Vorticity, Statistical Mechanics, and Monte Carlo Simulation

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Springer Science & Business Media, Jul 28, 2007 - Mathematics - 290 pages
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This book is meant for an audience of advanced undergraduates and graduate students taking courses on the statistical mechanics approach to turbulent ?ows and on stochastic simulations. It is also suitable for the self-study of professionals involved in the research and modelling of large scale stochastic ?uid ?ows with a substantial vortical component. Several related ideas motivate the approach in this book, namely, the application of equilibrium statistical mechanics to two-dimensional and 2- dimensional ?uid ?ows in the spirit of Onsager [337], and Kraichnan [227], is taken to be a valid starting point, and the primary importance of non-linear convection e?ects combined with the gravitational and rotational properties of large scale strati?ed ?ows over the secondary e?ects of viscosity is assumed. The latter point is corroborated by the many successful studies of ?uid v- cosity which limit its e?ects to speci?c and narrow regions such as boundary layers, and to the initial and transient phases of the experiment such as in the Ekman layer and spin-up [154] [344].
 

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Contents

Introduction
1
12 Eulers Equation for Inviscid Fluid Flow
4
Probability
9
23 Random Variables and Distribution Functions
10
24 Expectation Values and Averages
12
25 Variance
15
26 Multiple Variables and Independence
17
27 Limiting Theorems
19
Monte Carlo Simulations of SpinLattice Models on the Sphere
133
82 Correlation Functions
137
83 The Mean Nearest Neighbor Parity
140
84 Distances
157
85 Remarks
159
Polyhedra and Ground States
161
92 FaceSplitting Operations
162
921 Centroid Splitting
163

28 Bayesian Probability
25
29 Remarks
26
Statistical Mechanics
28
33 Partition Functions
30
34 Constraints and Chemical Potentials
32
35 Partition Functions by Lagrange Multipliers
34
36 Microstate and Macrostates
35
37 Expectation Values
38
38 Thermodynamics from Z
39
39 Fluctuations
42
310 Applications
44
The Monte Carlo Approach
51
43 Markov Chains
53
44 Detailed Balance
55
46 Multiple Canonical Constraints
57
47 Ensemble Averages
58
48 Random Number Generation
62
482 Multiple Recursive and Fibonacci Generators
63
484 Inverse Congruential Generators
64
485 Combined Generators
65
Spectral Methods
66
53 Basis Functions
70
54 Minimizers
72
55 Fourier transforms
73
56 Spherical Harmonics
77
Discrete Models in Fluids
79
62 Eulers Equations for Fluid Flows
80
63 Nbody Hamiltonians
83
64 Symplectic Variables
86
65 Coordinates and Stereographic Projection
89
66 Dynamics on the Plane
91
67 Dynamics on the Sphere
103
68 Remarks
113
SpinLattice Models
115
72 Statistical Mechanics of Vortex SpinLattice Models
116
73 The Lattice Model
117
731 Point Strength
118
732 Normalized Strength
121
74 Negative Temperatures
124
75 Phase Transitions
125
76 EnergyEnstrophyCirculation Model
127
77 Solution of the Spherical Ising Model for Γ 0
128
922 Geodesic Splitting
164
923 Noncommutivity
165
93 Polyhedral Families
168
94 Polyhedral Families Versus Vortex Gas
169
941 Pairwise Interaction Energies
170
95 Energy of Split Faces
173
951 Tetrahedron Splittings
174
952 130 Vortices
176
953 Octahedron Splittings
178
954 258 Vortices
180
955 Icosahedron Splittings
182
956 642 Vortices
184
Mesh Generation
187
102 The Vortex Gas on the Sphere
188
103 Radial Distribution Function
190
104 Vortex Gas Results
192
105 Rigid Bodies
206
106 Spherical Codes
209
Statistical Mechanics for a Vortex Gas
213
112 The Vortex Gas on the Plane
214
1121 Trapped Slender Vortex Filaments
219
113 The Discretized Model
222
114 Extremizing E
224
TwoLayer QuasiGeostrophic Models
232
123 Governing Equations
234
124 Numerical Models
240
125 Numerical Vortex Statistics
241
Coupled Barotropic Vorticity Dynamics on a Rotating Sphere
245
132 The Coupled Barotropic Vorticity Rotating Sphere System
246
1321 Physical Quantities of the Coupled Flow Rotating Sphere Model
247
133 Rotating versus Nonrotating Sphere Variational Results
249
134 EnergyEnstrophy Theory for Barotropic Flows
250
Nonrotating Sphere
251
1342 Rotating Sphere
253
135 Statistical Mechanics
254
1352 Gaussian EnergyEnstrophy Model
255
136 Spherical Model
256
137 Monte Carlo Simulations of the Spherical Model
257
139 Remarks
260
References
262
Index
283
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