Macroscopic Limits of Quantum Systems: Munich, Germany, March 30 - April 1, 2017

263 Pages·2018·3.26 MB·English

by  Daniela Cadamuro

Springer Proceedings in Mathematics & Statistics
Daniela Cadamuro
Maximilian Duell
Wojciech Dybalski
Editors
Sergio Simonella   
Macroscopic
Limits of
Quantum
Systems
Munich, Germany, March 30 - April 1,
2017
Springer Proceedings in Mathematics & Statistics
Volume 270
Springer Proceedings in Mathematics & Statistics
This book series features volumes composed of selected contributions from
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Daniela Cadamuro Maximilian Duell
(cid:129)
Wojciech Dybalski Sergio Simonella
(cid:129)
Editors
Macroscopic Limits
of Quantum Systems
Munich, Germany, March 30 - April 1, 2017
123
Editors
Daniela Cadamuro Wojciech Dybalski
Institut für Theoretische ZentrumMathematik, M5
PhysikUniversitätLeipzig Technische UniversitätMünchen
Leipzig, Germany Garching,Germany
Maximilian Duell SergioSimonella
ZentrumMathematik, M5 UMPAUMR 5669CNRS
Technische UniversitätMünchen Écolenormale supérieure
Garching,Germany Lyon,France
ISSN 2194-1009 ISSN 2194-1017 (electronic)
SpringerProceedings in Mathematics& Statistics
ISBN978-3-030-01601-2 ISBN978-3-030-01602-9 (eBook)
https://doi.org/10.1007/978-3-030-01602-9
LibraryofCongressControlNumber:2018957047
MathematicsSubjectClassification(2010): 82C10,35Q82,81Qxx,32A70,11F72,49-XX
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Preface
“It is amazing how different the world appears on different scales.” Moving from
the subatomic up to the intergalactic world, we cross several physical levels
apparently ruled by very diverse laws. The connection between steps of this hier-
archy is a never-ending source of discoveries and mathematical challenges.
Our incipit is borrowed from the book by Herbert Spohn on the large-scale
dynamics of interacting particles, which isan outstanding reference in the fieldfor
almost 30 years. Nowadays, research activity is intensively focused on quantum
systems. The present volume collects contributions from a number of experts and
aims to be a survey of current directions of investigation. Most of the authors
presented their research during the 3-day workshop “Macroscopic Limits of
Quantum Systems”, held in spring 2017 at the Technical University of Munich,
whichwasattendedbynearly60peoplefromseveralinstitutionsandcountries.The
conference was also an occasion to celebrate the achievements of Herbert and his
reception of the Max Planck Medal.
The main subject of the workshop was the mathematical investigation of col-
lectivephenomenaemergingfromquantumtheoryatobservablescales.Ontheside
of bosonic systems, the problem of Bose–Einstein condensation, ground-state
properties of Bogoliubov Hamiltonians and the Gross–Pitaevskii limit were cov-
ered. Concerning fermionic systems, the rigorous validation of Hartree–Fock
equations and the motion of tracer particles were discussed. Moreover, supercon-
ductivityandmathematicalaspectsoftheBCStheorywerepresented,aswellasthe
statisticalmechanicsoftheuniformelectrongasanditsrelationtoJellium.Further
importanttopicswereinthescopeoftheworkshop:thejustificationandproperties
of quantum Boltzmann-type equations, the theory of indirect measurements, the
strong-coupling limit for the polaron model, Peierls substitution in perturbed
periodic systems, and the adiabatic theorem for many-body systems.
This proceedings volume is a selection of works from leading scientists
including a large part of (regrettably, not all) the speakers of the workshop, and
proposing an overview offields in which the current research is most active. The
bookwillservebothadvancedgraduatestudentsasanintroductiontoresearch,and
specialists in mathematical physics, as it presents technical details in an accessible
v
vi Preface
way. As is well known, to prove rigorous results in quantum physics requires the
development of powerful tools within abstract mathematical theories. Here, we
encounter fruitful connections to functional analysis, spectral theory, statistical
mechanics,renormalizationgrouptechniques,andvariationalcalculus.Thisbookis
also an account of that remarkable effort.
Leipzig, Germany Daniela Cadamuro
Garching, Germany Maximilian Duell
Garching, Germany Wojciech Dybalski
Lyon, France Sergio Simonella
July 2018
Contents
Kinetic Theory and Thermalization of Weakly Interacting
Fermions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Jani Lukkarinen
The BCS Critical Temperature in a Weak External Electric
Field via a Linear Two-Body Operator . . . . . . . . . . . . . . . . . . . . . . . . . 29
Rupert L. Frank and Christian Hainzl
Effective Dynamics of Two Tracer Particles Coupled to a Fermi
Gas in the High-Density Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
Maximilian Jeblick, David Mitrouskas and Peter Pickl
Mean-Field Evolution of Fermions with Singular Interaction. . . . . . . . . 81
Chiara Saffirio
Recent Advances in the Theory of Bogoliubov Hamiltonians . . . . . . . . . 101
Marcin Napiórkowski
Multi-scale Analysis in the Occupation Numbers of Particle States:
An Application to Three-Modes Bogoliubov Hamiltonians. . . . . . . . . . . 123
Alessandro Pizzo
Blow-Up Profile of Rotating 2D Focusing Bose Gases . . . . . . . . . . . . . . 145
Mathieu Lewin, Phan Thành Nam and Nicolas Rougerie
Gross–Pitaevskii Evolution for Bose–Einstein Condensates . . . . . . . . . . 171
Benjamin Schlein
Mean-Field Limits of Particles in Interaction with Quantized
Radiation Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
Nikolai Leopold and Peter Pickl
vii
viii Contents
Taking Inspiration from Quantum-Wave Analogies—Recent
Results for Photonic Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
Max Lein
The Localization Dichotomy for Gapped Periodic Systems and Its
Relevance for Macroscopic Transport . . . . . . . . . . . . . . . . . . . . . . . . . . 243
Gianluca Panati
Kinetic Theory and Thermalization
of Weakly Interacting Fermions
JaniLukkarinen
Abstract Weakly interacting quantum fluids allow for a natural kinetic theory
descriptionwhichtakesintoaccountthefermionicorbosonicnatureoftheinteract-
ingparticles.Inthesimplestcases,onearrivesattheBoltzmann–Nordheimequations
forthereduceddensitymatrixofthefluid.Wediscussheretworelatedtopics:the
kinetictheoryofthefermionicHubbardmodel,inwhichconservationoftotalspin
resultsinanadditionalVlasov-typetermintheBoltzmannequation,andtherelation
betweenkinetictheoryandthermalization.
1 Introduction
Kinetictheorydescribesmotionwhichistransportdominatedinthesensethattyp-
icallythesolutionstothekineticequationscorrespondtoconstantvelocity,i.e.bal-
listic,motioninterceptedbycollisionswhosefrequencyisorderoneonthekinetic
space–time scales. Weakly interacting quantum fluids provide one such example
system,asdiscussedindetailin[1].
Wefocushereononeparticularcaseofaweaklyinteractingquantumfluid,the
caseofweaklyinteractingfermionshoppingonalattice.Suchamodelwouldarise
physicallyasadescriptionofafluidofelectronsinacrystalbackgroundpotential.
For our purposes, this model has also other attractive properties, namely, it has
interestingnon-trivialkinetictheorywithrelativelyfewtechnicalandmathematical
difficulties.Muchofthediscussionbelowcanbestraightforwardlyadaptedtobosonic
lattice systems, at least for initial data which exclude formation of Bose–Einstein
condensate. For more details about such extensions, we refer to [1]; for instance,
Remark 2.3 summarises the changes and new properties which arise for fermions
andbosonsmovingnotonalattice,butinthecontinuumR3.
B
J.Lukkarinen( )
DepartmentofMathematicsandStatistics,UniversityofHelsinki,P.O.Box68,
00014Helsinginyliopisto,Finland
e-mail:jani.lukkarinen@helsinki.fi
©SpringerNatureSwitzerlandAG2018 1
D.Cadamuroetal.(eds.),MacroscopicLimitsofQuantumSystems,
SpringerProceedingsinMathematics&Statistics270,
https://doi.org/10.1007/978-3-030-01602-9_1