Quantum signatures of chaos:
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| Main Authors: | , , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Cham
Springer
[2018]
|
| Edition: | Fourth edition |
| Series: | Springer series in synergetics
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| Subjects: | |
| Online Access: | Inhaltstext http://www.springer.com/ Inhaltsverzeichnis |
| Physical Description: | xxvi, 659 Seiten Illustrationen, Diagramme |
| ISBN: | 9783319975795 331997579X |
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| 653 | |a dissipative systems | ||
| 653 | |a level dynamics | ||
| 653 | |a quantum localization | ||
| 653 | |a quantum time evolution | ||
| 653 | |a universal spectral fluctuations | ||
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Record in the Search Index
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| adam_text |
Contents 1 Introduction. References. 1 13 2 Time Reversal and Unitary Symmetries. 15 15 16 17 20 21 22 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.11 2.12 2.13 2.14 2.15 2.16 Autonomous Classical Flows. Spinless Quanta. Spin-1/2 Quanta. Hamiltonians Without T Invariance. T Invariant Hamiltonians, T2 = 1. Unitary Symmetries. Kramers’ Degeneracy for T Invariant Hamiltonians, T2 = -l. T Invariant Hamiltonians with T2 = — 1 and Additional Unitary Symmetries. ^Invariance with T2 — -1 and no UnitarySymmetries. Nonconventional Time Reversal. Stroboscopic Maps for Periodically Driven Systems. Time Reversal for Maps. Canonical Transformations for FloquetOperators.
Universality. Universality for the Kicked Top: The Level Spacing Distribution. Beyond Dyson’s Threefold Way . 2.16.1 Spectral Mirror Symmetries. 2.16.2 Universality in Non-standard Symmetry Classes . 2.16.3 Hamiltonian Matrix Structures and Canonical Transformations for Non-standard Symmetries. 2.16.4 Physical Realizations of Non-standard Symmetry Classes in Fermionic Systems . 58 24 25 28 31 33 35 38 41 43 45 46 49 51 xix
Contents XX 2 . í 6.5 Quantum Mechanical Realisation of Non-standard Symmetry Classes in Finite Dimensional Hilbert Spaces: Two Coupled Tops. 2 .16.6 Non-standard Universality for Two Coupled Tops. 2.17 Problems . References. Level Repulsion. 3.1 Preliminaries. 3 Symmetric Versus Nonsymmetric H or F. Kramers’Degeneracy. Universality Classes of Level Repulsion. Nonstandard Symmetry Classes. 3.5.1 The Chiral Symmetry Classes. 3.5.2 Classes CI and BDIIIV. 3.5.3 Classes BDV and C. 3.6 Experimental Observation of Level Repulsion. 3.7 Problems . References. 3.2 3.3 3.4 3.5 4 5 61 66 68 69 71 71 72 74 77 78 79 80 80 81 82 83 85 85 85 4.3
Einstein-Brillouin-Keller Approximation. 87 4.4 Level Crossings for Integrable Systems. 89 4.5 Poissonian Level Sequences. 90 4.6 Superposition of Independent Spectra. 91 4.7 Periodic Orbits and the Semiclassical Density of Levels. 93 4.8 Level Density Fluctuations for Integrable Systems. 99 4.9 Exponential Spacing Distribution for Integrable Systems. 106 4.10 Equivalence of Different Unfoldings. 107 4.11 Problems. 108 References. 109 Level Clustering. 4.1 Preliminaries. 4.2 Invariant Tori of Classically Integrable Systems. Random-Matrix Theory. 5.1 5.2 5.3 Preliminaries. Gaussian Ensembles of Hermitian Matrices. 5.2.1 Gaussian Orthogonal Ensemble. 5.2.2 Gaussian Unitary Ensemble. 5.2.3
Gaussian Symplectic Ensemble. 5.2.4 Arbitrary Matrix Dimension. Riemann Geometry for Matrix Ensembles: Invariance of the Volume Element. 118 5.3.1 Change of Variables on a Manifold. Ill Ill 112 112 114 114 116 118
xxi Contents Geometry for Matrix Ensembles: Invariance of the Volume Element dN H. 5.4 More of Riemann Geometry for Matrix Ensembles: Eigenvalue Distributions. 120 5.5 Eigenvalue Distributions for the Non-standardSymmetry Classes. 5.6 Level Spacing Distributions (Wigner Surmizes). 5.7 Average Level Density (GUE). 5.8 Dyson’s Circular Ensembles. 5.8.1 Circular Unitary Ensemble. 5.8.2 COEandCSE. 5.8.3 Poissonian Ensemble. 5.9 Same Spectral Correlations in Circular and Gaussian Ensembles for N — oo. 132 5.10 Eigenvector Distributions. 5.10.1 Single-Vector Density . 5.10.2 Joint Density of Eigenvectors. 5.11 Ergodicity of the Level Density. 5.12 Asymptotic Level Spacing Distributions. 5.13 Determinants as Gaussian Grassmann Integrals. 5.14 Two-Point Correlations of the Level
Density. 5.14.1 Two-Point Correlator and Form Factor. 5.14.2 Form Factor for the Poissonian Ensemble. 5.14.3 Form Factor for the CUE. 5.14.4 Form Factor for the COE. 5.14.5 Form Factor for the CSE . 5.15 Newton’s Relations. 5.15.1 Traces Versus Secular Coefficients. 5.15.2 Solving Newton’s Relations. 5.16 Selfinversiveness and Riemann-Siegel Lookalike. 5.17 Higher Correlations of the Level Density. 5.17.1 Correlation and Cumulant Functions. 5.17.2 Ergodicity of the CUE Form Factor. 5.17.3 Ergodicity of the CUE Two-Point Correlator. 5.17.4 Joint Density of Traces of Large CUE Matrices. 5.18 Correlations of Secular Coefficients. 5.19 Unfolding Spectra. 5.20 Fidelity of Kicked Tops to Random-Matrix Theory. 5.21 Problems . References. 5.3.2 6 122
124 125 126 127 130 132 133 133 136 137 141 151 156 156 158 158 161 165 167 167 170 172 174 174 176 179 183 185 191 193 200 202 205 Preliminaries. 205 Semicircle Law for the Gaussian Unitary Ensemble. 206 6.2.1 The Green Function and Its Average. 206 Supersymmetry and Sigma Model for Random Matrices. 6.1 6.2 119
xxii Contents An Aside: Complex Conjugation of Grassmann Variables. 207 6.2.3 The GUE Average. 208 6.2.4 Doing the Superintegral. 209 6.2.5 Two Remaining Saddle-Point Integrals. 211 6.3 Superalgebra. 215 6.3.1 Motivation and Generators of Grassmann Algebras . 215 6.3.2 Supervectors, Supermatrices. 215 6.3.3 Superdeterminants. 218 6.3.4 Complex Scalar Product, Hermitian and Unitary Supermatrices. 221 6.3.5 Diagonalizing Supermatrices. 222 6.4 Superintegrals. 223 6.4.1 Some Bookkeeping for Ordinary Gaussian Integrals. 223 6.4.2 Recalling Grassmann Integrals. 224 6.4.3 Gaussian Superintegrals. 226 6.4.4 Some Properties of General Superintegrals. 227 6.4.5 Integrals over Supermatrices, Parisi-Sourlas-Efetov-Wegner Theorem. 229 6.4.6 Asymptotic Analysis of SuSy Integrals: Massive and Zero
Modes. 231 6.5 The Semicircle Law Revisited . 234 6.6 The Two-Point Function of the Gaussian Unitary Ensemble. 238 6.6.1 Generating Function. 239 6.6.2 Unitarity vs Pseudo-Unitarity and Superanalytic Hubbard-Stratonovich Transformation. 241 6.6.3 Efetov’s Sigma Model. 244 6.6.4 Rational Parametrization. 247 6.7 Two-Point Functions of the Circular Ensembles. 250 6.7.1 Generating Function. 251 6.8 Zero Dimensional Sigma Model for the CUE. 254 6.8.1 Color-Flavor Transformation. 254 6.8.2 The Q Manifold. 257 6.8.3 Riemann Geometry for Supermatrix Ensembles: Flat Measure d{Q) = d{Z,Ž). 259 6.8.4 Proof of the Color-Flavor Transformation (6.8.2). 261 6.8.5 Evaluation of the Generating Function and the Two-Point Correlator for the CUE. 266 6.8.6 Evaluation of the Grassmann Integral Q. 270 6.9 The Zero Dimensional Sigma Model for COE and CSE. 273 6.10 Universality of Spectral Fluctuations: Non-Gaussian Ensembles. 280
6.10.1 Delta Functions of Grassmann Variables. 281 6.10.2 Generating Function. 282 6.2.2
Contents xxiii 6.11 Problems . 286 References. 287 7 Ballistic Sigma Model for Individual Unitary Maps and Graphs. 289 Preliminaries. Generation Function for the Two-Point Correlator. Reduction to the Zero Dimensional Sigma Model and Condition of Validity. 7.4 Perturbative Account of Fluctuations. 7.4.1 Fluctuations on the Q Manifold. 7.4.2 Small Parameters. 7.4.3 Semiclassical Limit. 7.4.4 Conditions for Universal Behavior. 7.5 Quantum Graphs. 7.5.1 Directed Graphs and Their Spectra. 7.5.2 The Sigma Model Approach to Spectral Statistics. References. 7.1 7.2 7.3 8 9 289 290 291 292 292 293 294 296 297 297 299 302 303 8.1 Preliminaries. 303 8.2 Localization in Anderson’s Hopping
Model. 305 8.3 The Kicked Rotator as a Variant of Anderson’s Model. 307 8.4 Lloyd’s Model. 314 8.5 The Classical Diffusion Constant as the Quantum Localization Length. 320 8.6 Absence of Localization for the Kicked Top. 321 8.7 The Rotator as a Limiting Case of the Top. 332 8.8 Banded Random Matrices. 333 8.8.1 Banded Matrices Modelling Thick Wires. 333 8.8.2 Inverse Participation Ratio and Localization Length. 335 8.8.3 Sigma Model. 337 8.8.4 Implementing the One Dimensional Sigma Model . 341 8.9 Sigma Model for the Kicked Rotor. 353 8.9.1 A Rotor Without Time Reversal Invariance. 353 8.9.2 Inverse Participation Ratio. 354 8.9.3 Sigma Model. 355 8.9.4 Slow Modes. 357 8.10 Problems. 361 References. 362 Quantum
Localization. Classical Hamiltonian Chaos. 9.1 9.2 9.3 9.4 9.5 Preliminaries. Phase Space, Hamilton’s Equations and All That. Action as a Generating Function. Linearized Flow and Its Jacobian Matrix . Liouville Picture. 365 365 365 367 369 370
Contents xxiv Symplectic Structure. 371 Lyapunov Exponents. 373 Stretching Factors and Local Stretching Rates. 374 Poincaré Map. 376 Stroboscopic Maps of Periodically Driven Systems. 378 Varieties of Chaos; Mixing and Effective Equilibration. 379 The Sum Rule of Hannay and Ozorio de Almeida. 380 9.12.1 Maps. 381 9.12.2 Flows. 382 9.13 Propagator and Zeta Function. 384 9.14 Exponential Stability of the Boundary Value Problem. 387 9.15 Sieber-Richter Self-Encounter and Partner Orbit. 388 9.15.1 Non-technical Discussion. 388 9.15.2 Quantitative Discussion of 2-Encounters . 391 9.16 /-Encounters and Orbit Bunches. 399 9.17 Densities of Arbitrary Encounter Sets. 404 9.18 Concluding Remarks. 406 9.19 Problems. 406
References. 406 9.6 9.7 9.8 9.9 9.10 9.11 9.12 10 Semiclassical Roles for CJassical Orbits. 409 10.1 Preliminaries. 409 10.2 Van Vleck Propagator. 409 10.2.1 Maps. 411 10.2.2 Flows. 416 10.3 Gutzwiller’s Trace Formula. 420 10.3.1 Maps. 420 10.3.2 Flows. 425 10.3.3 Weyl’s Law. 431 10.3.4 Limits of Validity and Outlook. 432 10.4 Lagrangian Manifolds and Maslov Theory. 434 10.4.1 Lagrangian Manifolds. 434 10.4.2 Elements of Maslov Theory. 441 10.4.3 Maslov Indices as Winding Numbers. 445 10.5 Riemann-Siegel Look-Alike. 450 10.6 Spectral Two-Point
Correlator. 456 10.6.1 Real and Complex Correlator. 457 10.6.2 Local Energy Average. 458 10.6.3 Generating Function. 460 10.6.4 Periodic-Orbit Representation. 461 10.7 Diagonal Approximation. 466 10.7.1 Unitary Class. 466 10.7.2 Orthogonal Class. 468 10.8 Off-Diagonal Contributions, Unitary Symmetry Class . 468 10.8.1 Structures of Pseudo-Orbit Quadruplets. 471
Contents xxv Diagrammatic Rules. Example of Structure Contributions: A Single 2-Encounter. 475 10.8.4 Cancellation of All Encounter Contributions for the Unitary Class . 476 10.9 Semiclassical Construction of a Sigma Model, Unitary Symmetry Class. 479 10.9.1 Matrix Elements for Ports and Contraction Lines for Links. 479 10.9.2 Wick’s Theorem and Link Summation. 10.9.3 Signs. 10.9.4 Proof of Contraction Rules, Unitary Case. 10.9.5 Emergence of a Sigma Model. 10.10 Semiclassical Construction of a Sigma Model, Orthogonal Symmetry Class. 490 10.10.1 Structures. 10.10.2 Leading-Order Contributions. 10.10.3 Symbols for Ports and Contraction Lines for Links. 10.10.4 Gauss and Wick. 10.10.5 Signs. 10.10.6 Proof of Contraction Rules, Orthogonal Case. 10.10.7 Sigma Model. 10.11
Outlook. 10.12 Mixed Phase Space. 10.13 Problems . References. 473 Level Dynamics. 11.1 Preliminaries. 11.2 Fictitious Particles (Pechukas-Yukawa Gas). 11.3 Conservation Laws. 11.4 Intermultiplet Crossings. 11.5 Level Dynamics for Classically Integrable Dynamics. 11.6 Two-Body Collisions. 11.7 Ergodicity of Level Dynamics and Universality of Spectral Fluctuations. 11.7.1 Ergodicity. 11.7.2 Collision Time. 11.7.3 Universality. 11.8 Equilibrium Statistics. 11.9 Random-Matrix Theory as Equilibrium Statistical
Mechanics . 11.9.1 General Strategy. 11.9.2 A Typical Coordinate Integral. 11.9.3 Influence of a Typical Constantof the Motion. 11.9.4 The General Coordinate Integral. 511 511 513 519 521 522 528 10.8.2 10.8.3 11 481 483 486 487 491 492 494 495 496 498 500 501 502 505 507 529 529 531 532 533 536 536 541 547 548
Contents xxvi 11.9.5 Concluding Remarks . 11.10 Dynamics of Rescaled Energy Levels . 11.11 Level Curvature Statistics. 11.12 Level Velocity Statistics. 11.13 Dyson’s Brownian-Motion Model. 11.14 Local and Global Equilibrium in Spectra. 11.15 Problems. References. 12 Dissipative Systems. 12.1 Preliminaries. 12.2 Hamiltonian Embeddings. 12.3 Time-Scale Separation for Probabilities and Coherences. 12.4 Dissipative Death of Quantum Recurrences. 12.5 Complex Energies and Quasi-Energies. 12.6 Different Degrees of Level Repulsion for Regular and Chaotic Motion. 611 12.7 Poissonian Random Process in the Plane. 12.8 Ginibre’s Ensemble of Random Matrices. 12.8.1 Normalizing the Joint
Density . 12.8.2 The Density of Eigenvalues. 12.8.3 The Reduced Joint Densities. 12.8.4 The Spacing Distribution. 12.9 General Properties of Generators. 12.10 Universality of Cubic Level Repulsion. 12.10.1 Antiunitary Symmetries. 12.10.2 Microreversibility . 12.11 Dissipation of Quantum Localization. 12.11.1 Zaslavsky’s Map. 12.11.2 Damped Rotator. 12.11.3 Destruction of Localization. 12.12 Problems. References. Index 550 551 555 563 567 577 585 588 591 591 591 597 600 609 614 616 617 619 621 622 628 632 632 634 640 640 643 646 649 652 655 |
| any_adam_object | 1 |
| author | Haake, Fritz 1941-2019 Gnutzmann, Sven 1970- Kuś, Marek 1955- |
| author_GND | (DE-588)1196383804 (DE-588)12236550X (DE-588)1201420512 |
| author_facet | Haake, Fritz 1941-2019 Gnutzmann, Sven 1970- Kuś, Marek 1955- |
| author_role | aut aut aut |
| author_sort | Haake, Fritz 1941-2019 |
| author_variant | f h fh s g sg m k mk |
| building | Verbundindex |
| bvnumber | BV045868335 |
| classification_rvk | UK 7800 UK 7600 |
| ctrlnum | (OCoLC)1222399721 (DE-599)DNB1161817514 |
| discipline | Physik |
| edition | Fourth edition |
| format | Book |
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| id | DE-604.BV045868335 |
| illustrated | Illustrated |
| indexdate | 2024-07-20T06:38:07Z |
| institution | BVB |
| institution_GND | (DE-588)1064344704 |
| isbn | 9783319975795 331997579X |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-031251681 |
| oclc_num | 1222399721 |
| open_access_boolean | |
| owner | DE-703 DE-355 DE-BY-UBR DE-188 DE-83 |
| owner_facet | DE-703 DE-355 DE-BY-UBR DE-188 DE-83 |
| physical | xxvi, 659 Seiten Illustrationen, Diagramme |
| publishDate | 2018 |
| publishDateSearch | 2018 |
| publishDateSort | 2018 |
| publisher | Springer |
| record_format | marc |
| series2 | Springer series in synergetics |
| spelling | Haake, Fritz 1941-2019 Verfasser (DE-588)1196383804 aut Quantum signatures of chaos Fritz Haake, Sven Gnutzmann, Marek Kuś Fourth edition Cham Springer [2018] © 2018 xxvi, 659 Seiten Illustrationen, Diagramme txt rdacontent n rdamedia nc rdacarrier Springer series in synergetics Chaostheorie (DE-588)4009754-7 gnd rswk-swf Quantenchaos (DE-588)4130849-9 gnd rswk-swf Quantentheorie (DE-588)4047992-4 gnd rswk-swf Quantenmechanik (DE-588)4047989-4 gnd rswk-swf Random matrix theory classical Hamiltonian chaos dissipative systems level dynamics quantum localization quantum time evolution universal spectral fluctuations Quantenchaos (DE-588)4130849-9 s DE-604 Quantenmechanik (DE-588)4047989-4 s Chaostheorie (DE-588)4009754-7 s 1\p DE-604 Quantentheorie (DE-588)4047992-4 s 2\p DE-604 Gnutzmann, Sven 1970- Verfasser (DE-588)12236550X aut Kuś, Marek 1955- Verfasser (DE-588)1201420512 aut Springer International Publishing (DE-588)1064344704 pbl Erscheint auch als Online-Ausgabe 978-3-319-97580-1 (DE-604)BV045913139 Vorangegangen ist 9783642054273 X:MVB text/html http://deposit.dnb.de/cgi-bin/dokserv?id=96876331c2eb415d95fda10142a8278a&prov=M&dok_var=1&dok_ext=htm Inhaltstext X:MVB http://www.springer.com/ Digitalisierung UB Regensburg - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=031251681&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 2\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Haake, Fritz 1941-2019 Gnutzmann, Sven 1970- Kuś, Marek 1955- Quantum signatures of chaos Chaostheorie (DE-588)4009754-7 gnd Quantenchaos (DE-588)4130849-9 gnd Quantentheorie (DE-588)4047992-4 gnd Quantenmechanik (DE-588)4047989-4 gnd |
| subject_GND | (DE-588)4009754-7 (DE-588)4130849-9 (DE-588)4047992-4 (DE-588)4047989-4 |
| title | Quantum signatures of chaos |
| title_auth | Quantum signatures of chaos |
| title_exact_search | Quantum signatures of chaos |
| title_full | Quantum signatures of chaos Fritz Haake, Sven Gnutzmann, Marek Kuś |
| title_fullStr | Quantum signatures of chaos Fritz Haake, Sven Gnutzmann, Marek Kuś |
| title_full_unstemmed | Quantum signatures of chaos Fritz Haake, Sven Gnutzmann, Marek Kuś |
| title_short | Quantum signatures of chaos |
| title_sort | quantum signatures of chaos |
| topic | Chaostheorie (DE-588)4009754-7 gnd Quantenchaos (DE-588)4130849-9 gnd Quantentheorie (DE-588)4047992-4 gnd Quantenmechanik (DE-588)4047989-4 gnd |
| topic_facet | Chaostheorie Quantenchaos Quantentheorie Quantenmechanik |
| url | http://deposit.dnb.de/cgi-bin/dokserv?id=96876331c2eb415d95fda10142a8278a&prov=M&dok_var=1&dok_ext=htm http://www.springer.com/ http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=031251681&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT haakefritz quantumsignaturesofchaos AT gnutzmannsven quantumsignaturesofchaos AT kusmarek quantumsignaturesofchaos AT springerinternationalpublishing quantumsignaturesofchaos |