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Lecture Notes in Physics 967 Yves Pomeau Minh-Binh Tran Statistical Physics of Non Equilibrium Quantum Phenomena Lecture Notes in Physics Volume 967 FoundingEditors WolfBeiglböck,Heidelberg,Germany JürgenEhlers,Potsdam,Germany KlausHepp,Zürich,Switzerland Hans-ArwedWeidenmüller,Heidelberg,Germany SeriesEditors MatthiasBartelmann,Heidelberg,Germany Roberta Citro, Salerno, Italy PeterHänggi,Augsburg,Germany MortenHjorth-Jensen,Oslo,Norway MaciejLewenstein,Barcelona,Spain AngelRubio,Hamburg,Germany ManfredSalmhofer,Heidelberg,Germany WolfgangSchleich,Ulm,Germany StefanTheisen,Potsdam,Germany JamesD.Wells,AnnArbor,MI,USA GaryP.Zank,Huntsville,AL,USA The Lecture Notes in Physics The series Lecture Notes in Physics (LNP), founded in 1969, reports new devel- opmentsin physicsresearch and teaching-quicklyand informally,but with a high qualityand the explicitaim to summarizeand communicatecurrentknowledgein anaccessibleway.Bookspublishedinthisseriesareconceivedasbridgingmaterial between advanced graduate textbooks and the forefront of research and to serve threepurposes: • to be a compact and modern up-to-date source of reference on a well-defined topic • to serve as an accessible introduction to the field to postgraduate students and nonspecialistresearchersfromrelatedareas • to be a source of advanced teaching material for specialized seminars, courses andschools Bothmonographsandmulti-authorvolumeswillbeconsideredforpublication. Editedvolumesshould however consistofaverylimitednumberofcontributions only.ProceedingswillnotbeconsideredforLNP. VolumespublishedinLNParedisseminatedbothinprintandinelectronicfor- mats,theelectronicarchivebeingavailableatspringerlink.com.Theseriescontent isindexed,abstractedandreferencedbymanyabstractingandinformationservices, bibliographicnetworks,subscriptionagencies,librarynetworks,andconsortia. Proposalsshouldbe sent to a memberof the EditorialBoard, ordirectly to the managingeditoratSpringer: Dr LisaScalone SpringerNature PhysicsEditorialDepartment Tiergartenstrasse17 69121Heidelberg,Germany [email protected] Moreinformationaboutthisseriesathttp://www.springer.com/series/5304 Yves Pomeau • Minh-Binh Tran Statistical Physics of Non Equilibrium Quantum Phenomena 123 Yves Pomeau Minh-Binh Tran LadHyX Department of Mathematics École Polytechnique Southern Methodist University Palaiseau Cedex, France Dallas, TX, USA ISSN 0075-8450 ISSN 1616-6361 (electronic ) Lecture Notes in Physics ISBN 978-3-030-34393-4 ISBN 978-3-030-34394-1 (eB ook) https://doi.org/10.1007/978-3-030-34394-1 Mathematics Subject Classification (2010): 82C40, 82C10 © Springer Nature Switzerland AG 2019 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland Foreword There are a number of excellent books on the kinetic theory of classical and quantum gases, but this book is unique. The focus of the first part of the book is a kinetic theory that describes the interaction of single atoms with a radiation field. It deals with radiation from two- and three-level atoms and the statistical theory of shelving. Shelving refers to a situation where an excited atomic state, with a very long lifetime, can shut off the emission of light in a decay channel of another excited state. The focus of the second part of the book is the kinetic theory of Bose– Einstein condensates (BEC) at ultra-low temperatures where the excitation spectrum is phonon-like, rather than particle-like. This brings a new type of contribution to collision processes in the BEC. The kinetic theory used to describe the hydrodynamic processes in the ultra-low temperature BEC uses an expansion of distribution functions in powers of the wave vector of hydrodynamic processes (an approach first suggested by Résibois). This approach to the kinetic theory of the BEC provides a very elegant way to compute the transport processes in the BEC. Theauthorsarewellqualifiedtowritethisbook.YvesPomeauhasalong history of exemplary (and award winning) work in kinetic theory, and Minh- BinhTran isamathematicianwhose work hasfocusedon thederivationand mathematical properties of kinetic equations. The book provides a summary of the applications of kinetic theory to atomic systems interacting with radiation fields, and excitations interacting with very low temperature coherent phases of matter. The topics discussed in this book are not discussed in a similar manner elsewhere, and therefore the book provides an important ad- dition to the literature on the kinetic theory of matter-radiation interaction and the kinetic theory of BECs. Professor of Physics Linda Reichl Co-Director of the Center for Complex Quantum Systems University of Texas at Austin, USA v Preface ThetransitionfromNewtonianmechanicstoquantummechanicsintheearly years of the twentieth century was a major step in the progress of our understanding of the world. This transition involved more than just a change of equations – it led to a deep change in our understanding of the limits of human knowledge. It included from the very beginning a statistical interpre- tation of the theory. Statistical concepts were originally introduced to provide a classical (i.e. not quantum-theoretic) description of complex systems with many degrees of freedom, such as a volume of fluid including a very large number of molecules. These large systems cannot be fully described and/or predicted since no human being has enough computational power to solve Newton’s equations1 where the initial data (position and velocity) involve too many particles. In the limit of a dilute gas Boltzmann (cf. [41, 42]) found the right theory for describing the evolution of a large number of particles interacting by short range two-body forces. In this theory, kinetic equations are a primary tool. Kinetic equations modelize systems made up of a large number of particles (gases, plasma, etc.) by a distribution function in the phase space of particles, based on the modeling assumption that there are so many particles that the whole system can be treated as a continuum. The phase space under investigation includes not only microscopic variables, which describe the state of the particles, but also macroscopic variables, which depend on time and on position in physical space. As a consequence, kinetic equations accuratelydescribe thedynamicsof dilute gasesand provideconnections between macroscopic observable and microscopic properties. Kinetic equations take into account collisions and more generally interactions between particles, all phenomena which change their momentum and kinetic energy. After 1Wereluctantlyusetheword“equations”herebecauseNewtonneverwrotedownordinary differential equations of classical mechanics in the modern sense. As is well known, he solveddynamicalproblemsbyusingelegantgeometricalmethods,replacingwhatwenow callcalculus.Forinstance,thesolutionofthetwo-bodyproblemwithageneralspherical potentialwasobtainedbyreplacingintegralsbycalculationsofarea. vii viii Preface a collision, on one hand, the number of particles with the previous momentum will decrease; on the other hand, the number of particles with the new momentumwillincrease.Inordertomodelizetheeffectsofparticlecollisions on the probability distribution of momentum, one includes in the equations forthisprobabilitytermsthatareintegralsoverthedistributionfunction,as wasdonebyBoltzmannfordilutegases.Thesenewtermsarecalledcollision integrals, which introduce conserved quantities that are used to construct hydrodynamic balance equations and parameterize the equilibrium distribution. Based on collision integrals, we can obtain the rates of dissipation and relaxationandexpressionsforthetransportcoefficientsinthelimitofaslow dependence of the hydrodynamic parameters following the general method invented by Enskog and Hilbert [85]. Quantumkinetictheoryisdevelopedinthesecondpartofthisbook,which is devoted to the extension of Boltzmann’s classical (non-quantum) kinetic theory of a dilute gas of quantum bosons. This is the source of many in- teresting fundamental questions, particularly because, if the temperature is low enough, such a gas is known to have at equilibrium a transition, the Bose–Einstein transition, where a finite portion of the particles stay in the quantum ground state. Chapter 9 is devoted to the Bose–Einstein transition at equilibrium in the limit of a small, but non-vanishing interaction between particles. This shows in particular that the transition is first order although it is second order without interaction. The existence of this transition obvi- ouslyraisesthequestionofthewaythis“condensate”growsdynamicallyfrom zero as the temperature is lowered and how it interacts with itself and with the rest of the particles. To be more precise, attempts to extend the Boltz- mann equation, to study quantum gases, using arguments similar to those used to derive the classical Boltzmann equation, resulted in the Boltzmann– Nordheim kinetic equation. Nordheim derived [125] the kinetic equations for a dilute gas of bosons and of fermions by using methods similar to those of Boltzmann. In the literature the equation for bosons is sometimes called the Uehling–Uhlenbeck equation [159]. This equation is for the dynamics of the momentum distribution as the Wigner transform of the one-particle density matrix.BelowtheBose–EinsteintransitiontemperatureT ,whichdepends BE on the density of the gas, this Wigner function becomes singular at zero momentum,whichcorrespondstoafinitefractionofparticlesinthecondensate. Therefore, one needs to modify the Boltzmann–Nordheim (B-N) equation to take into account this singular part in the Wigner transform. For a homogeneous system, deriving a coupled system of equations for the condensate fraction and the excited states is not too difficult. On the other hand, the non-homogeneous case is more subtle: at zero temperature the dynamics of a weakly interacting Bose gas is described by the Gross–Pitaevskii (G-P) equation when out of equilibrium (note that we use the word “temperature” a bit loosely since it does not strictly have a meaning in ‘out of equilibrium’ situations–itwouldbemoreaccuratetoreplaceitbyanaverageenergyper atom). Therefore, for a dilute Bose gas at finite temperature (a temperature between absolute zero 0K and the BE transition temperature), the system is Preface ix described by a coupled system of B-N and G-P equations. In such a system of trapped Bose gases at finite temperature, the quantum Boltzmann equation corresponds to the evolution of the density distribution function of the thermal cloud and the G-P is the equation of the condensate. However, this is not the only case where a kind of kinetic theory is neces- sary to describe the dynamics of a quantum system. In its early days quantum mechanics dealt almost exclusively with the interaction between atoms or molecules and the electromagnetic (EM) field. Even with a single isolated atom this introduces a system with infinitely many degrees of freedom, the modes of the radiated EM field if the atom is, as is often the case, in un- bounded vacuum or at least in a big enough box. Following recent work by one of the authors (YP) with Martine Le Berre and Jean Ginibre, we introduce in the first part of this book a kinetic equation, of Kolmogorov type, which is needed to describe a rather standard physical situation, namely an isolated atom (actually, in experiments, an ion) under the effect of a classical pumping EM field which keeps it in its excited state(s) together with the random emission of fluorescence photons putting this atom back into its groundstate.Whenreducedtoatwo-levelsystem,thisyieldsarathersimple Kolmogorovkineticequationdescribingthefluctuationsofthequantumstate of the atom, whereas the case of a three-level system is far more complex in general and will be dealt with in a limit pertinent to some experiments. This book provides an introduction to these systems for both mathemati- cians and theoretical physicists who are interested in the topic. Yves Pomeau Minh-Binh Tran Acknowledgements Y. Pomeau thanks Martine Le Berre and Jean Ginibre for many discussions and their invaluable help. M.-B. Tran would like to thank Ricardo Alonso, Alejandro Aceves, Stanislav Boldyrev, Gheorghe Craciun, J. Robert Dorfman, Miguel Escobedo, Pierre Germain, Alexandru D. Ionescu, Shi Jin, SergeyNazarenko,AlanC.Newell,ToanT.Nguyen,LeslieM.Smith,Benno Rumpf, Avy Soffer, Herbert Spohn and Eitan Tadmor for their important help and fruitful discussions. He would especially like to express his grati- tudetoLindaE.ReichlandEnriqueZuazuafortheirilluminatingadviceand careful reading of the manuscript, and to Irene M. Gamba for her great sup- port over the years. M.-B. Tran is supported by NSF Grants DMS-1814149, DMS-1854453 and a Sam Taylor Fellowship. xi