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Đây là bộ sách tiếng anh về chuyên ngành vật lý gồm các lý thuyết căn bản và lý liên quan đến công nghệ nano ,công nghệ vật liệu ,công nghệ vi điện tử,vật lý bán dẫn. Bộ sách này thích hợp cho những ai đam mê theo đuổi ngành vật lý và muốn tìm hiểu thế giới vũ trụ và hoạt độn ra sao.
Chaos: Classical and Quantum Part I: Deterministic Chaos Predrag Cvitanovi´c – Roberto Artuso – Ronnie Mainieri – Gregor Tanner – G´abor Vattay – Niall Whelan – Andreas Wirzba —————————————————————- ChaosBook.org/version11.8, Aug 30 2006 printed August 30, 2006 ChaosBook.org comments to: predrag@nbi.dk ii Contents Part I: Classical chaos Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . xiv 1 Overture 1 1.1 Why ChaosBook? . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Chaos ahead . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 The future as in a mirror . . . . . . . . . . . . . . . . . . . 4 1.4 A game of pinball . . . . . . . . . . . . . . . . . . . . . . . . 9 1.5 Chaos for cyclists . . . . . . . . . . . . . . . . . . . . . . . . 14 1.6 Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 1.7 From chaos to statistical mechanics . . . . . . . . . . . . . . 22 1.8 A guide to the literature . . . . . . . . . . . . . . . . . . . . 23 guide to exerci se s 26 - resum´e 27 - references 28 - exercises 30 2 Go with t he flow 31 2.1 Dynamical systems . . . . . . . . . . . . . . . . . . . . . . . 31 2.2 Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.3 Computing trajectories . . . . . . . . . . . . . . . . . . . . . 39 resum´e 40 - references 40 - exercises 42 3 Do it again 45 3.1 Poincar´e sections . . . . . . . . . . . . . . . . . . . . . . . . 45 3.2 Constructing a Poincar´e section . . . . . . . . . . . . . . . . 48 3.3 Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 resum´e 53 - references 53 - exercises 55 4 Local stability 57 4.1 Flows transport neighborhoods . . . . . . . . . . . . . . . . 57 4.2 Linear flows . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 4.3 Stability of flows . . . . . . . . . . . . . . . . . . . . . . . . 64 4.4 Stability of maps . . . . . . . . . . . . . . . . . . . . . . . . 67 resum´e 70 - references 70 - exercises 72 5 Newtonian dynamics 73 5.1 Hamiltonian flows . . . . . . . . . . . . . . . . . . . . . . . 73 5.2 Stability of Hamiltonian flows . . . . . . . . . . . . . . . . . 75 5.3 Symplectic maps . . . . . . . . . . . . . . . . . . . . . . . . 77 references 80 - exercises 82 iii iv CONTENTS 6 Billiards 85 6.1 Billiard dynamics . . . . . . . . . . . . . . . . . . . . . . . . 85 6.2 Stability of billiards . . . . . . . . . . . . . . . . . . . . . . 88 resum´e 91 - references 91 - exercises 93 7 Get straight 95 7.1 Changing coordinates . . . . . . . . . . . . . . . . . . . . . 95 7.2 Rectification of flows . . . . . . . . . . . . . . . . . . . . . . 97 7.3 Classical dyn amics of collinear h elium . . . . . . . . . . . . 98 7.4 Rectification of maps . . . . . . . . . . . . . . . . . . . . . . 102 resum´e 104 - references 104 - exercises 106 8 Cycle stability 107 8.1 Stability of period ic orbits . . . . . . . . . . . . . . . . . . . 107 8.2 Cycle stabilities are cycle invariants . . . . . . . . . . . . . 110 8.3 Stability of Poincar´e map cycles . . . . . . . . . . . . . . . . 112 8.4 Rectification of a 1-dimensional periodic orbit . . . . . . . . 112 8.5 Smooth conjugacies and cycle stability . . . . . . . . . . . . 114 8.6 Neighborhood of a cycle . . . . . . . . . . . . . . . . . . . . 114 resum´e 116 - references 116 - exercises 118 9 Transporting densities 119 9.1 Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 9.2 Perron-Frobenius operator . . . . . . . . . . . . . . . . . . . 121 9.3 Invariant measures . . . . . . . . . . . . . . . . . . . . . . . 123 9.4 Density evolution for infinitesimal times . . . . . . . . . . . 126 9.5 Liouville operator . . . . . . . . . . . . . . . . . . . . . . . . 129 resum´e 131 - references 132 - exercises 133 10 Averaging 137 10.1 Dynamical averaging . . . . . . . . . . . . . . . . . . . . . . 137 10.2 Evolution operators . . . . . . . . . . . . . . . . . . . . . . 144 10.3 Lyapunov exponents . . . . . . . . . . . . . . . . . . . . . . 146 10.4 Why not just run it on a computer? . . . . . . . . . . . . . 150 resum´e 152 - references 153 - exercises 154 11 Qualitative dynamics, for pedestrians 157 11.1 Qualitative dynamics . . . . . . . . . . . . . . . . . . . . . . 157 11.2 A brief detour; recoding, symmetries, tilings . . . . . . . . . 162 11.3 Stretch and fold . . . . . . . . . . . . . . . . . . . . . . . . . 164 11.4 Kneading theory . . . . . . . . . . . . . . . . . . . . . . . . 169 11.5 Markov graphs . . . . . . . . . . . . . . . . . . . . . . . . . 171 11.6 Symbolic dynamics, basic notions . . . . . . . . . . . . . . . 173 resum´e 178 - references 178 - exercises 180 12 Qualitative dynamics, for cyclists 183 12.1 Going global: S table/unstable manifolds . . . . . . . . . . . 184 12.2 Horseshoes . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 12.3 Spatial ordering . . . . . . . . . . . . . . . . . . . . . . . . . 188 12.4 Pruning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 resum´e 194 - references 195 - exercises 199 CONTENTS v 13 Counting 203 13.1 Counting itineraries . . . . . . . . . . . . . . . . . . . . . . 203 13.2 Topological trace formula . . . . . . . . . . . . . . . . . . . 206 13.3 Determinant of a graph . . . . . . . . . . . . . . . . . . . . 208 13.4 Topological zeta function . . . . . . . . . . . . . . . . . . . 211 13.5 Counting cycles . . . . . . . . . . . . . . . . . . . . . . . . . 214 13.6 In finite partitions . . . . . . . . . . . . . . . . . . . . . . . . 218 13.7 Shadowing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 resum´e 222 - references 222 - exercises 224 14 Trace formulas 231 14.1 Trace of an evolution operator . . . . . . . . . . . . . . . . 231 14.2 A trace formula for maps . . . . . . . . . . . . . . . . . . . 233 14.3 A trace formula for fl ows . . . . . . . . . . . . . . . . . . . . 235 14.4 An asymptotic trace formula . . . . . . . . . . . . . . . . . 238 resum´e 240 - references 240 - exercises 242 15 Spectral determinants 243 15.1 Spectral determinants for maps . . . . . . . . . . . . . . . . 243 15.2 Spectral determinant for flows . . . . . . . . . . . . . . . . . 245 15.3 Dynamical zeta f unctions . . . . . . . . . . . . . . . . . . . 247 15.4 False zeros . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251 15.5 Spectral determinants vs. dynamical zeta functions . . . . . 251 15.6 All too m any eigenvalues? . . . . . . . . . . . . . . . . . . . 253 resum´e 255 - references 256 - exercises 258 16 Why do e s it work? 261 16.1 Linear maps: exact spectra . . . . . . . . . . . . . . . . . . 262 16.2 Evolution operator in a matrix repr esentation . . . . . . . . 266 16.3 Classical Fredholm theory . . . . . . . . . . . . . . . . . . . 269 16.4 Analyticity of spectral determinants . . . . . . . . . . . . . 271 16.5 Hyperbolic maps . . . . . . . . . . . . . . . . . . . . . . . . 276 16.6 The physics of eigenvalues and eigenfunctions . . . . . . . . 278 16.7 Troubles ahead . . . . . . . . . . . . . . . . . . . . . . . . . 281 resum´e 284 - references 284 - exercises 286 17 Fixed points, and how to get them 287 17.1 Where are the cycles? . . . . . . . . . . . . . . . . . . . . . 288 17.2 One-dimensional mappings . . . . . . . . . . . . . . . . . . 290 17.3 Multipoint shooting method . . . . . . . . . . . . . . . . . . 291 17.4 d-dimensional mappings . . . . . . . . . . . . . . . . . . . . 294 17.5 Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294 resum´e 298 - references 299 - exercises 301 18 Cycle expansions 305 18.1 Pseudocycles and shadowing . . . . . . . . . . . . . . . . . . 305 18.2 Construction of cycle expansions . . . . . . . . . . . . . . . 308 18.3 Cycle formulas for dynamical averages . . . . . . . . . . . . 312 18.4 Cycle expansions for finite alphabets . . . . . . . . . . . . . 316 18.5 Stability ordering of cycle expansions . . . . . . . . . . . . . 317 18.6 Dirichlet series . . . . . . . . . . . . . . . . . . . . . . . . . 320 vi CONTENTS resum´e 322 - references 323 - exercises 325 19 Why cycle? 329 19.1 Escape rates . . . . . . . . . . . . . . . . . . . . . . . . . . . 329 19.2 Natural measure in terms of periodic orbits . . . . . . . . . 332 19.3 Flow conservation sum rules . . . . . . . . . . . . . . . . . . 333 19.4 Corr elation f unctions . . . . . . . . . . . . . . . . . . . . . . 334 19.5 Trace formulas vs. level sums . . . . . . . . . . . . . . . . . 336 resum´e 338 - references 338 - exercises 339 20 Thermodynamic formalism 341 20.1 R´enyi entropies . . . . . . . . . . . . . . . . . . . . . . . . . 341 20.2 Fractal dimensions . . . . . . . . . . . . . . . . . . . . . . . 346 resum´e 349 - references 350 - exercises 351 21 Intermit tency 353 21.1 Intermittency everywhere . . . . . . . . . . . . . . . . . . . 354 21.2 Intermittency for pedestrians . . . . . . . . . . . . . . . . . 357 21.3 Intermittency for cyclists . . . . . . . . . . . . . . . . . . . 369 21.4 BER zeta functions . . . . . . . . . . . . . . . . . . . . . . . 375 resum´e 379 - references 379 - exercises 381 22 Discrete symmetries 385 22.1 Preview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386 22.2 Discrete symmetries . . . . . . . . . . . . . . . . . . . . . . 390 22.3 Dynamics in the f undamental domain . . . . . . . . . . . . 392 22.4 Factorizations of dynamical zeta functions . . . . . . . . . . 396 22.5 C 2 factorization . . . . . . . . . . . . . . . . . . . . . . . . . 398 22.6 C 3v factorization: 3-disk game of pinball . . . . . . . . . . . 400 resum´e 403 - references 404 - exercises 406 23 Deterministic diffusion 409 23.1 Diffusion in period ic arrays . . . . . . . . . . . . . . . . . . 410 23.2 Diffusion induced by chains of 1-d maps . . . . . . . . . . . 414 23.3 Marginal stability and anomalous diffusion . . . . . . . . . . 422 resum´e 426 - references 427 - exercises 429 24 Irrationally winding 431 24.1 Mode locking . . . . . . . . . . . . . . . . . . . . . . . . . . 432 24.2 Local theory: “Golden mean” renormalization . . . . . . . . 438 24.3 Global theory: Thermodynamic averaging . . . . . . . . . . 440 24.4 Hausdorff dimension of irrational windings . . . . . . . . . . 442 24.5 Thermodynamics of Farey tree: Farey model . . . . . . . . 444 resum´e 449 - references 449 - exercises 452 CONTENTS vii Part II: Quantum chaos 25 Prologue 455 25.1 Quantum pinball . . . . . . . . . . . . . . . . . . . . . . . . 456 25.2 Quantization of helium . . . . . . . . . . . . . . . . . . . . . 458 guide to literature 459 - references 460 - 26 Quant um mechanics, briefly 461 exercises 466 27 WKB quantization 467 27.1 WKB ansatz . . . . . . . . . . . . . . . . . . . . . . . . . . 467 27.2 Method of s tationary phase . . . . . . . . . . . . . . . . . . 470 27.3 WKB quantization . . . . . . . . . . . . . . . . . . . . . . . 471 27.4 Beyond the quadratic saddle point . . . . . . . . . . . . . . 473 resum´e 475 - references 475 - exercises 477 28 Semiclassical evolution 479 28.1 Hamilton-Jacobi theory . . . . . . . . . . . . . . . . . . . . 479 28.2 Semiclassical propagator . . . . . . . . . . . . . . . . . . . . 488 28.3 Semiclassical Green’s function . . . . . . . . . . . . . . . . . 491 resum´e 498 - references 499 - exercises 501 29 Noise 505 29.1 Deterministic transport . . . . . . . . . . . . . . . . . . . . 506 29.2 Brownian difussion . . . . . . . . . . . . . . . . . . . . . . . 507 29.3 Weak noise . . . . . . . . . . . . . . . . . . . . . . . . . . . 508 29.4 Weak noise approximation . . . . . . . . . . . . . . . . . . . 510 resum´e 512 - references 512 - 30 Semiclassical quantization 515 30.1 Trace formula . . . . . . . . . . . . . . . . . . . . . . . . . . 515 30.2 Semiclassical spectral determinant . . . . . . . . . . . . . . 520 30.3 One-dof systems . . . . . . . . . . . . . . . . . . . . . . . . 522 30.4 Two-dof systems . . . . . . . . . . . . . . . . . . . . . . . . 523 resum´e 524 - references 525 - exercises 528 31 Relaxation for cyclists 529 31.1 Fictitious time relaxation . . . . . . . . . . . . . . . . . . . 530 31.2 Discrete iteration r elaxation method . . . . . . . . . . . . . 536 31.3 Least action method . . . . . . . . . . . . . . . . . . . . . . 538 resum´e 542 - references 542 - exercises 544 32 Quant um scattering 545 32.1 Density of states . . . . . . . . . . . . . . . . . . . . . . . . 545 32.2 Quantum mechanical scattering m atrix . . . . . . . . . . . . 549 32.3 Kr ein-Friedel-Lloyd formula . . . . . . . . . . . . . . . . . . 550 32.4 Wigner time d elay . . . . . . . . . . . . . . . . . . . . . . . 553 references 555 - exercises 558 viii CONTENTS 33 Chaotic multiscattering 559 33.1 Quantum mechanical scattering matrix . . . . . . . . . . . . 560 33.2 N -scatterer spectral determinant . . . . . . . . . . . . . . . 563 33.3 Semiclassical 1-disk scattering . . . . . . . . . . . . . . . . . 567 33.4 From quantum cycle to semiclassical cycle . . . . . . . . . . 574 33.5 Heisenberg uncertainty . . . . . . . . . . . . . . . . . . . . . 577 34 Helium atom 579 34.1 Classical dynamics of collinear helium . . . . . . . . . . . . 580 34.2 Chaos, symbolic dynamics and periodic orbits . . . . . . . . 581 34.3 Local coordinates, fundamental matrix . . . . . . . . . . . . 586 34.4 Getting ready . . . . . . . . . . . . . . . . . . . . . . . . . . 588 34.5 Semiclassical quantization of collinear helium . . . . . . . . 589 resum´e 598 - references 599 - exercises 600 35 Diffraction distraction 603 35.1 Quantum eavesdropping . . . . . . . . . . . . . . . . . . . . 603 35.2 An application . . . . . . . . . . . . . . . . . . . . . . . . . 609 resum´e 616 - references 616 - exercises 618 Epilogue 619 Index 624 CONTENTS ix Part III: Appendices on ChaosBook.org A A brief history of chaos 639 A.1 Chaos is born . . . . . . . . . . . . . . . . . . . . . . . . . . 639 A.2 Chaos grows up . . . . . . . . . . . . . . . . . . . . . . . . . 643 A.3 Chaos with us . . . . . . . . . . . . . . . . . . . . . . . . . . 644 A.4 Death of the Old Quantum Theory . . . . . . . . . . . . . . 648 references 650 - B Infinite-dimensional flows 651 C Stability of Hamiltonian flows 655 C.1 Symplectic invariance . . . . . . . . . . . . . . . . . . . . . 655 C.2 Monodromy matrix for Hamiltonian flows . . . . . . . . . . 656 D Implementing evolution 659 D.1 Koopmania . . . . . . . . . . . . . . . . . . . . . . . . . . . 659 D.2 Implementing evolution . . . . . . . . . . . . . . . . . . . . 661 references 664 - exercises 665 E Symbolic dynamics techniques 667 E.1 Topological zeta functions for infinite subshifts . . . . . . . 667 E.2 Prime factorization for dynamical itineraries . . . . . . . . . 675 F Counting itineraries 681 F.1 Counting curvatures . . . . . . . . . . . . . . . . . . . . . . 681 exercises 683 G Finding cycles 685 G.1 Newton-Raphson method . . . . . . . . . . . . . . . . . . . 685 G.2 Hybrid Newton-Raphson / relaxation method . . . . . . . . 686 H Applications 689 H.1 Evolution operator for Lyapunov exponents . . . . . . . . . 689 H.2 Advection of vector fields by chaotic flows . . . . . . . . . . 694 references 698 - exercises 700 I Discrete symmetries 701 I.1 Preliminaries and definitions . . . . . . . . . . . . . . . . . . 701 I.2 C 4v factorization . . . . . . . . . . . . . . . . . . . . . . . . 706 I.3 C 2v factorization . . . . . . . . . . . . . . . . . . . . . . . . 711 I.4 H´enon map symmetries . . . . . . . . . . . . . . . . . . . . 713 I.5 Symmetries of th e symbol square . . . . . . . . . . . . . . . 714 J Convergence of spectral determinants 715 J.1 Curvature expansions: geometric picture . . . . . . . . . . . 715 J.2 On importance of pruning . . . . . . . . . . . . . . . . . . . 718 J.3 Ma-the-matical caveats . . . . . . . . . . . . . . . . . . . . . 719 J.4 Estimate of the nth cumulant . . . . . . . . . . . . . . . . . 720 x CONTENTS K Infinite dimensional operators 723 K.1 Matrix-valued functions . . . . . . . . . . . . . . . . . . . . 723 K.2 Operator n orms . . . . . . . . . . . . . . . . . . . . . . . . . 725 K.3 Trace class and Hilbert-Schmidt class . . . . . . . . . . . . . 726 K.4 Determinants of trace class operators . . . . . . . . . . . . . 728 K.5 Von Koch matrices . . . . . . . . . . . . . . . . . . . . . . . 732 K.6 Regularization . . . . . . . . . . . . . . . . . . . . . . . . . 733 references 735 - L Statistical mechanics recycled 737 L.1 The thermodynamic limit . . . . . . . . . . . . . . . . . . . 737 L.2 Ising models . . . . . . . . . . . . . . . . . . . . . . . . . . . 739 L.3 Fisher d roplet model . . . . . . . . . . . . . . . . . . . . . . 743 L.4 Scaling functions . . . . . . . . . . . . . . . . . . . . . . . . 748 L.5 Geometrization . . . . . . . . . . . . . . . . . . . . . . . . . 752 resum´e 759 - references 760 - exercises 762 M Noise/quantum corrections 765 M.1 Period ic orbits as integrable systems . . . . . . . . . . . . . 765 M.2 The Birkhoff normal form . . . . . . . . . . . . . . . . . . . 769 M.3 Bohr-Sommerfeld quantization of periodic orbits . . . . . . 770 M.4 Quantum calculation of corrections . . . . . . . . . . . . . 772 references 779 - N Solutions 781 O Projects 827 O.1 Deterministic diffusion, zig-zag map . . . . . . . . . . . . . 829 O.2 Deterministic diffusion, sawtooth map . . . . . . . . . . . . 836 [...]... only one prime cycle for each cyclic permutation class For example, p = 0011 = 1001 = 1100 = 0110 is prime, but 0101 = 01 is not By the chain rule for derivatives the stability of a cycle is the same everywhere along the orbit, so each prime cycle of length np contributes np terms to the sum (1.7) Hence (1.7) can be rewritten as Γ(z) = np p ∞ r=1 z np | p | r = p n p tp , 1 − tp tp = z np | p | (1.8)... bounces among the pinball machine’s disks, and only high-school level Euclidean geometry is needed to describe its trajectory A physicist’s pinball game is the game of pinball stripped to its bare essentials: three equidistantly placed reflecting disks in a plane, figure 1.1 A physicist’s pinball is free, frictionless, pointlike, spin-less, perfectly elastic, and noiseless Point-like pinballs are shot... uniquely specified by its position and momentum at a given instant, and no two distinct phase space trajectories can intersect Their projections onto arbitrary subspaces, however, can and do intersect, in rather unilluminating ways In the pinball example the problem is that we are looking at the projections of a 4-dimensional phase space trajectories onto a 2-dimensional subspace, the configuration space... examples of 3-disk cycles: (a) 12123 and 13132 are mapped into each other by the flip across 1 axis Similarly (b) 123 and 132 are related by flips, and (c) 1213, 1232 and 1323 by rotations (d) The cycles 121212313 and 121212323 are related by rotaion and time reversal These symmetries are discussed in more detail in chapter 22 (from ref [1.2]) p sin φ1 (s1 ,p1 ) a s1 p sin φ2 (s2 ,p2 ) s2 φ1 s1 φ1 (a) p sin... contributed, and the field followed no single line of development; rather one sees many interwoven strands of progress In retrospect many triumphs of both classical and quantum physics seem a stroke of luck: a few integrable problems, such as the harmonic oscillator and the Kepler problem, though “non-generic”, have gotten us very far The success has lulled us into a habit of expecting simple solutions to simple... line: A pinball game is used to motivate and illustrate most of the concepts to be developed in ChaosBook Throughout the book indicates that the section requires a hearty stomach and is probably best skipped on first reading fast track points you where to skip to tells you where to go for more depth on a particular topic indicates an exercise that might clarify a point in the text 1 2 CHAPTER 1 OVERTURE... and in addition had remembrance and intelligence enough to consider all the circumstances and to take them into account, he would be a prophet and would see the future in the present as in a mirror Leibniz chose to illustrate his faith in determinism precisely with the type of physical system that we shall use here as a paradigm of chaos His claim is wrong in a deep and subtle way: a state of a physical... initial data, the dynamics is predictable only up to a finite Lyapunov time 1 TLyap ≈ − ln |δx/L| , λ (1.1) despite the deterministic and, for Baron Leibniz, infallible simple laws that rule the pinball motion A positive Lyapunov exponent does not in itself lead to chaos One could try to play 1- or 2-disk pinball game, but it would not be much of a game; trajectories would only separate, never to meet again... chaos has in this context taken on a narrow technical meaning If a deterministic system is locally unstable (positive Lyapunov exponent) and globally mixing (positive entropy) - figure 1.3 - it is said to be chaotic While mathematically correct, the definition of chaos as “positive Lyapunov + positive entropy” is useless in practice, as a measurement of these quantities is intrinsically asymptotic and. .. spatially extended systems in terms of recurrent spatiotemporal patterns Pictorially, dynamics drives a given spatially extended system through a repertoire of unstable patterns; as we watch a turbulent system evolve, every so often we catch a glimpse of a familiar pattern: ChaosBook.org/version11.8, Aug 30 2006 intro - 10jul2006 appendix B 8 CHAPTER 1 OVERTURE =⇒ other swirls =⇒ For any finite spatial . A physicist’s pinball is free, frictionless, point- like, spin-less, perfectly elastic, and noiseless. Point-like pinballs are shot intro - 10jul2006 ChaosBook.org/version11.8,. page 5; Ya.B. Pesin for the remarks quoted on page 647; M.A. Porter for the quotations on page 19 and page 647; E.A. Spiegel for quotations on page 1 and