Quantum transport : modelling analysis and asymptotics : lectures given at the C.I.M.E. Summer School held in Cetraro, Italy, September 11-16, 2006 / by Grégoire Allaire ... [and others] ; editors, Naoufel Ben Abdallah, Giovanni Frosali.

Format:

Book

Conference/Event

Contributor:

Allaire, Grégoire.

Ben-Abdallah, Naoufel.

Frosali, Giovanni.

Conference Name:

C.I.M.E. Session "Quantum Transport: Modelling Analysis and Asymptotics" (2006 : Cetraro, Italy)

Series:

Lecture notes in mathematics (Springer-Verlag) ; 1946.

Lecture notes in mathematics, 0075-8434 ; 1946

Language:

English

Subjects (All):

Quantum electrodynamics--Mathematical models--Congresses.

Quantum electrodynamics.

Evolution equations--Asymptotic theory--Congresses.

Evolution equations.

Quantum theory--Mathematical models--Congresses.

Quantum theory.

Transport theory--Mathematical models--Congresses.

Transport theory.

Transport theory--Mathematical models.

Quantum theory--Mathematical models.

Evolution equations--Asymptotic theory.

Mathematical models.

Genre:

Conference papers and proceedings.

Physical Description:

xiv, 251 pages : illustrations ; 24 cm.

Place of Publication:

Berlin : Springer, [2008]

Summary:

The CIME Summer School held in Cetraro, Italy, in 2006 addressed researchers interested in the mathematical study of quantum transport models. In this volume, a result of the above mentioned Summer School, four leading specialists present different aspects of quantum transport modelling. Allaire introduces the periodic homogenization theory, with a particular emphasis on applications to the Schrodinger equation. Arnold focuses on several quantum evolution equations that are used for quantum semiconductor device simulations. Degond presents quantum hydrodynamic and diffusion models starting from the entropy minimization principle. Hou provides the state-of-the-art survey of the multiscale analysis, modelling and simulation of transport phenomena. The volume contains accurate expositions of the main aspects of quantum transport modelling and provides an excellent basis for researchers in this field.

Contents:

Periodic Homogenization and Effective Mass Theorems for the Schrodinger Equation / Gregoire Allaire 1

2 Asymptotic Expansions in Periodic Homogenization 2

3 Two-Scale Convergence 7

4 Application to Homogenization 11

5 Bloch Waves 13

6 Schrodinger Equation in Periodic Media 19

7 Semiclassical Analysis and WKB Ansatz 20

8 Homogenization Without Drift 23

9 Generalization with Drift 30

10 Homogenized System of Equations 33

11 Localization 36

Mathematical Properties of Quantum Evolution Equations / Anton Arnold 45

1 Quantum Transport Models for Semiconductor Nano-Devices 47

1.1 Quantum Waveguide with Adjustable Cavity 48

1.2 Resonant Tunneling Diode 52

2 Linear Schrodinger Equation 59

2.1 Free Schrodinger Group 59

2.2 Smoothing Effects and Gain of Integrability in R[superscript N] 60

2.3 Potentials, Inhomogeneous Equation 61

2.4 Strichartz Estimates 63

3 Schrodinger-Poisson Analysis in R[superscript 3] 64

3.1 H[superscript 1]-Analysis 65

3.2 L[superscript 2]-Analysis 66

3.3 Schrodinger-Poisson Systems 69

4 Density Matrices 70

4.1 Framework, Trace Class Operators 70

4.2 Macroscopic Quantities 71

4.3 Time Evolution of Closed/Hamiltonian Systems 74

4.4 Von Neumann-Poisson Equation in R[superscript 3] 76

5 Wigner Function Models 78

5.1 Wigner Functions 78

5.2 Linear Wigner-Fokker-Planck: Well-Posedness 80

5.3 Linear Wigner-Fokker-Planck: Large Time Behavior 82

5.4 Wigner-Poisson-Fokker-Planck: Global Solutions in R[superscript 3] 85

6 Open Quantum Systems in Lindblad Form 90

6.1 Lindblad Form 91

6.2 Quantum Fokker-Planck Equation 96

7 Wigner Boundary Value Problems 100

7.1 1D Stationary Boundary Value Problem 100

7.2 Exponential Convergence to Steady State 104

Quantum Hydrodynamic and Diffusion Models Derived from the Entropy Principle / Pierre Degond, Samy Gallego, Florian Mehats, Christian Ringhofer 111

2 Quantum Kinetic Equations: An Introduction 112

2.1 Quantum Statistical Mechanics of Nonequilibrium Systems 112

2.2 N-Particle Quantum System 117

2.3 Quantum Methods: A Brief and Incomplete Summary 120

2.4 Hydrodynamic Limits: A Review 122

3 Quantum Hydrodynamic Models Derived from the Entropy Principle 124

3.1 Quantum Setting 124

3.2 QHD via Entropy Minimization 125

3.3 Quantum Isothermal Euler Model 133

4 Quantum Diffusion Models 144

4.1 Quantum Energy-Transport Model 144

4.2 Quantum Drift-Diffusion Model 152

Multiscale Computations for Flow and Transport in Heterogeneous Media / Yalchin Efendiev, Thomas Y. Hou 169

2 Review of Homogenization Theory 171

2.1 Homogenization Theory for Elliptic Problems 171

2.1.1 Special Case: One-Dimensional Problem 172

2.1.2 Multiscale Asymptotic Expansions 173

2.1.3 Justification of Formal Expansions 175

2.1.4 Boundary Corrections 176

2.2 Convection of Microstructure 176

2.2.1 Nonlocal Memory Effect of Homogenization 178

3 Numerical Upscaling Based on Multiscale Finite Element Methods 179

3.1 Multiscale Finite Element Methods for Elliptic PDEs 181

3.2 Convergence Analysis of MsFEM 182

3.2.1 Error Estimates (h < [epsilon]) 183

3.2.2 Error Estimates (h > [epsilon]) 185

3.3 The Oversampling Technique 188

3.4 Performance and Implementation Issues 189

3.4.1 Cost and Performance 190

3.4.2 MsFEM for Problems with Scale Separation 191

3.4.3 Convergence and Accuracy 191

3.5 Brief Overview of Mixed Finite Element and Finite Volume Element Methods 192

3.5.1 Control Volume Multiscale Finite Element Method 192

3.5.2 Mixed Multiscale Finite Element Methods 194

3.6 Applications 195

3.6.1 Flow in Porous Media 195

3.6.2 Fine Scale Recovery 198

3.6.3 Scale-Up of One-Phase Flows 200

3.7 MsFEM Using Limited Global Information 205

3.7.1 Motivation 205

3.7.2 Modified Multiscale Finite Volume Element Method Using Limited Global Information 210

3.7.3 Mixed Multiscale Finite Element Methods 211

3.7.4 Numerical Results 212

3.7.5 Galerkin Finite Element Methods with Limited Global Information 215

3.7.6 Extensions of Galerkin Finite Element Methods with Limited Global Information 220

3.7.7 Mixed Finite Element Methods with Limited Global Information 222

4 Multiscale Finite Element Methods for Nonlinear Partial Differential Equations 223

4.1 Multiscale Finite Volume Element Method (MsFVEM) 225

4.2 Examples of V[subscript epsilon superscript h] 226

4.3 Convergence of MsFEM for Nonlinear Partial Differential Equations 227

4.4 Multiscale Finite Element Methods for Nonlinear Parabolic Equations 228

4.5 Numerical Results 230

4.6 Generalizations of MsFEM and Some Remarks 235

5 Multiscale Simulations of Two-Phase Immiscible Flow in Adaptive Coordinate System 237.

Notes:

Includes bibliographical references.

ISBN:

9783540795735

3540795731

354079574X

9783540795742

OCLC:

225449610