Vorticity, statistical mechanics, and Monte Carlo simulation / Chjan Lim, Joseph Nebus.

Format:

Book

Author/Creator:

Lim, Chjan.

Contributor:

Nebus, Joseph.

Series:

Springer monographs in mathematics 1439-7382

Springer monographs in mathematics, 1439-7382

Language:

English

Subjects (All):

Statistical mechanics.

Mathematical physics.

Fluid mechanics.

Physical Description:

xiv, 290 pages : illustrations ; 24 cm.

Place of Publication:

New York : Springer, 2007.

Summary:

This book is drawn from across many active fields of mathematics and physics, and has connections to atmospheric dynamics, spherical codes, graph theory, constrained optimization problems, Markov Chains, and Monte Carlo methods. It addresses how to access interesting, original, and publishable research in statistical modeling of large-scale flows and several related fields. The authors of this book explicitly reach around the major branches of mathematics and physics, showing how the use of a few straightforward approaches can create a cornucopia of intriguing questions and provide the tools to answer them.

In this book, the reader will learn how to research a topic and how to understand statistical mechanics treatments of fluid dynamics. Of particular interest should be the application of Monte Carlo methods to problems like the dispersal of points on the sphere, the phase transitions of inviscid fluid flows in models that increasingly approximate the conditions of actual planetary atmospheres, and the treatment of negative-absolute temperatures and the effects these extremely high-energy states have on fluid flows. Special attention is given to spherical models as well.

This book is intended for upper-level undergraduate or beginning level graduate courses of mathematics and physics. It will also be of use to readers interested in statistical mechanics methods applied to fluid mechanics problems. Readers will gain an understanding of how to synthesize new mathematics by applying familiar tools in new ways, and developing new tools to fit particular applications.

Contents:

1.1 Statistical Mechanics and Vortex Problems 1

1.2 Euler's Equation for Inviscid Fluid Flow 4

2 Probability 9

2.2 Basic Notions 9

2.3 Random Variables and Distribution Functions 10

2.4 Expectation Values and Averages 12

2.5 Variance 15

2.6 Multiple Variables and Independence 17

2.7 Limiting Theorems 19

2.8 Bayesian Probability 25

2.9 Remarks 26

3 Statistical Mechanics 29

3.2 Ensembles 29

3.3 Partition Functions 30

3.4 Constraints and Chemical Potentials 32

3.5 Partition Functions by Lagrange Multipliers 34

3.6 Microstate and Macrostates 35

3.7 Expectation Values 38

3.8 Thermodynamics from Z 39

3.9 Fluctuations 42

3.10 Applications 44

4 The Monte Carlo Approach 51

4.2 Microstates and Macrostates 51

4.3 Markov Chains 53

4.4 Detailed Balance 55

4.5 The Metropolis Rule 55

4.6 Multiple Canonical Constraints 57

4.7 Ensemble Averages 58

4.8 Random Number Generation 62

4.8.1 Linear Congruence Generators 62

4.8.2 Multiple Recursive and Fibonacci Generators 63

4.8.3 Add-with-Carry and Subtract-with-Carry Generators 63

4.8.4 Inverse Congruential Generators 64

4.8.5 Combined Generators 65

5 Spectral Methods 67

5.2 Inner Products 67

5.3 Basis Functions 70

5.4 Minimizers 72

5.5 Fourier transforms 73

5.6 Spherical Harmonics 77

6 Discrete Models in Fluids 79

6.2 Euler's Equations for Fluid Flows 80

6.3 N-body Hamiltonians 83

6.4 Symplectic Variables 86

6.5 Coordinates and Stereographic Projection 89

6.6 Dynamics on the Plane 91

6.7 Dynamics on the Sphere 103

6.8 Remarks 113

7 Spin-Lattice Models 115

7.2 Spin-lattice Models 116

7.3 The Lattice Model 117

7.3.1 Point Strength 118

7.3.2 Normalized Strength 121

7.4 Negative Temperatures 124

7.5 Phase Transitions 125

7.6 Energy-Enstrophy-Circulation Model 127

7.7 Solution of the Spherical Ising Model for [Characters not reproducible] = 0 128

8 Monte Carlo Simulations 133

8.2 Correlation Functions 137

8.3 The Mean Nearest Neighbor Parity 140

8.4 Distances 157

8.5 Remarks 159

9 Polyhedra and Ground States 161

9.2 Face-Splitting Operations 162

9.2.1 Centroid Splitting 163

9.2.2 Geodesic Splitting 164

9.2.3 Noncommutivity 165

9.3 Polyhedral Families 168

9.4 Polyhedral Families Versus Vortex Gas 169

9.4.1 Pairwise Interaction Energies 170

9.5 Energy of Split Faces 173

9.5.1 Tetrahedron Splittings 174

9.5.2 130 Vortices 176

9.5.3 Octahedron Splittings 178

9.5.4 258 Vortices 180

9.5.5 Icosahedron Splittings 182

9.5.6 642 Vortices 184

10 Mesh Generation 187

10.2 The Vortex Gas on the Sphere 188

10.3 Radial Distribution Function 190

10.4 Vortex Gas Results 192

10.5 Rigid Bodies 206

10.6 Spherical Codes 209

11 Statistical Mechanics for a Vortex Gas 213

11.2 The Vortex Gas on the Plane 214

11.2.1 Trapped Slender Vortex Filaments 219

11.3 The Discretized Model 222

11.4 Extremizing E 224

11.5 Variational Problem on the Radius 226

12 Two-Layer Quasi-Geostrophic Models 233

12.2 Two-Layer Quasi-Geostrophic Models 233

12.3 Governing Equations 234

12.4 Numerical Models 240

12.5 Numerical Vortex Statistics 241

13 A Coupled Barotropic Flow - Rotating Solid Sphere System 245

13.2 The Coupled Barotropic Vorticity - Rotating Sphere System 246

13.2.1 Physical Quantities of the Coupled Flow - Rotating Sphere Model 247

13.3 Variational Results 249

13.4 Energy-Enstrophy Theory for Barotropic Flows 250

13.4.1 Non-Rotating Extremals 251

13.4.2 Rotating Sphere 253

13.5 Statistical Mechanics 254

13.5.1 Derivation of Spin-Lattice Hamiltonians 254

13.5.2 Gaussian Energy-Enstrophy Model 255

13.6 Spherical Model 256

13.7 Monte Carlo Simulations of the Spherical Model 257

13.8 The Vector Spherical Model for Barotropic Statistics 257

13.9 Remarks 260.

Notes:

Includes bibliographical references (pages [263]-282) and index.

ISBN:

0387350756

9780387350752

OCLC:

76949134

Publisher Number:

9780387350752