Introduction to computational methods in many body physics:
Gespeichert in:
| Format: | Buch |
|---|---|
| Sprache: | English |
| Veröffentlicht: |
Paramus, N.J.
Rinton Press
2006
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XV, 400 S. Ill., graph. Darst. |
| ISBN: | 1589490096 9781589490093 |
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| 008 | 060317s2006 ad|| |||| 00||| eng d | ||
| 020 | |a 1589490096 |9 1-58949-009-6 | ||
| 020 | |a 9781589490093 |9 978-1-58949-009-3 | ||
| 035 | |a (OCoLC)65175442 | ||
| 035 | |a (DE-599)BVBBV021515508 | ||
| 040 | |a DE-604 |b ger |e rakwb | ||
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| 049 | |a DE-19 |a DE-703 |a DE-83 | ||
| 050 | 0 | |a QC174.17.P7 | |
| 084 | |a SK 955 |0 (DE-625)143274: |2 rvk | ||
| 245 | 1 | 0 | |a Introduction to computational methods in many body physics |c Eds.: Michael Bonitz ... |
| 264 | 1 | |a Paramus, N.J. |b Rinton Press |c 2006 | |
| 300 | |a XV, 400 S. |b Ill., graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 650 | 4 | |a Many-body problem |x Numerical solutions | |
| 700 | 1 | |a Bonitz, Michael |d 1960- |e Sonstige |0 (DE-588)120375176 |4 oth | |
| 856 | 4 | 2 | |m Digitalisierung UB Bayreuth |q application/pdf |u http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014732071&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |3 Inhaltsverzeichnis |
| 999 | |a oai:aleph.bib-bvb.de:BVB01-014732071 | ||
Datensatz im Suchindex
| _version_ | 1804135254035791872 |
|---|---|
| adam_text | Contents
Authors
v
Preface
vii
Chapter
1
Introduction
1
1.1
Preliminary remarks
................................. 1
1.2
Density operator.
Von
Neumann equation
..................... 1
1.3
Solution of the
von Neumann/Liouville
equation
................. 4
1.4
BBGKY-hierarchy
.................................. 4
1.4.1
Reduced density operators. Equations of motion
............. 4
1.5
Basic representations of the hierarchy
....................... 6
1.5.1
Coordinate representation
.......................... 6
1.5.2
Wigner representation
............................ 8
1.5.3
Classical kinetic equations. Quantum corrections
............. 10
1.5.4
Spatially homogeneous systems. Momentum representation
....... 11
1.6
Relation to equilibrium correlation functions
................... 14
References
.......................................... 16
PART I Dynamics of Classical Many-particle Systems
17
Chapter
2
Classical Particle Simulations
19
Hartmut Ruhl
2.1
Synopsis
........................................ 19
2.2
Introduction
...................................... 20
2.3
The physics model
.................................. 25
2.3.1
Governing equations
............................. 25
2.3.2
The Boltzmann collision operator
...................... 26
2.3.3
The collision integral for immobile ions
.................. 29
2.4
The numerical approach
............................... 30
2.4.1
Normalization
................................ 31
x
Contents
2.4.2
Maxwell s equations
............................. 32
2.4.2.1
The FDTD scheme
........................ 32
2.4.2.2
Periodic boundary conditions
.................. 34
2.4.2.3
Radiating boundary conditions
................. 34
2.4.3
The
Vlasov
equation
............................. 39
2.4.3.1
The distribution function
..................... 39
2.4.3.2
Equations of motion
....................... 40
2.4.3.3
Periodic boundary conditions
.................. 44
2.4.3.4
Reflecting boundary conditions
................. 45
2.4.4
The
Vlasov-
Boltzmann equation
...................... 46
2.4.4.1
Equations of motion
....................... 46
2.4.4.2
The collisional model
....................... 48
2.4.4.3
Scattering angles for Rutherford scattering
........... 52
2.4.4.4
Relativistic binary kinematics
.................. 53
2.4.4.5
Required time resolution and grid size
............. 55
2.4.5
Currents
.................................... 55
2.4.5.1
Mass distribution of a quasi-particle
............... 55
2.4.5.2
The current conserving scheme
................. 56
2.4.6
Energy conservation
............................. 59
2.5
The simulation code PSC
.............................. 60
2.5.1
Details of the code
.............................. 61
2.5.1.1
Name conventions for important fields
............. 61
2.5.1.2
The modules of the PSC
..................... 61
2.5.1.3
Time progression in the PSC
................... 65
2.5.1.4
TCSH scripts for data processing
................ 65
2.5.1.5
IDL scripts
............................. 66
2.5.1.6
PBS batch scripts
......................... 68
2.5.2
Required hardware and software
...................... 68
2.5.2.1
Server hardware
.......................... 69
2.5.2.2
Operating system
......................... 69
2.5.2.3
Fortran compilers
......................... 69
2.5.2.4
Message passing software
..................... 69
2.5.2.5
Graphics software
......................... 69
2.5.2.6
The batch system
......................... 69
2.6
Examples
....................................... 70
2.6.1
Basics of nonlinear plasma optics
...................... 71
2.6.1.1
Simulations of laser propagation in vacuum
.......... 77
2.6.1.2
Simulations of relativistic self-focusing in 2D
.......... 79
2.6.2
Wakefields in plasma
............................. 82
2.6.2.1
Wake field simulations in 2D
................... 93
2.6.2.2
Simulation of self-
modulatio
n
of laser pulses in 2D
...... 97
2.6.3
Aspects of plasma absorption
........................ 100
2.6.3.1
A kinetic model for sharp edged plasma
............ 102
Contents xi
2.6.3.2
A fluid model for sharp edged plasma
.............. 106
2.6.3.3
Simulation of laser-matter interaction under oblique incidence
107
2.6.3.4
Absorption at high laser intensities
............... 110
2.7
Summary
.......................................
Ill
2.8
The open source project PSC
............................ 112
References
.......................................... 113
PART II Correlated Quantum Systems in Equilibrium and Non-
equilibrium
121
Chapter
3
Density Functional Theory
123
George F. Bertsch and Kazuhiro Yabana
3.1
Introduction
......................................123
3.1.1
Units and notation
..............................124
3.1.2
Hartree-Fock theory
.............................125
3.1.3
Homogeneous electron gas
..........................127
3.1.3.1
Free electrons
...........................127
3.1.3.2
Exchange energy
.........................128
3.2
What is density functional theory?
.........................129
3.2.1
Hohenberg-Kohn theorem
..........................129
3.2.2
A simple example: the Thomas-Fermi theory
...............131
3.2.2.1
Variational equation of Thomas-Fermi theory
.........131
3.2.2.2
Thomas-Fermi atom
.......................132
3.2.2.3
An example
............................133
3.3
Kohn-Sham theory
..................................134
3.3.1
Local density approximation
........................135
3.3.2
Spin and the local spin density approximation
..............136
3.3.3
The generalized gradient approximation
..................136
3.4
Numerical methods for the Kohn-Sham equation
.................138
3.4.1
Exact Exchange
...............................141
3.4.2
O(N) methods
................................141
3.5
Some applications and limitations of DFT
.....................142
3.5.1
Two examples of condensed matter
.....................143
3.5.2
Vibrations
...................................144
3.5.3
NMR chemical shifts
.............................144
3.6
Limitations of DFT
.................................146
3.7
Time-dependent density functional theory: the equations
............147
3.7.1
Optical properties
..............................148
3.7.1.1
/-sum rule
.............................149
3.7.2
Methods to solve the TDDFT equations
..................151
3.7.2.1
Linear response formula
.....................153
3.7.3
Dynamic polarizability
............................153
3.7.4
Dielectric function
..............................154
xii Contents
3.8 TDDFT:
numerical aspects
.............................156
3.8.1
Configuration matrix method
........................156
3.8.2
Linear response method
...........................158
3.8.3
Sternheimer method
.............................158
3.8.4
Real time method
..............................158
3.9
Applications of TDDFT
...............................160
3.9.1
Simple metal clusters
............................ 161
3.9.2
Carbon structures
.............................. 163
3.9.3
Diamond
................................... 164
3.9.4
Other applications
.............................. 164
3.9.5
Limitations
.................................. 166
References
.......................................... 167
Chapter
4
Generalized Quantum Kinetic Equations
171
Dirk Semkat and Michael Bonitz
4.1
Introduction
......................................171
4.2
Idea of second quantization
.............................175
4.3
Real-time Green s functions
.............................179
4.3.1
Basic definitions
...............................179
4.3.2
Calculation of macrophysical quantities
..................181
4.4
Derivation of the Kadanoff-Baym/Keldysh equations
...............182
4.4.1
The Marti
η
-Schwinger
hierarchy*
.....................183
4.4.2
Decoupling of the hierarchy*
........................185
4.4.3
Kadanoff-Baym/Keldysh equations (KBE)
................187
4.4.4
Determination of the self-energy
......................190
4.4.4.1
Mean-field effects. Hartree-Fock (HF) approximation
.....191
4.4.4.2
Particle scattering. Second Born approximation
........192
4.5
Single-time kinetic equations
............................193
4.5.1
Time-diagonal Kadanoff-Baym equation
..................193
4.5.2
Reconstruction problem and spectral function
...............194
4.5.3
Quantum Landau equation
.........................194
4.6
Numerical procedure
.................................197
4.6.1
Strategy of solving a kinetic equation
...................197
4.6.2
Solving the KBE. Overview
.........................198
4.6.3
Program structure
..............................200
4.6.4
Discretization
.................................202
4.6.5
Computation of the self-energies
......................203
4.6.5.1
Self-energies in second Born approximation
...........203
4.6.5.2
Hartree-Fock self-energy
.....................205
4.6.6
Computation of the time integrals
.....................205
4.6.7
Solution of the differential equation
....................206
4.6.8
Reconstruction
oí
g^
.............................207
4.7
Numerical Results
............................... 207
Contents xiii
4.7.1 Model
system
.................................207
4.7.2 Evolution
of the correlation functions and of statistical quantities
.... 207
4.7.3
Evolution of the spectral function
.....................213
4.7.4
Influence of initial correlations*
.......................218
4.7.5
Multi-component systems
..........................219
4.7.6
Systems containing heavy particles
.....................220
4.8
Interband Kadanoff-Baym
equations
........................222
4.8.1
The multi-band KBE. Application to semiconductors
..........222
4.8.2
Application to plasma oscillations
.....................223
4.9
Survey of numerical applications to other systems
.................227
Appendix: Self-energy in
Т
-matrix approximation
...................229
References
..........................................231
PART III First-principle Approaches to Correlated Quantum
Systems
235
Chapter
5
Classical and Quantum Monte Carlo
237
Alexei Filinov and Michael Bonitz
5.1
Classical systems and the Monte Carlo method
..................237
5.1.1
Introduction
.................................237
5.1.2
Monte Carlo integration
...........................238
5.1.2.1
Random quantities
........................238
5.1.2.2
Statistical tests
..........................240
(a) The Kendall tests
.....................241
5.1.2.3
Practical applications of statistical tests
............244
5.1.2.4
Monte Carlo integration
.....................257
(a) Straightforward sampling
................257
(b) Error of
MC
integration
.................261
(c) Efficiency criteria of the
MC
method
..........263
(d) Importance sampling
...................264
5.1.3
Practical realizations of Monte Carlo integration
.............266
5.1.4
Monte Carlo integration in statistical physics
...............272
5.1.4.1
Observables
in statistical mechanics
...............272
5.1.4.2
Metropolis method. Markov chain
................274
5.1.5
Statistical ensembles
.............................276
5.1.5.1
Canonical ensemble
........................276
5.1.5.2
Microcanonical ensemble
.....................277
5.1.5.3
Grand canonical ensemble
....................279
5.1.6
Practical applications of the Metropolis algorithm
............280
5.1.6.1
Simulations of the 2D Ising Model (canonical ensemble)
. . . 280
5.1.6.2
2D Lennard-Jones fluid
......................290
5.2
Path Integral Monte Carlo
..............................294
5.2.1
Density matrix and group property
.....................294
xiv
Contents
5.2.1.1
Quantum statistics. Spin effects
................. 298
5.2.1.2
Fermion sign problem
....................... 300
5.2.1.3
Mapping onto an effective classical system
.......... 300
5.2.1.4
Generalized Metropolis algorithm
................ 302
5.2.1.5
Monte Carlo moves in PIMC
................. 304
(a) Displacement of a whole particle
............304
(b) Single slice moves
.....................304
(c) Multilevel moves
.....................305
(d) Sampling of permutations
................308
(e) Acceptance Ratio
.....................312
(f) Summary.
........................312
5.2.1.6
Monte Carlo algorithm for PIMC simulations
.........313
5.2.2
Improved high-temperature N—particle density matrix
..........320
5.2.2.1
Calculation of the pair density matrix
..............322
(a) Matrix squaring technique
................322
(b) Variational approach for the density matrix
......323
(c) First order perturbation expansion for the pair den¬
sity matrix. Off-diagonal and diagonal Kelbg potential.
324
5.2.3
Calculation of physical
observables
.....................327
5.2.4
Simulations of macroscopic systems. Finite-size effects
..........330
5.2.4.1 Ewald
transformation
.......................330
5.2.4.2
Pseudopotentials in quantum simulations
............332
5.2.4.3
Elimination of finite-size errors
.................334
(a) Twist averaged boundary conditions
..........335
(b) Periodic Coulomb interaction
..............335
(c) Finite-size scaling
.....................336
5.2.5
Applications of PIMC
............................337
5.2.5.1
2D Coulomb clusters
.......................337
5.2.5.2
Binding energies of
excitons, trions,
biexcitons in semiconduc¬
tor nanostructures
.........................341
5.2.6
Conclusion and outlook
...........................345
References
..........................................346
Chapter
6
Quantum Molecular Dynamics
351
Alexei Filinov, Vladimir Filinov, Yurii Lozovik and Michael Bonitz
6.1
Introduction
...................................... 351
6.2
Quantum distribution functions
........................... 352
6.2.1
Wigner function
............................... 354
6.2.2
Husimi function
............................... 355
6.2.3
Evolution equation
.............................. 355
6.3
Quantum dynamics I: Method of Wigner trajectories
............... 356
6.3.1
Wigner representation
............................ 356
6.3.2
Wigner trajectories
.............................. 358
Contents xv
6.3.3
Inclusion
of quantum
statistics
.......................359
6.3.4
Simulation algorithm
.............................360
6.3.5
Application: Tunneling of interacting identical particles
.........361
6.3.5.1
Theoretical model
.........................363
6.3.5.2
Initial parameters
.........................364
6.3.5.3
Reaction probabilities
.......................365
6.3.5.4
Tunneling times
..........................367
6.3.5.5
Conclusion
.............................368
6.4
Quantum dynamics II: Monte Carlo approach to ensembles of quantum trajec¬
tories
.........................................368
6.4.1
Integral form of the Wigner-
Liou
ville
equation
..............368
6.4.2
Method of characteristics
..........................369
6.4.3
Quantum corrections of higher order
.................... 372
6.4.4
Calculation of linear functionals of the Wigner function by Monte Carlo
sampling
...................................373
6.4.4.1
Algorithm for generation of an ensemble of quantum trajectories374
6.4.5
Calculation of averages of physical
observables
..............375
6.5
Wigner function in the canonical ensemble
.....................376
6.5.1
Time correlation functions
..........................376
6.5.2
Initial conditions for the Wigner-Liouville equation
............377
6.5.3
Integral equations
..............................379
6.5.4
Quantum dynamics
.............................380
6.5.5
Application: Interacting electrons in
а о
ne-
dimensional random array
of scatterers. Anderson localization
.....................384
6.5.5.1
Theoretical model and numerical aspects
............384
6.5.5.2
Numerical results
.........................387
6.5.5.3
Temporal quantum momentum-momentum correlation func¬
tions
................................387
6.5.5.4
Fourier transform of the momentum-momentum time corre¬
lation functions
..........................388
6.5.5.5
Conclusion
.............................390
6.5.6
Outlook
....................................391
References
..........................................391
Access to program examples
394
Index
395
|
| adam_txt |
Contents
Authors
v
Preface
vii
Chapter
1
Introduction
1
1.1
Preliminary remarks
. 1
1.2
Density operator.
Von
Neumann equation
. 1
1.3
Solution of the
von Neumann/Liouville
equation
. 4
1.4
BBGKY-hierarchy
. 4
1.4.1
Reduced density operators. Equations of motion
. 4
1.5
Basic representations of the hierarchy
. 6
1.5.1
Coordinate representation
. 6
1.5.2
Wigner representation
. 8
1.5.3
Classical kinetic equations. Quantum corrections
. 10
1.5.4
Spatially homogeneous systems. Momentum representation
. 11
1.6
Relation to equilibrium correlation functions
. 14
References
. 16
PART I Dynamics of Classical Many-particle Systems
17
Chapter
2
Classical Particle Simulations
19
Hartmut Ruhl
2.1
Synopsis
. 19
2.2
Introduction
. 20
2.3
The physics model
. 25
2.3.1
Governing equations
. 25
2.3.2
The Boltzmann collision operator
. 26
2.3.3
The collision integral for immobile ions
. 29
2.4
The numerical approach
. 30
2.4.1
Normalization
. 31
x
Contents
2.4.2
Maxwell's equations
. 32
2.4.2.1
The FDTD scheme
. 32
2.4.2.2
Periodic boundary conditions
. 34
2.4.2.3
Radiating boundary conditions
. 34
2.4.3
The
Vlasov
equation
. 39
2.4.3.1
The distribution function
. 39
2.4.3.2
Equations of motion
. 40
2.4.3.3
Periodic boundary conditions
. 44
2.4.3.4
Reflecting boundary conditions
. 45
2.4.4
The
Vlasov-
Boltzmann equation
. 46
2.4.4.1
Equations of motion
. 46
2.4.4.2
The collisional model
. 48
2.4.4.3
Scattering angles for Rutherford scattering
. 52
2.4.4.4
Relativistic binary kinematics
. 53
2.4.4.5
Required time resolution and grid size
. 55
2.4.5
Currents
. 55
2.4.5.1
Mass distribution of a quasi-particle
. 55
2.4.5.2
The current conserving scheme
. 56
2.4.6
Energy conservation
. 59
2.5
The simulation code PSC
. 60
2.5.1
Details of the code
. 61
2.5.1.1
Name conventions for important fields
. 61
2.5.1.2
The modules of the PSC
. 61
2.5.1.3
Time progression in the PSC
. 65
2.5.1.4
TCSH scripts for data processing
. 65
2.5.1.5
IDL scripts
. 66
2.5.1.6
PBS batch scripts
. 68
2.5.2
Required hardware and software
. 68
2.5.2.1
Server hardware
. 69
2.5.2.2
Operating system
. 69
2.5.2.3
Fortran compilers
. 69
2.5.2.4
Message passing software
. 69
2.5.2.5
Graphics software
. 69
2.5.2.6
The batch system
. 69
2.6
Examples
. 70
2.6.1
Basics of nonlinear plasma optics
. 71
2.6.1.1
Simulations of laser propagation in vacuum
. 77
2.6.1.2
Simulations of relativistic self-focusing in 2D
. 79
2.6.2
Wakefields in plasma
. 82
2.6.2.1
Wake field simulations in 2D
. 93
2.6.2.2
Simulation of self-
modulatio
n
of laser pulses in 2D
. 97
2.6.3
Aspects of plasma absorption
. 100
2.6.3.1
A kinetic model for sharp edged plasma
. 102
Contents xi
2.6.3.2
A fluid model for sharp edged plasma
. 106
2.6.3.3
Simulation of laser-matter interaction under oblique incidence
107
2.6.3.4
Absorption at high laser intensities
. 110
2.7
Summary
.
Ill
2.8
The open source project PSC
. 112
References
. 113
PART II Correlated Quantum Systems in Equilibrium and Non-
equilibrium
121
Chapter
3
Density Functional Theory
123
George F. Bertsch and Kazuhiro Yabana
3.1
Introduction
.123
3.1.1
Units and notation
.124
3.1.2
Hartree-Fock theory
.125
3.1.3
Homogeneous electron gas
.127
3.1.3.1
Free electrons
.127
3.1.3.2
Exchange energy
.128
3.2
What is density functional theory?
.129
3.2.1
Hohenberg-Kohn theorem
.129
3.2.2
A simple example: the Thomas-Fermi theory
.131
3.2.2.1
Variational equation of Thomas-Fermi theory
.131
3.2.2.2
Thomas-Fermi atom
.132
3.2.2.3
An example
.133
3.3
Kohn-Sham theory
.134
3.3.1
Local density approximation
.135
3.3.2
Spin and the local spin density approximation
.136
3.3.3
The generalized gradient approximation
.136
3.4
Numerical methods for the Kohn-Sham equation
.138
3.4.1
Exact Exchange
.141
3.4.2
O(N) methods
.141
3.5
Some applications and limitations of DFT
.142
3.5.1
Two examples of condensed matter
.143
3.5.2
Vibrations
.144
3.5.3
NMR chemical shifts
.144
3.6
Limitations of DFT
.146
3.7
Time-dependent density functional theory: the equations
.147
3.7.1
Optical properties
.148
3.7.1.1
/-sum rule
.149
3.7.2
Methods to solve the TDDFT equations
.151
3.7.2.1
Linear response formula
.153
3.7.3
Dynamic polarizability
.153
3.7.4
Dielectric function
.154
xii Contents
3.8 TDDFT:
numerical aspects
.156
3.8.1
Configuration matrix method
.156
3.8.2
Linear response method
.158
3.8.3
Sternheimer method
.158
3.8.4
Real time method
.158
3.9
Applications of TDDFT
.160
3.9.1
Simple metal clusters
. 161
3.9.2
Carbon structures
. 163
3.9.3
Diamond
. 164
3.9.4
Other applications
. 164
3.9.5
Limitations
. 166
References
. 167
Chapter
4
Generalized Quantum Kinetic Equations
171
Dirk Semkat and Michael Bonitz
4.1
Introduction
.171
4.2
Idea of second quantization
.175
4.3
Real-time Green's functions
.179
4.3.1
Basic definitions
.179
4.3.2
Calculation of macrophysical quantities
.181
4.4
Derivation of the Kadanoff-Baym/Keldysh equations
.182
4.4.1
The Marti
η
-Schwinger
hierarchy*
.183
4.4.2
Decoupling of the hierarchy*
.185
4.4.3
Kadanoff-Baym/Keldysh equations (KBE)
.187
4.4.4
Determination of the self-energy
.190
4.4.4.1
Mean-field effects. Hartree-Fock (HF) approximation
.191
4.4.4.2
Particle scattering. Second Born approximation
.192
4.5
Single-time kinetic equations
.193
4.5.1
Time-diagonal Kadanoff-Baym equation
.193
4.5.2
Reconstruction problem and spectral function
.194
4.5.3
Quantum Landau equation
.194
4.6
Numerical procedure
.197
4.6.1
Strategy of solving a kinetic equation
.197
4.6.2
Solving the KBE. Overview
.198
4.6.3
Program structure
.200
4.6.4
Discretization
.202
4.6.5
Computation of the self-energies
.203
4.6.5.1
Self-energies in second Born approximation
.203
4.6.5.2
Hartree-Fock self-energy
.205
4.6.6
Computation of the time integrals
.205
4.6.7
Solution of the differential equation
.206
4.6.8
Reconstruction
oí
g^
.207
4.7
Numerical Results
. 207
Contents xiii
4.7.1 Model
system
.207
4.7.2 Evolution
of the correlation functions and of statistical quantities
. 207
4.7.3
Evolution of the spectral function
.213
4.7.4
Influence of initial correlations*
.218
4.7.5
Multi-component systems
.219
4.7.6
Systems containing heavy particles
.220
4.8
Interband Kadanoff-Baym
equations
.222
4.8.1
The multi-band KBE. Application to semiconductors
.222
4.8.2
Application to plasma oscillations
.223
4.9
Survey of numerical applications to other systems
.227
Appendix: Self-energy in
Т
-matrix approximation
.229
References
.231
PART III First-principle Approaches to Correlated Quantum
Systems
235
Chapter
5
Classical and Quantum Monte Carlo
237
Alexei Filinov and Michael Bonitz
5.1
Classical systems and the Monte Carlo method
.237
5.1.1
Introduction
.237
5.1.2
Monte Carlo integration
.238
5.1.2.1
Random quantities
.238
5.1.2.2
Statistical tests
.240
(a) The Kendall tests
.241
5.1.2.3
Practical applications of statistical tests
.244
5.1.2.4
Monte Carlo integration
.257
(a) Straightforward sampling
.257
(b) Error of
MC
integration
.261
(c) Efficiency criteria of the
MC
method
.263
(d) Importance sampling
.264
5.1.3
Practical realizations of Monte Carlo integration
.266
5.1.4
Monte Carlo integration in statistical physics
.272
5.1.4.1
Observables
in statistical mechanics
.272
5.1.4.2
Metropolis method. Markov chain
.274
5.1.5
Statistical ensembles
.276
5.1.5.1
Canonical ensemble
.276
5.1.5.2
Microcanonical ensemble
.277
5.1.5.3
Grand canonical ensemble
.279
5.1.6
Practical applications of the Metropolis algorithm
.280
5.1.6.1
Simulations of the 2D Ising Model (canonical ensemble)
. . . 280
5.1.6.2
2D Lennard-Jones fluid
.290
5.2
Path Integral Monte Carlo
.294
5.2.1
Density matrix and group property
.294
xiv
Contents
5.2.1.1
Quantum statistics. Spin effects
. 298
5.2.1.2
Fermion sign problem
. 300
5.2.1.3
Mapping onto an "effective" classical system
. 300
5.2.1.4
Generalized Metropolis algorithm
. 302
5.2.1.5
Monte Carlo "moves" in PIMC
. 304
(a) Displacement of a whole particle
.304
(b) Single slice moves
.304
(c) Multilevel moves
.305
(d) Sampling of permutations
.308
(e) Acceptance Ratio
.312
(f) Summary.
.312
5.2.1.6
Monte Carlo algorithm for PIMC simulations
.313
5.2.2
Improved high-temperature N—particle density matrix
.320
5.2.2.1
Calculation of the pair density matrix
.322
(a) Matrix squaring technique
.322
(b) Variational approach for the density matrix
.323
(c) First order perturbation expansion for the pair den¬
sity matrix. Off-diagonal and diagonal Kelbg potential.
324
5.2.3
Calculation of physical
observables
.327
5.2.4
Simulations of macroscopic systems. Finite-size effects
.330
5.2.4.1 Ewald
transformation
.330
5.2.4.2
Pseudopotentials in quantum simulations
.332
5.2.4.3
Elimination of finite-size errors
.334
(a) Twist averaged boundary conditions
.335
(b) Periodic Coulomb interaction
.335
(c) Finite-size scaling
.336
5.2.5
Applications of PIMC
.337
5.2.5.1
2D Coulomb clusters
.337
5.2.5.2
Binding energies of
excitons, trions,
biexcitons in semiconduc¬
tor nanostructures
.341
5.2.6
Conclusion and outlook
.345
References
.346
Chapter
6
Quantum Molecular Dynamics
351
Alexei Filinov, Vladimir Filinov, Yurii Lozovik and Michael Bonitz
6.1
Introduction
. 351
6.2
Quantum distribution functions
. 352
6.2.1
Wigner function
. 354
6.2.2
Husimi function
. 355
6.2.3
Evolution equation
. 355
6.3
Quantum dynamics I: Method of Wigner trajectories
. 356
6.3.1
Wigner representation
. 356
6.3.2
Wigner trajectories
. 358
Contents xv
6.3.3
Inclusion
of quantum
statistics
.359
6.3.4
Simulation algorithm
.360
6.3.5
Application: Tunneling of interacting identical particles
.361
6.3.5.1
Theoretical model
.363
6.3.5.2
Initial parameters
.364
6.3.5.3
Reaction probabilities
.365
6.3.5.4
Tunneling times
.367
6.3.5.5
Conclusion
.368
6.4
Quantum dynamics II: Monte Carlo approach to ensembles of quantum trajec¬
tories
.368
6.4.1
Integral form of the Wigner-
Liou
ville
equation
.368
6.4.2
Method of characteristics
.369
6.4.3
Quantum corrections of higher order
. 372
6.4.4
Calculation of linear functionals of the Wigner function by Monte Carlo
sampling
.373
6.4.4.1
Algorithm for generation of an ensemble of quantum trajectories374
6.4.5
Calculation of averages of physical
observables
.375
6.5
Wigner function in the canonical ensemble
.376
6.5.1
Time correlation functions
.376
6.5.2
Initial conditions for the Wigner-Liouville equation
.377
6.5.3
Integral equations
.379
6.5.4
Quantum dynamics
.380
6.5.5
Application: Interacting electrons in
а о
ne-
dimensional random array
of scatterers. Anderson localization
.384
6.5.5.1
Theoretical model and numerical aspects
.384
6.5.5.2
Numerical results
.387
6.5.5.3
Temporal quantum momentum-momentum correlation func¬
tions
.387
6.5.5.4
Fourier transform of the momentum-momentum time corre¬
lation functions
.388
6.5.5.5
Conclusion
.390
6.5.6
Outlook
.391
References
.391
Access to program examples
394
Index
395 |
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| index_date | 2024-07-02T14:20:32Z |
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| spelling | Introduction to computational methods in many body physics Eds.: Michael Bonitz ... Paramus, N.J. Rinton Press 2006 XV, 400 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Many-body problem Numerical solutions Bonitz, Michael 1960- Sonstige (DE-588)120375176 oth Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014732071&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Introduction to computational methods in many body physics Many-body problem Numerical solutions |
| title | Introduction to computational methods in many body physics |
| title_auth | Introduction to computational methods in many body physics |
| title_exact_search | Introduction to computational methods in many body physics |
| title_exact_search_txtP | Introduction to computational methods in many body physics |
| title_full | Introduction to computational methods in many body physics Eds.: Michael Bonitz ... |
| title_fullStr | Introduction to computational methods in many body physics Eds.: Michael Bonitz ... |
| title_full_unstemmed | Introduction to computational methods in many body physics Eds.: Michael Bonitz ... |
| title_short | Introduction to computational methods in many body physics |
| title_sort | introduction to computational methods in many body physics |
| topic | Many-body problem Numerical solutions |
| topic_facet | Many-body problem Numerical solutions |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014732071&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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