Effective Evolution Equations from Quantum Dynamics

97 Pages·2016·1.326 MB·English

by  Niels Benedikter, Marcello Porta, Benjamin Schlein (auth.)

SPRINGER BRIEFS IN MATHEMATICAL PHYSICS 7
Niels Benedikter
Marcello Porta
Benjamin Schlein
Effective
Evolution
Equations
from Quantum
Dynamics
123
SpringerBriefs in Mathematical Physics
Volume 7
Series editors
Nathanaël Berestycki, Cambridge, UK
Mihalis Dafermos, Cambridge, UK
Tohru Eguchi, Tokyo, Japan
Atsuo Kuniba, Tokyo, Japan
Matilde Marcolli, Pasadena, USA
Bruno Nachtergaele, Davis, USA
More information about this series at http://www.springer.com/series/11953
Niels Benedikter Marcello Porta
(cid:129)
Benjamin Schlein
Effective Evolution
Equations from Quantum
Dynamics
123
Niels Benedikter Benjamin Schlein
Department ofMathematical Sciences Institute of Mathematics
University of Copenhagen University ofZurich
Copenhagen Zurich
Denmark Switzerland
Marcello Porta
Institute of Mathematics
University of Zurich
Zurich
Switzerland
ISSN 2197-1757 ISSN 2197-1765 (electronic)
SpringerBriefs inMathematical Physics
ISBN978-3-319-24896-7 ISBN978-3-319-24898-1 (eBook)
DOI 10.1007/978-3-319-24898-1
LibraryofCongressControlNumber:2015950862
SpringerChamHeidelbergNewYorkDordrechtLondon
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Preface
In these notes we review the material presented at the summer school on
“Mathematical Physics, Analysis and Stochastics” held at the University of
HeidelberginJuly2014.Weconsiderthetimeevolutionofquantumsystemsandin
particular the rigorous derivation of effective equations approximating the
many-body Schrödinger dynamics in certain physically interesting regimes.
We would like to thank Manfred Salmhofer and Christoph Kopper for orga-
nizing the summer school “Mathematical Physics, Analysis and Stochastics” and
for encouraging us to write up these notes. The work of Marcello Porta and of
Benjamin Schlein has been supported by the ERC grant MAQD-240518. Niels
BenedikterhasbeenpartiallysupportedbytheERCgrantCoMBoS-239694andby
the ERC Advanced grant 321029.
Copenhagen Niels Benedikter
Zurich Marcello Porta
Zurich Benjamin Schlein
v
Contents
1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 Mean-Field Regime for Bosonic Systems . . . . . . . . . . . . . . . . . . . . 7
3 Coherent States Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
4 Fluctuations Around Hartree Dynamics. . . . . . . . . . . . . . . . . . . . . 31
5 The Gross-Pitaevskii Regime. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
6 Mean-Field Regime for Fermionic Systems . . . . . . . . . . . . . . . . . . 57
7 Dynamics of Fermionic Quasi-Free Mixed States. . . . . . . . . . . . . . 79
Appendix A: The Role of Correlations in the Gross-Pitaevskii
Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
vii
Chapter 1
Introduction
Systems of interest in physics and other natural sciences can be described at the
microscopic and the macroscopic level. Microscopically, a system is described in
termsofitselementaryconstituentsandtheirfundamentalinteractions.Whilesucha
descriptionisveryaccurate,itistypicallynotwell-suitedforcomputationsbecause
ofthelargenumberofdegreesoffreedom.Examplesofmicroscopictheoriesinclude
Newton’s theory of classical mechanics, Schrödinger’s quantum mechanics, quan-
tumelectrodynamicsandEinstein’sgeneralrelativity.1Ontheotherhand,amacro-
scopic description of the system does not resolve the constituents and only takes
intoaccounteffectiveinteractions.Itfocusesonmacroscopicallyobservablequan-
titieswhicharisefromthecollectivebehaviorofthesystemandareofinterestfor
the observer. Such a description is less accurate but it is much more accessible to
computations.ExamplesofmacroscopictheoriesareBoltzmann’skinetictheoryof
gases, the Navier-Stokes and the Euler equations of hydrodynamics, the Hartree
and Hartree-Fock theory, the BCS theory of superconductors and superfluids, the
Ginzburg-Landautheory,theGross-PitaevskiitheoryofBose-Einsteincondensation
andtheVlasovtheoryofplasmaphysics.
Because of the great importance of effective macroscopic theories for making
qualitativeandquantitativepredictionsaboutthebehaviorofphysicallyinteresting
systems,akeygoalofstatisticalmechanicsistounderstandtheiremergencefrom
microscopictheoriesinappropriatescalingregimes(alsocalledlimits,eventhough
weoftenthinkoftheparameterasbeinglargebutfinite).Heremathematicalphysics
can and should play a central role to put the effective theories, which are often
1Of course we do not claim that these theories are absolutely fundamental from the view of a
physicist.Itwouldbemorecorrecttoconsiderthemasdifferentlevelsbetweenfundamentaland
effective,andwhichtheorywecalleffectiveandwhichfundamentaldependsonthepairweare
lookingat.Forexample,wecouldalsoconsiderNewtonianmechanicsasamacroscopictheory
arising as an effective theory from quantum mechanics. On the next level we could view non-
relativisticquantummechanicsasaneffectivetheoryarisingfromtheStandardModel.
©TheAuthor(s)2016 1
N.Benedikteretal.,EffectiveEvolutionEquationsfromQuantumDynamics,
SpringerBriefsinMathematicalPhysics,DOI10.1007/978-3-319-24898-1_1
2 1 Introduction
obtained merely by heuristic and phenomenological arguments, on solid grounds
andunderstandtherangeandthelimitsoftheirvalidity.
Letuspresenttwoexamplesofphysicalsystemswhichcanbedescribedbyeffec-
tiveequationsthatcanberigorouslyderivedfrommicroscopictheoriesinappropriate
limits.
Largeatomsandmolecules.Weconsideraquantum-mechanicalsystemofNelec-
tronsand M nucleiofcharges Z1,...,ZM >0lo(cid:2)catedatpositions R1,...,RM ∈
R3.Weassumethesystemtobeneutral,i.e. N = M Z .Forsimplicitywework
i=1 i
intheBorn-Oppenheimerapproximation,i.e.wekeepthenucleifixed(onlytheelec-
tronsaredynamicalparticlesinthisapproximation).Atzerotemperaturethesystem
isinitsgroundstate,withenergy
E(N)= min (cid:4)ψ,H ψ(cid:5) (1.1)
N
ψ∈L2(R3N):(cid:3)ψ(cid:3)=1
a
where H denotestheHamiltonoperator
N
(cid:4) (cid:5)
(cid:3)N (cid:3)M (cid:3)N (cid:3)M
Z 1 Z Z
H = −Δ − i + + i j . (1.2)
N xj |x − R | |x −x | |R − R |
j=1 i=1 j i i<j i j i<j i j
NoticethatH actsonthesubspaceL2(R3N)ofL2(R3N)consistingofallfunc-
N a
tions that are antisymmetric with respect to permutations of the N electrons. (Of
course E(N) and H also depend on M, on the charges Z ,...,Z and on the
N 1 M
positions R ,...,R ). Observe that the last term on the r.h.s. of (1.2) is just a
1 M
constant representing the interaction among the nuclei. Already for N (cid:6) 20 it is
extremelydifficulttocomputethegroundstateenergy E(N)numericallysincethe
eigenvalueequationonehastosolveisapartialdifferentialequationin3N coupled
variables.
ThomasandFermipostulatedalreadyintheearlystagesofquantummechanics
attheendofthe1920sthatthegroundstateenergy E(N)canbeapproximatedby
E(N)(cid:6) E (N)= inf E (ρ) (1.3)
TF TF
ρ≥0,(cid:3)ρ(cid:3)1=N
withtheThomas-Fermifunctional
(cid:6) (cid:6)
3 (cid:3)M ρ(x)
E (ρ)= c ρ5/3(x)dx − Z dx
TF 5 TF i |x − R |
i=1 i
(cid:6)
1 ρ(x)ρ(y) (cid:3)M Z Z
+ dxdy+ i j .
2 |x −y| |R − R |
i<j i j
Noticethatonther.h.s.of(1.3)wearelookingforafunctionρ ∈ L1(R3);asa
consequence,intermsofnumericalcomputations,theminimizationproblem(1.3)
1 Introduction 3
is much simpler than the original problem (1.1) despite the fact that the resulting
Euler-Lagrangeequationsarenonlinear.
In[1]LiebandSimonprovedthattheapproximation(1.3)becomesexactinthe
limitoflargeN.Moreprecisely,theyshowedthat,whileE(N)andE (N)areboth
TF
proportionalto N7/3,
|E(N)−E (N)|≤CN7/3−1/30
TF
foranappropriateconstantC >0.Thisgivesamathematicallyrigorousderivation
oftheThomas-Fermitheory,andittellsushowbigN shouldbeinorderforE (N)
TF
to be a good approximation of the true quantum energy (later, better bounds have
beenobtained[2–4]:oneknowsthattheerrorwithrespecttoThomas-Fermitheory
isoftheorder N2 inthelimitoflarge N).
Kinetictheoryofdilutegases.ConsidernowagasofN classicalparticlesmoving
accordingtoNewton’sequations
x˙ (t)=v (t)
j j
(cid:3)N
v˙ (t)=− ∇V(x (t)−x (t)) (1.4)
j i j
i(cid:9)=j
for j =1,...,N.HereV isashortrange(compactlysupported),regularpotential.
Equation (1.4) is a system of 6N coupled ordinary differential equations. Given
appropriateinitialdata,itisknowntohaveauniquesolutionforallt ∈R.However,
sincethenumberofparticles N istypicallyextremelylarge,itisalmostimpossible
todeducefrom(1.4)interestingqualitativeorquantitativepropertiesofthesolution.
At the beginning of the twentieth century Boltzmann, based on clever heuristic
arguments,proposedtodescribethedynamicsofthegasbythenonlinearequation
∂ f (x,v)+v·∇ f (x,v)
t (cid:6)t (cid:6) x t
(cid:7) (cid:8)
= dv(cid:11) dωB(v−v(cid:11);ω) f (x,v )f (x,v(cid:11) )− f (x,v)f (x,v(cid:11)) (1.5)
t out t out t t
S2
forthephase-spacedensity f (x,v),whichshouldmeasurethenumberofparticles
t
attimet thatarelocatedclosetox ∈R3andhavevelocityclosetov ∈R3.Here
v =v+ω·(v(cid:11)−v)ω
out
v(cid:11) =v(cid:11)−ω·(v(cid:11)−v)ω
out
are the velocity of two particles emerging from the collision of two particles with
velocities v,v(cid:11) and collision vector ω ∈ S2. Boltzmann’s equation is a partial dif-
ferential equation in only six variables, and is therefore much more accessible to
computationsthan(1.4).