## Vorticity, Statistical Mechanics, and Monte Carlo SimulationThis 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

1 | |

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 |

283 | |

### Other editions - View all

Vorticity, Statistical Mechanics, and Monte Carlo Simulation Chjan C. Lim,Joseph Nebus No preview available - 2006 |

Vorticity, Statistical Mechanics, and Monte Carlo Simulation Chjan C. Lim,Joseph Nebus No preview available - 2010 |

### Common terms and phrases

angular momentum approximation barotropic flows Barotropic Vorticity configuration constant constraint coordinates Coupled Barotropic defined density derivative differential discrete distance domain dynamics enstrophy entropy equal equation equilibrium statistical expectation value finite fluid flow free energy free particles geodesic split given Hamiltonian infinitely integral interaction inverse temperature inviscid kinetic energy Latitude-Longitude lattice linear Markov chain mean field mean field theory mean nearest neighbor mesh sites method Metropolis-Hastings algorithm microstate minimizers Monte Carlo simulations motion nearest neighbor parity negative temperatures number of mesh number of points octahedron pair partition function phase space phase transition plane point vortices polyhedra polyhedron probability properties quantity radial distribution function radius random variable relative enstrophy rotating sphere spherical model spin-lattice models statistical equilibrium statistical mechanics symmetry theorem theory total circulation triangle vector vortex gas problem vorticity field zero

### Popular passages

Page 1 - In my entire scientific life, extending over forty-five years, the most shattering experience has been the realization that an exact solution of Einstein's equations of general relativity, discovered by the New Zealand mathematician, Roy Kerr, provides the absolutely exact representation of untold numbers of massive black holes that populate the universe.

Page 279 - PG SAFFMAN, The approach of a vortex pair to a plane surface in an inviscid fluid, J.

Page 266 - Nato Advanced Study Institute on Nonlinear Equations in Physics and Mathematics (Istanbul, August 1977) (ed.

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Page 264 - R. Raczka, Theory of Group Representations and Applications, PWN - Polish Scientific Publishers, Warszawa 1977.

### References to this book

Proceedings of the Workshop Collective Phenomena in Macroscopic Systems ... Giuseppe Bertin,R. Pozzoli,M. Rome No preview available - 2007 |