Computational physics: simulation of classical and quantum systems
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| 1. Verfasser: | |
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| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Berlin [u.a.]
Springer
2010
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| Online-Zugang: | Inhaltstext Inhaltsverzeichnis |
| Beschreibung: | XV, 319 S. graph. Darst. |
| ISBN: | 9783642139895 |
Internformat
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| 100 | 1 | |a Scherer, Philipp O. J. |e Verfasser |0 (DE-588)110745108 |4 aut | |
| 245 | 1 | 0 | |a Computational physics |b simulation of classical and quantum systems |c Philipp O. J. Scherer |
| 264 | 1 | |a Berlin [u.a.] |b Springer |c 2010 | |
| 300 | |a XV, 319 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
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Datensatz im Suchindex
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IMAGE 1
CONTENTS
PART I NUMERICAL METHODS
1 ERROR ANALYSIS 3
1.1 MACHINE NUMBERS AND ROUNDING ERRORS 3
1.2 NUMERICAL ERRORS OF ELEMENTARY FLOATING POINT OPERATIONS 6 1.2.1
NUMERICAL EXTINCTION 6
1.2.2 ADDITION 7
1.2.3 MULTIPLICATION 8
1.3 ERROR PROPAGATION 8
1.4 STABILITY OF ITERATIVE ALGORITHMS 11
1.5 EXAMPLE: ROTATION 12
1.6 TRUNCATION ERROR 13
PROBLEMS 13
2 INTERPOLATION 15
2.1 INTERPOLATING FUNCTIONS 15
2.2 POLYNOMIAL INTERPOLATION 16
2.2.1 LAGRANGE POLYNOMIALS 16
2.2.2 NEWTON'S DIVIDED DIFFERENCES 17
2.2.3 INTERPOLATION ERROR 18
2.2.4 NEVILLE METHOD 20
2.3 SPLINE INTERPOLATION 21
2.4 MULTIVARIATE INTERPOLATION 25
PROBLEMS 26
3 NUMERICAL DIFFERENTIATION 29
3.1 SIMPLE FORWARD DIFFERENCE 29
3.2 SYMMETRICAL DIFFERENCE QUOTIENT 30
3.3 EXTRAPOLATION METHODS 31
3.4 HIGHER DERIVATIVES 33
3.5 MORE DIMENSIONS 34
PROBLEMS 35
BIBLIOGRAFISCHE INFORMATIONEN HTTP://D-NB.INFO/1002961580
DIGITALISIERT DURCH
IMAGE 2
X CONTENTS
4 NUMERICAL INTEGRATION 37
4. 1 EQUIDISTANT SAMPLE POINTS 37
4.1.1 NEWTON-COTES RULES 38
4.1.2 NEWTON-COTES EXPRESSIONS FOR AN OPEN INTERVAL 39 4.1.3 COMPOSITE
NEWTON-COTES FORMULAS 40
4.1.4 EXTRAPOLATION METHOD (ROMBERG INTEGRATION) 40 4.2 OPTIMIZED SAMPLE
POINTS 42
4.2.1 CLENSHAW-CURTIS EXPRESSIONS 42
4.2.2 GAUSSIAN INTEGRATION 43
PROBLEMS 45
5 SYSTEMS OF INHOMOGENEOUS LINEAR EQUATIONS 47
5. 1 GAUSSIAN ELIMINATION METHOD 47
5.1.1 PIVOTING 50
5.1.2 DIRECT LU DECOMPOSITION 51
5.2 QR DECOMPOSITION 51
5.3 LINEAR EQUATIONS WITH TRIDIAGONAL MATRIX 53
5.4 CYCLIC TRIDIAGONAL SYSTEMS 55
5.5 ITERATIVE SOLUTION OF INHOMOGENEOUS LINEAR EQUATIONS 56 5.5.1
GENERAL TREATMENT 56
5.5.2 JACOBI METHOD 57
5.5.3 GAUSS-SEIDEL METHOD 57
5.5.4 DAMPING AND SUCCESSIVE OVER-RELAXATION 58
5.6 CONJUGATE GRADIENTS 59
PROBLEMS 60
6 ROOTS AND EXTREMAL POINTS 63
6. 1 ROOT FINDING 63
6.1.1 BISECTION 63
6.1.2 REGULA FALSI METHOD 64
6.1.3 NEWTON-RAPHSON METHOD 65
6.1.4 SECANT METHOD 66
6.1.5 ROOTS OF VECTOR FUNCTIONS 66
6.2 OPTIMIZATION WITHOUT CONSTRAINTS 67
6.2.1 STEEPEST DESCENT METHOD 68
6.2.2 CONJUGATE GRADIENT METHOD 68
6.2.3 NEWTON-RAPHSON METHOD 69
6.2.4 QUASI-NEWTON METHODS 69
PROBLEMS 70
7 FOURIER TRANSFORMATION 73
7. 1 DISCRETE FOURIER TRANSFORMATION 74
7.1.1 TRIGONOMETRIE INTERPOLATION 75
7.1.2 REAL-VALUED FUNCTIONS 77
IMAGE 3
CONTENTS
7.1.3 APPROXIMATE CONTINUOUS FOURIER TRANSFORMATION 77 7.2 ALGORITHMS 78
7.2.1 GOERTZEL'S ALGORITHM 79
7.2.2 FAST FOURIER TRANSFORMATION 80
PROBLEMS 84
8 RANDOM NUMBERS AND MONTE CARLO METHODS 87
8. 1 SOME BASIC STATISTICS 87
8.1.1 PROBABILITY DENSITY AND CUMULATIVE PROBABILITY DISTRIBUTION 87
8.1.2 EXPECTATION VALUES AND MOMENTS 88
8.1.3 MULTIVARIATE DISTRIBUTIONS 92
8.1.4 CENTRAL LIMIT THEOREM 93
8.1.5 EXAMPLE: BINOMIAL DISTRIBUTION 93
8.1.6 AVERAGE OF REPEATED MEASUREMENTS 94
8.2 RANDOM NUMBERS 95
8.2.1 THE METHOD BY MARSAGLIA AND ZAMANN 96
8.2.2 RANDOM NUMBERS WITH GIVEN DISTRIBUTION 96
8.2.3 EXAMPLES 97
8.3 MONTE CARLO INTEGRATION 99
8.3.1 NUMERICAL CALCULATION OF TT 99
8.3.2 CALCULATION OF AN INTEGRAL 100
8.3.3 MORE GENERAL RANDOM NUMBERS 101
8.4 MONTE CARLO METHOD FOR THERMODYNAMIC AVERAGES 102 8.4.1 SIMPLE
(MINDED) SAMPLING 102
8.4.2 IMPORTANCE SAMPLING 103
8.4.3 METROPOLIS ALGORITHM 104
PROBLEMS 106
9 EIGENVALUE PROBLEMS 1 09
9. 1 DIRECT SOLUTION 109
9.2 JACOBI METHOD 109
9.3 TRIDIAGONAL MATRICES 111
9.4 REDUCTION TO A TRIDIAGONAL MATRIX 111
9.5 LARGE MATRICES 114
PROBLEMS 115
10 DATA FITTING 117
10.1 LEAST SQUARE FIT 117
10.1.1 LINEAR LEAST SQUARE FIT 119
10.1.2 LEAST SQUARE FIT USING ORTHOGONALIZATION 120 10.2 SINGULAR VALUE
DECOMPOSITION 123
PROBLEMS 127
IMAGE 4
XII CONTENTS
11 EQUATIONS OF MOTION 129
11.1 STATE VECTOR OF A PHYSICAL SYSTEM 129
11.2 TIME EVOLUTION OF THE STATE VECTOR 130
11.3 EXPLICIT FORWARD EULER METHOD 132
11.4 IMPLICIT BACKWARD EULER METHOD 134
11.5 IMPROVED EULER METHODS 135
11.6 TAYLOR SERIES METHODS 137
11.7 RUNGE-KUTTA METHODS 138
11.7.1 SECOND-ORDER RUNGE-KUTTA METHOD 138
11.7.2 THIRD-ORDER RUNGE-KUTTA METHOD 138
11.7.3 FOURTH-ORDER RUNGE-KUTTA METHOD 139
11.8 QUALITY CONTROL AND ADAPTIVE STEP-SIZE CONTROL 140
11.9 EXTRAPOLATION METHODS 141
11.10 MULTISTEP METHODS 142
11.10.1 EXPLICIT MULTISTEP METHODS 142
11.10.2 IMPLICIT MULTISTEP METHODS 143
11.10.3 PREDICTOR-CORRECTOR METHODS 144
11.11 VERLET METHODS 144
11.11.1 LIOUVILLE EQUATION 144
11.11.2 SPLIT OPERATOR APPROXIMATION 145
11.11.3 POSITION VERLET METHOD 146
11.11.4 VELOCITY VERLET METHOD 146
11.11.5 STANDARD VERLET METHOD 147
11.11.6 ERROR ACCUMULATION FOR THE STANDARD VERLET METHOD . . 149
11.11.7 LEAP FROG METHOD 149
PROBLEMS 150
PART II SIMULATION OF CLASSICAL AND QUANTUM SYSTEMS
12 ROTATIONAL MOTION 157
12.1 TRANSFORMATION TO A BODY FIXED COORDINATE SYSTEM 157 12.2
PROPERTIES OF THE ROTATION MATRIX 158
12.3 PROPERTIES OF W, CONNECTION WITH THE VECTOR OF ANGULAR VELOCITY .
160 12.4 TRANSFORMATION PROPERTIES OF THE ANGULAR VELOCITY 161 12.5
MOMENTUM AND ANGULAR MOMENTUM 163
12.6 EQUATIONS OF MOTION OF A RIGID BODY 163
12.7 MOMENTS OF INERTIA 164
12.8 EQUATIONS OF MOTION FOR A ROTOR 165
12.9 EXPLICIT SOLUTIONS 165
12.10 LOSS OF ORTHOGONALITY 167
12.11 IMPLICIT METHOD 168
12.12 EXAMPLE: FREE SYMMETRIE ROTOR 170
12.13 KINETIC ENERGY OF A ROTOR 171
12.14 PARAMETRIZATION BY EULER ANGLES 172
IMAGE 5
CONTENTS
12.15 CAYLEY-KLEIN PARAMETERS, QUATERNIONS, EULER PARAMETERS 172 12.16
SOLVING THE EQUATIONS OF MOTION WITH QUATERNIONS 176 PROBLEMS 176
13 SIMULATION OF THERMODYNAMIC SYSTEMS 179
13.1 FORCE FIELDS FOR MOLECULAR DYNAMICS SIMULATIONS 179 13.1.1
INTRAMOLECULAR FORCES 179
13.1.2 INTERMOLECULAR FORCES 180
13.1.3 APPROXIMATE SEPARATION OF ROTATION AND VIBRATIONS . . 180 13.2
SIMULATION OF A VAN DER WAALS SYSTEM 181
13.2.1 INTEGRATION OF THE EQUATIONS OF MOTION 181
13.2.2 BOUNDARY CONDITIONS AND AVERAGE PRESSURE 182 13.2.3 INITIAL
CONDITIONS AND AVERAGE TEMPERATURE 183 13.2.4 ANALYSIS OF THE RESULTS
183
13.3 MONTE CARLO SIMULATION 186
13.3.1 ONE-DIMENSIONAL ISING MODEL 186
13.3.2 TWO-DIMENSIONAL ISING MODEL 188
PROBLEMS 189
14 RANDOM WALK AND BROWNIAN MOTION 193
14.1 RANDOM WALK IN ONE DIMENSION 194
14.1.1 RANDOM WALK WITH CONSTANT STEP SIZE 195
14.2 THE FREELY JOINTED CHAIN 196
14.2.1 BASIC STATISTIC PROPERTIES 197
14.2.2 GYRATION TENSOR 199
14.2.3 HOOKEAN SPRING MODEL 200
14.3 LANGEVIN DYNAMICS 202
PROBLEMS 204
15 ELECTROSTATICS 207
15.1 POISSON EQUATION 207
HOMOGENEOUS DIELECTRIC MEDIUM 207
CHARGED SPHERE 209
VARIABLE S 210
DISCONTINUOUS E 211
SOLVATION ENERGY OF A CHARGED SPHERE 211
THE SHIFTED GRID METHOD 213
15.2 POISSON BOLTZMANN EQUATION FOR AN ELECTROLYTE 215
15.2.1 DISCRETIZATION OF THE LINEARIZED POISSON-BOLTZMANN EQUATION 216
15.3 BOUNDARY ELEMENT METHOD FOR THE POISSON EQUATION 216 15.3.1
INTEGRAL EQUATIONS FOR THE POTENTIAL 217
15.3.2 CALCULATION OF THE BOUNDARY POTENTIAL 219
15. 15. 15. 15.
15. 15.
.1 .2 .3
.4 .5 .6
IMAGE 6
XIV CONTENTS
15.4 BOUNDARY ELEMENT METHOD FOR THE LINEARIZED POISSON- BOLTZMANN
EQUATION 222
15.5 ELECTROSTATIC INTERACTION ENERGY (ONSAGER MODEL) 223 15.5.1
EXAMPLE: POINT CHARGE IN A SPHERICAL CAVITY 225 PROBLEMS 225
16 WAVES 229
16.1 ONE-DIMENSIONAL WAVES 229
16.2 DISCRETIZATION OF THE WAVE EQUATION 231
16.3 BOUNDARY VALUES 232
16.4 THE WAVE EQUATION AS AN EIGENVALUE PROBLEM 233
16.4.1 EIGENFUNCTION EXPANSION 233
16.4.2 APPLICATION TO THE DISCRETE ONE-DIMENSIONAL WAVE EQUATION 234
16.5 NUMERICAL INTEGRATION OF THE WAVE EQUATION 237
16.5.1 SIMPLE ALGORITHM 237
16.5.2 STABILITY ANALYSIS 238
16.5.3 ALTERNATIVE ALGORITHM WITH EXPLICIT VELOCITIES 240 16.5.4
STABILITY ANALYSIS 240
PROBLEMS 242
17 DIFFUSION 243
17.1 BASIC PHYSICS OF DIFFUSION 243
17.2 BOUNDARY CONDITIONS 244
17.3 NUMERICAL INTEGRATION OF THE DIFFUSION EQUATION 245
17.3.1 FORWARD EULER OR EXPLICIT RICHARDSON METHOD 245 17.3.2 STABILITY
ANALYSIS 245
17.3.3 IMPLICIT BACKWARD EULER ALGORITHM 247
17.3.4 CRANK-NICOLSON METHOD 248
17.3.5 ERROR ORDER ANALYSIS 249
17.3.6 PRACTICAL CONSIDERATIONS 250
17.3.7 SPLIT OPERATOR METHOD FOR D 1 DIMENSIONS 250 PROBLEMS 252
18 NONLINEAR SYSTEMS 253
18.1 ITERATED FUNCTIONS 253
18.1.1 FIXED POINTS AND STABILITY 254
18.1.2 THE LJAPUNOW EXPONENT 256
18.1.3 THE LOGISTIC MAP 257
18.1.4 FIXED POINTS OF THE LOGISTIC MAP 258
18.1.5 BIFURCATION DIAGRAM 259
18.2 POPULATION DYNAMICS 260
18.2.1 EQUILIBRIA AND STABILITY 260
18.2.2 THE CONTINUOUS LOGISTIC MODEL 262
IMAGE 7
CONTENTS XV
18.3 LOTKA-VOLTERRA MODEL 262
18.3.1 STABILITY ANALYSIS 263
18.4 FUNCTIONAL RESPONSE 265
18.4.1 HOLLING-TANNER MODEL 266
18.5 REACTION-DIFFUSION SYSTEMS 269
18.5.1 GENERAL PROPERTIES OF REACTION-DIFFUSION SYSTEMS . . . 269
18.5.2 CHEMICAL REACTIONS 270
18.5.3 DIFFUSIVE POPULATION DYNAMICS 270
18.5.4 STABILITY ANALYSIS 270
18.5.5 LOTKA-VOLTERRA MODEL WITH DIFFUSION 272
PROBLEMS 273
19 SIMPLE QUANTUM SYSTEMS 277
19.1 QUANTUM PARTICLE IN A POTENTIAL WELL 278
19.2 EXPANSION IN A FINITE BASIS 282
19.3 TIME-INDEPENDENT PROBLEMS 284
19.3.1 SIMPLE TWO-LEVEL SYSTEM 285
19.3.2 THREE-STATE MODEL (SUPEREXCHANGE) 286
19.3.3 LADDER MODEL FOR EXPONENTIAL DECAY 290
19.4 TIME-DEPENDENT MODELS 292
19.4.1 LANDAU-ZENER MODEL 293
19.4.2 TWO-STATE SYSTEM WITH TIME-DEPENDENT PERTURBATION . 293 19.5
DESCRIPTION OF A TWO-STATE SYSTEM WITH THE DENSITY MATRIX FORMALISM 297
19.5.1 DENSITY MATRIX FORMALISM 297
19.5.2 ANALOGY TO NUCLEAR MAGNETICRESONANCE 300 19.5.3 RELAXATION
PROCESSES-BLOCH EQUATIONS 302 PROBLEMS 307
APPENDIX 309
REFERENCES 311
INDEX 315 |
| any_adam_object | 1 |
| author | Scherer, Philipp O. J. |
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| dewey-ones | 530 - Physics |
| dewey-raw | 530.0285 |
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| dewey-sort | 3530.0285 |
| dewey-tens | 530 - Physics |
| discipline | Physik Informatik Mathematik |
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| indexdate | 2024-07-20T11:01:51Z |
| institution | BVB |
| isbn | 9783642139895 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-021197553 |
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| spelling | Scherer, Philipp O. J. Verfasser (DE-588)110745108 aut Computational physics simulation of classical and quantum systems Philipp O. J. Scherer Berlin [u.a.] Springer 2010 XV, 319 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Computerphysik (DE-588)4273564-6 gnd rswk-swf (DE-588)4123623-3 Lehrbuch gnd-content Computerphysik (DE-588)4273564-6 s DE-604 Erscheint auch als Online-Ausgabe 978-3-642-13990-1 text/html http://deposit.dnb.de/cgi-bin/dokserv?id=3484423&prov=M&dok_var=1&dok_ext=htm Inhaltstext DNB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=021197553&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Scherer, Philipp O. J. Computational physics simulation of classical and quantum systems Computerphysik (DE-588)4273564-6 gnd |
| subject_GND | (DE-588)4273564-6 (DE-588)4123623-3 |
| title | Computational physics simulation of classical and quantum systems |
| title_auth | Computational physics simulation of classical and quantum systems |
| title_exact_search | Computational physics simulation of classical and quantum systems |
| title_full | Computational physics simulation of classical and quantum systems Philipp O. J. Scherer |
| title_fullStr | Computational physics simulation of classical and quantum systems Philipp O. J. Scherer |
| title_full_unstemmed | Computational physics simulation of classical and quantum systems Philipp O. J. Scherer |
| title_short | Computational physics |
| title_sort | computational physics simulation of classical and quantum systems |
| title_sub | simulation of classical and quantum systems |
| topic | Computerphysik (DE-588)4273564-6 gnd |
| topic_facet | Computerphysik Lehrbuch |
| url | http://deposit.dnb.de/cgi-bin/dokserv?id=3484423&prov=M&dok_var=1&dok_ext=htm http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=021197553&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT schererphilippoj computationalphysicssimulationofclassicalandquantumsystems |