’avoir accepte´ d’en etreˆ les rapporteurs. J’ai ben´ efici´ e´ d’un excellent environnement de travail au LPTHE ; non seulement en raison de la qualite´ scientifique, mais egalement´ des qualites´ humaines de ses membres. Merci a` eux tous, et plus particulierement` a` Damien Bremont, Benoˆıt Estienne et Alberto Sicilia qui ont egay´ e´ mes deux premieres` annees´ de these` ainsi qu’a` Bernard Diu pour ses visites impromptues et toujours rafraˆıchissantes. Enfin, j’ai une pensee´ particuliere` pour Monica Guica˘ qui m’a gaiement accompagne´ ces deux dernieres` annees.´ Paris, le 30 juilllet 2010. v CONTENTS 1 Introduction 1 1.1 Systems coupled to an environment ..................... 3 1.2 Models and methods ............................. 9 1.3 Questions ................................... 18 2 Symmetries of Langevin Generating Functionals 25 2.1 Langevin equation ............................... 28 2.2 The MSRJD path-integral formalism ..................... 31 2.3 Equilibrium .................................. 38 2.4 Out of equilibrium .............................. 51 2.5 Conclusions .................................. 59 Appendices ..................................... 60 3 Scalings and Super-universality in Coarsening versus Glassy Dynamics 71 3.1 The models .................................. 73 3.2 The typical growing length .......................... 76 3.3 Fluctuations .................................. 85 3.4 Conclusions .................................. 88 4 Driven Quantum Coarsening 91 4.1 The model ................................... 94 4.2 The dynamics ................................. 98 4.3 The influence of the fermionic environment .................104 4.4 Results .....................................109 4.5 The current ..................................128 4.6 Conclusions ..................................131 Appendices .....................................132 5 Conclusions and Outlook 147 6 Synopsis 151 Bibliography 163 vii HAPTER C 1 INTRODUCTION Once upon a time... NTIL the second half of the nineteenth century, science studied macroscopic phenom- U ena that were directly perceptible by human senses, even though scientists were often led to enhance the sensory perception with the more objective measurements of instruments. For instance, microscopes allowed biologists to discover cells – the building blocks of life – and their inner structure. In the field of physics, scientists studied mechanics, electricity, optics, acoustics but also thermodynamics and states of matter. All these domains were considered independent in the nineteenth century. In particular, thermodynamicians were far from imagining that their theory would take its roots in mechanics. Despite its successes, macroscopic physics was condemned to eventually lose its fun- damental character to the benefit of microscopic physics. Indeed, the nineteenth century saw the accession at the scientific level of the antique philosophical idea of the atomic hy- pothesis introduced by Leucippus and his student Democritus in the fifth century B.C. [1]. The quantitative study of chemical reactions revealed some stoichiometric laws that John Dalton and Amedeo Avogadro interpreted very convincingly within the frame of the atomic hypothesis: reactants were aggregates of microscopic components [2, 3]. This hypothesis, which was first considered a convenient way of presenting results – since it was impossible at that time to directly prove the existence of atoms – progressively gained ground during the nineteenth century. 1 2 James C. Maxwell, who was at first reluctant to take position in favor of atoms, was the first to introduce probabilistic methods to compute the distribution of particle velocities in a gas in 1860 [4, 5]. In 1872, Ludwig Boltzmann set the building blocks of out-of-equilibrium statistical mechanics by introducing the so-called Boltzmann equation that describes the generic transport properties in a gas by taking into account the dynamics of collisions. In 1877, he was the first to give a probabilistic interpretation of the second principle of ther- 1 modynamics with its celebrated formula for the entropy S = kB ln Ω [6]. This resolved the paradox raised by Lord Kelvin (and relayed by Johann J. Loschmidt) that it seemed impossible to deduce irreversible phenomena from microscopic mechanical systems. In 1902, Josiah W. Gibbs formalized and generalized the previous results of J. C. Maxwell and L. Boltzmann without the use of molecular models in the first modern treaty of statistical physics [7]. Indeed, refusing to enter the debate about the very structure of matter, he reformulated statistical mechanics by introducing the concepts of canonical and grand canonical ensembles. Statistical physics was born and it was ready to be generalized to the study of quantum systems. In 1905, the same year he unified mechanics and electromagnetism with the theory of special relativity and proposed the quantization of light, Albert Einstein published an arti- cle [8] devoted to the observable consequences of statistical physics that he considered as a fundamental theory. Phenomena that occur at our scale are more or less direct consequences of underlying mechanisms involving microscopic constituents and their properties that one is entitled to study to get a fundamental understanding of the whole physical world. A. Einstein was the first, together with Marian von Smoluchovski, to understand that the con- tinual and irregular motion of small particles in water (observed first in 1828 by the botanist Robert Brown with pollen particles, then with inorganic materials [9]) is caused by the ther- mal agitation of the water molecules. In his 1905 article, he computed the fluctuations of the Brownian particles and showed that they can be tested experimentally. One year later, Jean B. Perrin conducted a series of refined experiments in which he measured the trajectories and velocities of grains of different sizes and masses in solution. By using A. Einstein’s theory, he showed, that one could obtain a precise estimate of the Avogadro number by dif- ferent methods. His experiments put a definitive end to the controversy around the atomic hypothesis [10]. The first theoretical insight into non-equilibrium statistical physics is due to Lars On- sager who, in 1931, worked out the classical thermodynamics of states very close to equi- librium [11, 12]. He established that the crossed effects in a physical system, for instance the coefficient that relates the heat flux to the pressure gradient and the one that relates the particle flux to the temperature gradient, are equal. These relations are now known as the Onsager reciprocal relations. Herbert B. Callen and Theodore A. Welton proved in 1951 1. This expression of the formula was given by Max K. E. L. Planck in 1900. 3 the so-called fluctuation-dissipation theorem which predicts the non-equilibrium behavior of a system – such as the irreversible dissipation of energy into heat – from its reversible fluctuations in thermal equilibrium [13]. The development of far from equilibrium
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