VORTICITY, STATISTICAL MECHANICS, AND MONTE CARLO SIMULATION

By PAUL WILLIAMS

CONTENTS INTRODUCTION STATISTICAL MECHANICS AND VORTEX PROBLEMS EULER’S EQUATION FOR INVISCID FLUID FLOW PROBABILITY BASIC NOTIONS RANDOM VARIABLES AND DISTRIBUTION FUNCTIONS EXPECTATION VALUES AND AVERAGES VARIANCE MULTIPLE VARIABLES AND INDEPENDENCE LIMITING THEOREMS BAYESIAN PROBABILITY REMARKS STATISTICAL MECHANICS ENSEMBLES PARTITION FUNCTIONS CONSTRAINTS AND CHEMICAL POTENTIALS PARTITION FUNCTIONS BY LAGRANGE MULTIPLIERS MICROSTATE AND MACROSTATES EXPECTATION VALUES THERMODYNAMICS FROM Z FLUCTUATIONS 3 10 APPLICATIONS THE MONTE CARLO APPROACH MICROSTATES AND MACROSTATES MARKOV CHAINS DETAILED BALANCE THE METROPOLIS RULE MULTIPLE CANONICAL CONSTRAINTS ENSEMBLE AVERAGES RANDOM NUMBER GENERATION LINEAR CONGRUENCE GENERATORS MULTIPLE RECURSIVE AND FIBONACCI GENERATORS ADD-WITH-CARRY AND SUBTRACT-WITH-CARRY GENERATORS INVERSE CONGRUENTIAL GENERATORS COMBINED GENERATORS SPECTRAL METHODS INNER PRODUCTS BASIS FUNCTIONS MINIMIZERS FOURIER TRANSFORMS SPHERICAL HARMONICS DISCRETE MODELS IN FLUIDS EULER’S EQUATIONS FOR FLUID FLOWS N-BODY HAMILTONIANS SYMPLECTIC VARIABLES COORDINATES AND STEREOGRAPHIC PROJECTION DYNAMICS ON THE PLANE DYNAMICS ON THE SPHERE SPIN-LATTICE MODELS SPIN-LATTICE MODELS THE LATTICE MODEL POINT STRENGTH NORMALIZED STRENGTH NEGATIVE TEMPERATURES PHASE TRANSITIONS ENERGY-ENSTROPHY-CIRCULATION MODEL SOLUTION OF THE SPHERICAL ISING MODEL FOR ? = 0 MONTE CARLO SIMULATIONS CORRELATION FUNCTIONS THE MEAN NEAREST NEIGHBOR PARITY DISTANCES POLYHEDRA AND GROUND STATES FACE-SPLITTING OPERATIONS CENTROID SPLITTING GEODESIC SPLITTING NONCOMMUTIVITY POLYHEDRAL FAMILIES POLYHEDRAL FAMILIES VERSUS VORTEX GAS PAIRWISE INTERACTION ENERGIES ENERGY OF SPLIT FACES TETRAHEDRON SPLITTINGS 130 VORTICES OCTAHEDRON SPLITTINGS 258 VORTICES ICOSAHEDRON SPLITTINGS 642 VORTICES MESH GENERATION 10 1 INTRODUCTION 10 2 THE VORTEX GAS ON THE SPHERE 10 3 RADIAL DISTRIBUTION FUNCTION 10 4 VORTEX GAS RESULTS 10 5 RIGID BODIES 10 6 SPHERICAL CODES STATISTICAL MECHANICS FOR A VORTEX GAS 11 1 INTRODUCTION 11 2 THE VORTEX GAS ON THE PLANE 11 2 1 TRAPPED SLENDER VORTEX FILAMENTS 11 3 THE DISCRETIZED MODEL 11 4 EXTREMIZING E 11 5 VARIATIONAL PROBLEM ON THE RADIUS TWO-LAYER QUASI-GEOSTROPHIC MODELS 12 1 INTRODUCTION 12 2 TWO-LAYER QUASI-GEOSTROPHIC MODELS 12 3 GOVERNING EQUATIONS 12 4 NUMERICAL MODELS 12 5 NUMERICAL VORTEX STATISTICS 13 1 INTRODUCTION 13 2 THE COUPLED BAROTROPIC VORTICITY – ROTATING SPHERE SYSTEM 13 2 1 PHYSICAL QUANTITIES OF THE COUPLED FLOW – ROTATING SPHERE MODEL 13 3 VARIATIONAL RESULTS 13 4 ENERGY-ENSTROPHY THEORY FOR BAROTROPIC FLOWS 13 4 1 NON-ROTATING EXTREMALS 13 4 2 ROTATING SPHERE 13 5 STATISTICAL MECHANICS 13 5 1 DERIVATION OF SPIN-LATTICE HAMILTONIANS 13 5 2 GAUSSIAN ENERGY-ENSTROPHY MODEL 13 6 SPHERICAL MODEL 13 7 MONTE CARLO SIMULATIONS OF THE SPHERICAL MODEL 13 8 THE VECTOR SPHERICAL MODEL FOR BAROTROPIC STATISTICS 13 9 REMARKS REFERENCES