MICROSCOPIC DYNAMICS OF PLASMAS AND CHAOS

By PAUL WHITE

CONTENTS BASIC PHYSICAL SETTING THE ORIGINAL N-BODY SYSTEM PHYSICAL CONTEXT THE PLASMA MODEL WAVE–PARTICLE RESONANCE: A PARADIGM OF CLASSICAL MECHANICS FROM N-BODY DYNAMICS TO WAVE–PARTICLE INTERACTION INTUITIVE DERIVATION OF THE SELF-CONSISTENT HAMILTONIAN LANGMUIR WAVES WITHOUT RESONANT PARTICLES DECOMPOSITION OF THE ?ELD AND PARTICLE MOTION—RELEVANT SMALL PARAMETERS COLLECTIVE DYNAMICS BOHM–GROSS MODES COUPLED MOTION OF QUASI-RESONANT PARTICLES WITH BOHM–GROSS MODES LAGRANGIAN FORMULATIO` REFERENCE STATES OF THE PLASM NON-EXISTENCE OF ZERO-?ELD STATES THERMAL DISTRIBUTION OF THE ELECTRIC ?ELD PHYSICAL SCALINGS AND ERROR ESTIMATES* FINAL FORM OF THE HAMILTONIAN HISTORICAL BACKGROUND AND NOTES DYNAMICS OF THE SMALL-AMPLITUDE WAVE–PARTICLE SYSTEM REFERENCE STATE WITH A VANISHING ELECTRIC ?ELD SMALL PERTURBATION TO THE REFERENCE STATE BALLISTIC SOLUTIONS WAVELIKE SOLUTIONS INITIAL VALUE PROBLEM DISPERSION RELATION FOR WAVELIKE MODES PHYSICAL INTERPRETATION: COLD BEAMS CASE OF A SINGLE BEAM CASE OF TWO BEAMS MANY COLD BEAMS OR A WARM BEAM LANDAU UNSTABLE MODE EIGENMODES AND INITIAL VALUE PROBLEM VAN KAMPEN MODES* RELATION WITH BALLISTIC EIGENMODES TRANSITION FROM COLD TO WARM BEAMS SYNCHRONIZATION OF PARTICLES WITH A WAVE SYNCHRONIZATION OF PARTICLES WITH A WAVELIKE MODE SYNCHRONIZATION OF PARTICLES DURING LANDAU DAMPING* FATE OF PARTICLES IN THE PRESENCE OF MANY INCOHERENT MODES 3 10 HISTORICAL BACKGROUND STATISTICAL DESCRIPTION OF THE SMALL-AMPLITUDE WAVE–PARTICLE DYNAMICS APPROACH USING PERTURBATION EXPANSION SECOND-ORDER PERTURBATION ANALYSIS EVOLUTION OF WAVES EVOLUTION OF PARTICLES FOKKER–PLANCK EQUATION FOR THE PARTICLES APPROACH USING FLOQUET EQUATION* LINK WITH TRADITIONAL DESCRIPTIONS LANDAU EFFECT FROM A VLASOVIAN POINT OF VIEW SPONTANEOUS EMISSION HAMILTONIAN CHAOS GEOMETRICAL TOOLS FOR HAMILTONIAN CHAOS POINCAR´E SURFACE OF SECTION CONSERVATION OF AREAS, SYMPLECTIC DYNAMICS AND ?UX* ACTION-ANGLE VARIABLES MOTION OF ONE PARTICLE IN THE PRESENCE OF TWO WAVES SMALL RESONANCE OVERLAP AND CANTORI MODERATE RESONANCE OVERLAP AND STOCHASTIC LAYERS PHYSICAL SUMMARY CONSTRUCTION OF ORBITS ORIGIN OF HIGHER-ORDER RESONANCES POINCAR´E AND KAM THEOREMS* HIGHER-ORDER RESONANCES FROM ACTION-ANGLE VARIABLES RENORMALIZATION FOR KAM TORI SIMPLE APPROACH TO RENORMALIZATION MORE EXPLICIT DERIVATION* STUDY OF THE RENORMALIZATION MAPPING THRESHOLDS AND EXPONENTS ORDER IN CHAOS WIGGLING ARMS OF THE X-POINT LARGE RESONANCE OVERLAP: NUMERICAL RESULTS LARGE RESONANCE OVERLAP: ANALYTICAL RESULTS* HISTORICAL BACKGROUND AND FURTHER COMMENTS DIFFUSION: THE CASE OF THE NON-SELF-CONSISTENT DYNAMICS MODEL HAMILTONIAN DIFFUSION AS A NUMERICAL FACT CONCEPT OF RESONANCE BOX HEURISTIC ANALYTICAL APPROACH NUMERICAL CHECK OF THE CONCEPT RIGOROUS APPROACH* SCALING PROPERTIES OF ?NITE-TIME DYNAMICS ORIGIN OF THE FORCE DECORRELATION LOCALITY AND RESONANCE BOXES FOR RANDOM PHASES CORRELATED PHASES RANDOM INITIAL POSITIONS ORIGIN OF THE CHAOTIC DIFFUSION LOCALITY AND DIFFUSION FOR MORE GENERAL HAMILTONIANS INITIAL QUASILINEAR DIFFUSION NON-CHAOTIC QUASILINEAR DIFFUSION INITIAL QUASILINEAR DIFFUSION UP TO CHAOS DIFFUSION COEF?CIENT IN THE CHAOTIC REGIME 6 10 DRAG COEF?CIENT 6 11 FOKKER–PLANCK EQUATION 6 12 HISTORICAL BACKGROUND SELF-CONSISTENT DYNAMICS IN THE DIFFUSIVE REGIME QUASILINEAR DIFFUSION COEF?CIENT SIMPLE DERIVATION OF THE QUASILINEAR EQUATIONS EVOLUTION OF WAVES* DRAG ON PARTICLES* JOINT EVOLUTION OF PARTICLES AND WAVES SATURATION OF THE WEAK-WARM-BEAM INSTABILITY TIME EVOLUTION OF THE SINGLE-WAVE–PARTICLE SYSTEM SELF-CONSISTENT NONLINEAR REGIME WITH A SINGLE WAVE INSTABILITY SATURATION NONLINEAR REGIME OF WAVE DAMPING UNSTABLE BEAM–WAVE SYSTEM COLD-BEAM–WAVE INSTABILITY LANDAU INSTABILITY COMPARISON WITH THE VLASOV-WAVE DESCRIPTION LINEAR AND NONLINEAR WAVE DAMPING EVOLUTION OF THE WAVE INTENSITY FINITE-N EFFECTS COMPETING WITH LANDAU DAMPING GIBBSIAN EQUILIBRIUM OF THE SINGLE-WAVE–PARTICLE SYSTEM* GIBBS MICROCANONICAL AND CANONICAL ENSEMBLES PARTITION FUNCTION THERMODYNAMIC DISCUSSION WEIGHT FUNCTION HIGH-TEMPERATURE PHASE THERMODYNAMICS AND WAVE INTENSITY PARTICLE DISTRIBUTION AND JOINT PROBABILITIES LOW-TEMPERATURE PHASE PHASE DIAGRAMS BERNSTEIN–GREENE–KRUSKAL STATE COMPARISON WITH NUMERICAL SIMULATIONS FURTHER COMMENTS 10 CONCLUSION 10 1 SUMMARY 10 2 OPEN ISSUES A CONTINUOUS AND DISCRETE FOURIER TRANSFORMS PERTURBED LATTICES AND DISCRETE FOURIER TRANSFORMS REPRESENTATION OF TRIGONOMETRIC FUNCTIONS LAPLACE AND FOURIER TRANSFORMS B SPECIAL FUNCTIONS EULER GAMMA FUNCTION BESSEL FUNCTIONS DIRAC DISTRIBUTION C PHASE SPACE STRUCTURE AND ORBITS D SYMPLECTIC STRUCTURE AND NUMERICAL INTEGRATION SYMPLECTIC DYNAMICS* NUMERICAL INTEGRATION SCHEMES SYMPLECTIC INTEGRATION OF HAMILTONIAN DYNAMICS E PROBABILITY AND STOCHASTIC PROCESSES GENERAL RANDOM VARIABLES AND VECTORS GAUSSIAN DISTRIBUTIONS FOKKER–PLANCK EQUATION F ESTIMATES FOR CHAPTER N-TIME CORRELATION FUNCTION OF C(?) AND S(?) ESTIMATE OF THE MEAN NUMBER OF VISITED BOXES INITIAL BROWNIAN MOTION DEPENDENCE OF AN ORBIT ON ONE PHASE ESTIMATE OF NON-QUASILINEAR TERMS G VLASOVIAN FORMULATION OF WAVE–PLASMA DYNAMICS VLASOV-WAVE PARTIAL DIFFERENTIAL SYSTEM FORMULATION DERIVATION FROM N-BODY DYNAMICS: KINETIC LIMIT FOURIER-TRANSFORMED VLASOV-WAVE SYSTEM ZERO-?ELD SOLUTIONS AND THEIR LINEAR STABILITY WAVELIKE MODES BALLISTIC MODES COMPLETE SOLUTION BERNSTEIN–GREENE–KRUSKAL MODES VLASOV–POISSON SYSTEM REFERENCES