(Ebook PDF) The Lattice Boltzmann Equation For Complex States of Flowing Matter 1st edition by Sauro Succi 0192538853 9780192538857 full chapters

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ISBN-10 :  0192538853 

ISBN-13 :  9780192538857 

Author:  Sauro Succi 

Flowing matter is all around us, from daily-life vital processes (breathing, blood circulation), to industrial, environmental, biological, and medical sciences. Complex states of flowing matter are equally present in fundamental physical processes, far remote from our direct senses, such as quantum-relativistic matter under ultra-high temperature conditions (quark-gluon plasmas). Capturing the complexities of such states of matter stands as one of the most prominent challenges of modern science, with multiple ramifications to physics, biology, mathematics, and computer science. 

The Lattice Boltzmann Equation: For Complex States of Flowing Matter 1st Table of contents:

Part I Kinetic Theory of Fluids
1 Why a Kinetic Theory of Fluids?
1.1 The Navier–Stokes Equation
1.1.1 Elementary Derivation of the Navier–Stokes Equations
1.1.2 Navier–Stokes Equations in Lagrangian Form
1.1.3 Navier–Stokes Equations in Coordinate Form
1.2 Computational Aspects of the Navier–Stokes Equations
1.2.1 Why Is the Reynolds Number so Large?
1.3 The Benefits of Kinetic Extra Dimensions
1.3.1 Molecular Streaming versus Fluid Advection
1.3.2 Molecular Relaxation versus Momentum Diffusivity
1.4 Summary
References
Exercises
2 Boltzmann’s Kinetic Theory
2.1 Atomistic Dynamics
2.2 Statistical Dynamics: Boltzmann and the BBGKY Hierarchy
2.3 The Born–Bogoliubov–Green–Kirkwood–Yvon (BBGKY) Hierarchy
2.4 Back to Boltzmann
2.4.1 Two-Body Scattering
2.4.2 Spatial Ordering in Dilute Gases
2.4.3 Two-Body Scattering Problem
2.4.4 Distinguished Potentials
2.4.5 Molecular Chaos (Stosszahlansatz)
2.5 Local and Global Equilibria
2.5.1 Local Equilibria and Equation of State
2.5.2 The Evershifting Battle
2.6 Summary
References
Exercises
3 Approach to Equilibrium, the H-Theorem and Irreversibility
3.1 Approach to Equilibrium: the Second Principle of Thermodynamics
3.2 Approach to Equilibrium, the H-Theorem
3.2.1 Sketch of the Proof of the H-Theorem
3.2.2 H-theorem and Irreversibility
3.3 Collisionless Vlasov Equilibria
3.4 The Boltzmann Equation in Modern Mathematics
3.5 Summary
References
Exercises
4 Transport Phenomena
4.1 Length Scales and Transport Phenomena
4.2 Mean-Free Path (Again!)
4.3 Transport Parameters: Macroscopic Picture
4.3.1 Mass Diffusivity
4.3.2 Momentum Diffusivity: Kinematic Viscosity
4.3.3 Energy Diffusivity: Thermal Conductivity
4.4 Transport Parameters: Kinetic Picture
4.4.1 The Structure of Dissipation: Micro versus Macro
4.5 Dimensionless Numbers
4.6 Beyond Fick’s Law: Non-Local and Nonlinear Transport
4.7 Summary
References
Exercises
5 From Kinetic Theory to Navier–Stokes Hydrodynamics
5.1 From Kinetic Theory to Hydrodynamics: Heuristic Derivation
5.1.1 Moment Balance Equations
5.1.2 Closures: the Enslaving Principle
5.2 The Hilbert expansion
5.3 Beyond Hilbert: the Chapman–Enskog Multiscale Expansion
5.3.1 Higher-Order Hydrodynamics
5.4 Summary
References
Exercises
6 Generalized Hydrodynamics Beyond Navier–Stokes
6.1 Grad’s Thirteen-Moment Theory
6.2 Grad’s Approach in (Some) More Detail
6.3 R13: Regularized Grad-13 Expansion
6.4 Linearized Collision Operator
6.4.1 The CSF Spectral Decomposition
6.5 Summary: Grad versus Chapman–Enskog
References
Exercises
7 Kinetic Theory of Dense Fluids
7.1 Cautionary Remark
7.2 Finite-Density Extension of the Boltzmann Equation
7.3 The BBGKY Hierarchy
7.3.1 Statistical Mechanics: The Liouville Equation
7.3.2 Liouville Theorem
7.3.3 Taking Averages
7.3.4 Many-Body Equilibria of the Liouville Equation
7.4 Reduced Liouville Equations and the BBGKY Hierarchy
7.5 Two-Body Closures
7.5.1 Short and Mid-Range Interactions
7.6 Theories of Correlation Functions
7.7 The Enskog Equation
7.8 Entry Computers: Molecular Dynamics
7.9 Summary
References
Exercises
8 Model Boltzmann Equations
8.1 On Models and Theory
8.2 Bhatnagar–Gross–Krook Model Equation
8.3 Generalized BGK Models
8.3.1 Multi-Relaxation BGK
8.3.2 Multi-Stage BGK
8.4 Discrete Velocity Models
8.5 BGK Is Not (Necessarily) a Dilute-Gas Approximation
8.6 Summary
References
Exercises
9 Stochastic Particle Dynamics
9.1 Langevin Dynamics
9.2 Brownian Motion
9.2.1 Link to the Diffusion Equation
9.3 A Note on Coarse-Graining: Boltzmann versus Langevin
9.4 More on Langevin Dynamics
9.4.1 The Kramers Problem and First Passage Time
9.4.2 Driven Systems
9.5 Beyond Brownian Motion
9.5.1 Correlated Noise and Memory Effects
9.5.2 Nonlinear Friction
9.5.3 Runaway Electrons in Plasmas
9.5.4 Active Matter
9.6 Back to Phase Space: The Fokker–Planck Equation
9.6.1 Fokker–Planck Equation for Nonlinear Processes
9.7 Summary
References
Exercises
10 Numerical Methods for the Kinetic Theory of Fluids
10.1 Introduction
10.2 Molecular Dynamics
10.3 Direct Simulation Monte Carlo
10.4 Mesoscale Particle Methods
10.5 Multi-Particle Collisions
10.6 Dissipative Particle Dynamics
10.7 Grid Methods for the Boltzmann Equation
10.8 Lattice Particle Methods
10.9 Summary
References
Exercise
Part II Lattice Kinetic Theory
11 Lattice Gas-Cellular Automata
11.1 Boolean Hydrodynamics
11.2 Fluids in Gridland: The Frisch–Hasslacher–Pomeau Automaton
11.3 Fluons in Action: LGCA Microdynamic Evolution
11.4 From LGCA to Navier–Stokes
11.4.1 Discrete Local Equilibria
11.5 Practical Implementation
11.6 Lattice Gas Diseases and How to Cure Them
11.6.1 Statistical Noise
11.6.2 Low Reynolds Number
11.6.3 Exponential Complexity
11.6.4 Spurious Invariants
11.7 Summary
References
Exercises
12 Lattice Boltzmann Models with Underlying Boolean Microdynamics
12.1 Nonlinear LBE
12.1.1 Lattice Quantum Fluids
12.2 The Quasilinear LBE
12.3 The Scattering Matrix Aij
12.4 Numerical Experiments
12.5 Summary
References
Exercises
13 Lattice Boltzmann Models without Underlying Boolean Microdynamics
13.1 LBE with Enhanced Collisions
13.2 Hydrodynamic and Ghost Fields
13.3 Field-Theoretical Analogies
13.3.1 Extra-Dimensions
13.3.2 Ghost-Busters
13.4 The Route to Navier–Stokes: Enslaving Assumption
13.5 The Mirage of Zero Viscosity
13.6 Early Numerical Simulations
13.7 Summary
References
Exercises
14 Lattice Relaxation Schemes
14.1 Single-Relaxation Time
14.2 LBGK Equilibria
14.3 LBGK versus Early LBE
14.4 Multiple–Relaxation Time (MRT)
14.4.1 Constructing the MRT Matrix
14.4.2 Enhanced MRT Stability
14.5 Further Relaxation Variants
14.5.1 Regularized LBGK
14.5.2 Cascaded LBE
14.6 Summary
References
Exercises
15 The Hermite–Gauss Route to LBE
15.1 LBE from Continuum-Kinetic Theory: The Hermite–Gauss Approach
15.2 Multi-Dimensions, Higher-Order Lattices and Separability
15.3 Non-Cartesian Quadrature Schemes
15.3.1 Personal Afterthoughts
15.4 Relation to Discrete-Velocity Models
15.5 Summary: Three Ways to Lattice Boltzmann
References
Exercises
16 LBE in the Framework of Computational-Fluid Dynamics
16.1 LBE and CFD
16.1.1 Causality
16.1.2 Accuracy
16.2 Stability
16.2.1 Consistency
16.2.2 Efficiency
16.2.3 Parallel Computing
16.2.4 Flexibility
16.3 Link to Fully Lagrangian Schemes
16.4 Summary
References
Exercises
Part III Fluid Dynamics Applications
17 Boundary Conditions
17.1 Initial Conditions
17.2 General Formulation of LBE Boundary Conditions
17.3 Survey of Various Boundary Conditions
17.3.1 Periodic Boundary Conditions
17.3.2 No-Slip Boundary Conditions
17.3.3 Free-Slip Boundary Conditions
17.3.4 Frictional Slip
17.3.5 Sliding Walls
17.3.6 Wall Equilibria
17.3.7 The Inamuro Method
17.3.8 A Word of Caution: Analytical Results
17.4 Moving Boundaries
17.5 Open Boundaries
17.6 Exactly Incompressible LBE Schemes
17.6.1 Remark
17.6.2 Zou-He Non-Equilibrium Bounce-Back
17.7 Curved Boundaries
17.7.1 Staircased Boundaries
17.7.2 Extrapolation Schemes
17.7.3 Multi-Reflection
17.7.4 The Surfel Method
17.7.5 Bouzidi–Lallemand Bounce-Back Rule
17.8 Summary
References
Exercises
18 Flows at Moderate Reynolds Numbers
18.1 Moderate Reynolds Flows in Simple Geometry
18.2 LBE Implementation
18.3 Boundary Conditions
18.4 Flows Past Obstacles
18.5 Hemodynamics
18.6 More on the Pressure Field: Poisson Freedom
18.7 Summary
References
Exercises
19 LBE Flows in Disordered Media
19.1 Introduction
19.2 Flows Through Porous Media
19.3 LBE Flows Through Porous Media
19.4 Setting Up the LBE Simulation
19.5 Deposition Algorithm
19.6 Early-days Numerical Simulations
19.7 Synthetic Matter and Multiscale Modeling
19.8 Some Recent Developments
19.9 Summary
References
Exercises
20 Lattice Boltzmann for Turbulent Flows
20.1 Fluid Turbulence
20.1.1 Two-Dimensional Turbulence
20.1.2 Turbulence and Kinetic Scales
20.2 Early LBE Simulations of Two-Dimensional Turbulence
20.2.1 Sub-Grid Scales and Numerical Stability
20.3 Three-Dimensional-Incompressible Turbulence
20.3.1 Three-Dimensional Channel Flow
20.3.2 Three-Dimensional Turbulence: Parallel Performance
20.4 Summary
References
Exercises
Part IV Lattice Kinetic Theory: Advanced Topics
21 Entropic Lattice Boltzmann
21.1 Rescueing the Second Principle
21.1.1 Linear Stability and Time Marching
21.1.2 Stability and Realizability
21.2 Lattice H-Theorem
21.2.1 Entropic LB in the Sub-Grid Regime
21.3 Entropy Minimization Strategies
21.4 Modern ELB Developments
21.5 Entropic Misunderstandings
21.6 Summary
References
Exercises
22 Thermohydrodynamic LBE Schemes
22.1 Isothermal and Athermal Lattices
22.2 Temperature in a Discrete World
22.2.1 Equation of State and Thermodynamic Consistency
22.3 Thermodynamic Equilibria and Multi-Energy Lattices
22.4 Extended Parametric Equilibria
22.4.1 Parametric Equilibria without Nonlinear Deviations
22.5 Thermohydrodynamic LB Schemes Get in trouble
22.6 Early Attempts to Rescue Thermal LBE
22.6.1 Tolerance to Realizability Violations
22.6.2 The Kinetic-Closure Approach
22.6.3 Non-Space-Filling Lattices
22.6.4 Models with Rest Energy
22.7 Systematic Third-Order Equilibria
22.8 Entropic Schemes with Energy Conservation
22.9 Of Fluons and Phonons: The Double-Distribution Method
22.9.1 Second-Order Time Marching
22.10 Hybrid Methods
22.11 Passive-Scalar Schemes
22.12 Further Variants for Compressible Flows
22.13 Boundary Conditions for Thermal Flows
22.14 Summary
22.15 Appendix: Parameters of the D1V5 and D2V16 Thermal model
References
Exercises
23 Out of Legoland: Geoflexible Lattice Boltzmann Equations
23.1 Coarse-Graining LBE
23.2 Finite Volume LBE
23.2.1 Piecewise-Constant Streaming
23.2.2 Piecewise-Linear Streaming
23.2.3 Piecewise-Linear Collision Operator
23.2.4 Piecewise-Parabolic Interpolation
23.3 Unstructured Lattice Boltzmann Equation
23.4 Finite Difference LBE
23.5 Interpolation-Supplemented LBE
23.6 Finite Element LBE
23.7 Native LBE Schemes on Irregular Grids
23.7.1 Implicit LBE Schemes
23.8 Multigrid Lattice Boltzmann Scheme
23.9 Summary
References
Exercises
24 Lattice Boltzmann for Turbulence Modeling
24.1 Sub-Grid Scale Modeling
24.2 Lattice Boltzmann Turbulence Models
24.3 Hydro-Kinetic: Eddy-Viscosity Smagorinsky Model
24.3.1 LBE Formulation of the Smagorinsky SGS Model
24.4 Two-Equation Models
24.5 Reynolds-Averaged Navier–Stokes (RANS)
24.6 Large Eddy Simulation
24.6.1 Kinetic Hydro: Lattice Boltzmann Large-Eddy Simulation
24.7 The Kinetic Approach to Fluid Turbulence, Again
24.7.1 Very Large-Eddy Simulation
24.8 Fully Kinetic Approach
24.8.1 Relation between Kinetic and Hydrodynamic Closures
24.9 Entropic Models: Domesticating the Ghosts
24.10 Wall–Turbulence Interactions
24.11 Summary
References
Exercises
Part V Beyond Fluid Dynamics: Complex States of Flowing Matter
25 LBE for Generalized Hydrodynamics
25.1 Introduction
25.2 Including Force Fields
25.2.1 Including Forces with a Force-Free Formalism: Driven Hydrodynamics
25.2.2 Shifted Hydrodynamics
25.2.3 Galilean Rescue
25.3 Generalized Lattice Hydrodynamics
25.3.1 Shifted Lattice Equilibria
25.3.2 Multi-Relaxation Time
25.4 Lattice Implementation of the Force Term: Details Matter!
25.5 Electromagnetic Forces
25.5.1 Two-Dimensional Magnetohydrodynamics
25.6 Toward Strong Interactions: Hermite Representation
25.7 Summary
References
Exercises
26 Lattice Boltzmann for reactive flows
26.1 Chemical Reactions
26.2 Lattice Boltzmann Schemes with Chemical Reactions
26.3 Advection-Diffusion-Reaction Equations
26.4 Critical Remarks
26.4.1 Hybrid Approach
26.4.2 Moment-Propagation Method
26.4.3 Presence Index Approach
26.5 Reactive LBE Applications
26.6 Summary
References
Exercises
27 Lattice Boltzmann for Non-Ideal Fluids
27.1 Introduction
27.2 Non-Ideal Equation of State and Surface Tension
27.2.1 Non-Ideal Equations of State
27.2.2 Surface Tension
27.3 Numerical Methods for Flows with Interfaces
27.4 LB Schemes for Multiphase Flows: Numerical Challenges
27.4.1 Low Density Ratios
27.4.2 Thick Interfaces
27.5 Chromodynamic Models
27.6 The Shan–Chen Pseudo-Potential Approach
27.7 The Free-Energy Approach
27.8 Free-Energy Finite-Difference Methods
27.9 Attempts toward Thermodynamic Consistency
27.9.1 Lattice Enskog Equation
27.9.2 Two-Body Liouville Equation
27.9.3 Merging LB with Dynamic Density Functional Theory
27.10 Assorted Multiphase LB Approaches
27.10.1 Projection Methods
27.10.2 Alternative Equations of State
27.10.3 Adaptive Equations of State
27.10.4 Merging LB with Flux-Limiting Techniques
27.10.5 Entropic Method for Multiphase Flows
27.11 Miscellaneous Multiphase LBE Applications
27.12 Summary
References
Exercises
28 Extensions of the Pseudo-Potential Method
28.1 Introduction
28.2 Shan–Chen Revisited
28.2.1 Equation of State
28.2.2 Interface Pressure and Surface Tension
28.2.3 Surface Tension
28.3 Shan–Chen Limitations and How to Soften Them
28.4 Surface Tension Independent of the Equation of State: Multirange Pseudo-Potentials
28.5 Vapor Versus Liquid Sound Speed: Alternative Equations of State
28.6 Large Density Ratios and Spurious Currents
28.6.1 Alternative Discretizations
28.6.2 Local Grid Refinement
28.6.3 Multi-Range Potentials, Again
28.7 Thick Interfaces
28.7.1 Weakly Broken Universality
28.8 Thermodynamic Consistency: Shan–Chen from Pseudo-Free Energy
28.9 Summary
References
Exercises
29 Lattice Boltzmann Models for Microflows
29.1 Basics of Microfluidics
29.2 Non-Hydrodynamic Behavior Beyond Navier–Stokes
29.2.1 Slip-Flow and Knudsen Layers
29.3 Multiphase Flows
29.3.1 The Moving Contact Line
29.4 LB without Chapman–Enskog: An Eluded No-Go?
29.5 Early Minimal Models: Density-Dependent Relaxation
29.6 High-Order Lattices
29.6.1 Counting Degrees of Freedom: Multi-Scalars versus Tensor Fields
29.6.2 Structural Organization: Multiscalars versus Tensors
29.6.3 Realizable Kinetic Boundary Conditions
29.7 Diffuse Boundary Conditions
29.7.1 Partial-Accomodation Kinetic Boundary Conditions
29.7.2 Volumetric Boundary Conditions
29.7.3 Non-Equilibrium Boundary Conditions
29.8 Regularization
29.9 Exact Solutions
29.10 Partial Summary
29.11 Multiphase Microflows
29.11.1 Fluid-Wall Potentials
29.11.2 Fluid-Wall Pseudo-Potentials
29.11.3 Heterogeneous Free-Energy Models
29.12 Further Miscellaneous Developments
29.12.1 Spurious Knudsen Layers
29.12.2 High-Order Boundary Conditions
29.12.3 Non-Uniform Grids
29.12.4 Thermal Microflows
29.13 Summary
References
Exercises
30 The Fluctuating Lattice Boltzmann
30.1 Nanoscale flows
30.2 The Fluctuating LBE
30.3 Some FLBE Liabilities
30.4 Thermal Equipartition at All Scales
30.5 Theoretical Foundations of the Fluctuating LB
30.6 FLB from Equilibrium Statistical Mechanics
30.7 The Boltzmann Number
30.7.1 Thermal versus Inertial Mass
30.8 FLBE from Fluctuating Kinetic Theory
30.9 Numerical Stability and Computational Considerations
30.10 Loose Ends and Future Developments
30.10.1 Validity of FDT Under Confinement
30.10.2 FLBE Under Strong Confinement
30.10.3 Handling Strong Fluctuations
30.11 Summary
References
Exercises
31 LB for Flows with Suspended Objects: Fluid–Solid Interactions
31.1 Introduction
31.2 Ladd’s Collision-Based Coupling Method
31.2.1 Numerical Tests
31.3 Improving on Ladd’s Model
31.3.1 Dealing with Impulsive Forces: the Cover-Uncover Transition
31.3.2 Concentrated Suspensions: Near-Contact Problems
31.3.3 Higher-Order Boundary Conditions
31.4 Force-Based Coupling Methods for Fluid-Structure Interactions
31.4.1 Flexible Bodies: Polymers in LB Flows
31.4.2 Force-Based Coupling: the “Raspberry” Model
31.4.3 A Few Words of Comment
31.5 The Immersed Boundary Method
31.5.1 Discrete Dirac’s Delta: Smoothed Particles
31.5.2 Merging LB with IBM: Some Applications
31.6 External Boundary Force: A Unified Framework for Force-Based Methods
31.6.1 Time Marching
31.7 Eulerian–Eulerian Multicomponent Models
31.8 Summary
References
Exercises
Part VI Beyond Newtonian Mechanics:Quantum and Relativistic Fluids
32 Quantum Lattice Boltzmann (QLB)
32.1 Fluid Formulation of Quantum Mechanics
32.2 The Fluid Formulation of the Schrödinger Equation
32.2.1 Relativistic Quantum Mechanics: the Dirac Equation
32.2.2 The Majorana Representation
32.2.3 Dirac to Schrödinger: the Enslaving Approximation
32.2.4 The Interacting Case
32.3 The Quantum LBE
32.4 Time Marching
32.4.1 The QLB Dispersion Relation and Lorentz Invariance
32.5 Numerical Tests in One Space Dimension
32.5.1 Free Particle Motion
32.5.2 Harmonic Oscillator
32.5.3 Scattering Over a Rectangular Barrier
32.6 Multi-Dimensional QLB
32.6.1 Quantum LBE: Move, Turn and Collide
32.7 QLB is a Genuine Dirac Solver
32.8 The Lattice Wigner Equation
32.9 Summary
32.10 Appendix: Spinors
References
Exercises
33 QLB for Quantum Many-Body and Quantum Field Theory
33.1 The Quantum N-Body Problem
33.2 Numerical Quantum Many-Body Methods
33.2.1 Quantum Monte Carlo Methods
33.2.2 Variational Monte Carlo
33.2.3 Matrix Eigenvalue Methods: Exact Diagonalization
33.2.4 Dynamic Minimization
33.2.5 Quantum LB as the “Perfect” Lattice Monte Carlo?
33.3 Electron-Density Functional Theory
33.3.1 Kinetic Approach to Electronic Density Functional Theory
33.4 Lattice Boltzmann for Quantum Field Theory
33.4.1 Second Quantization and Heisenberg Interaction Representation
33.4.2 Quantum Fields in Spatially Extended Systems
33.4.3 LB for (1+1) Quantum Field Theory
33.5 Quantum LB for Quantum Computing
33.5.1 Bits and Qubits
33.5.2 Hamiltonians and Circuits
33.6 Quantum Random Walks
33.6.1 QLB is a Quantum Random Walk
33.7 Quantum Simulators for Classical Kinetic Theory
33.7.1 Second-Quantized Kinetic Theory
33.7.2 Non-Unitary Hamiltonians
33.8 Summary
References
Exercises
34 Relativistic Lattice Boltzmann (RLB)
34.1 Relativistic Mechanics
34.2 Relativistic Kinetic Theory
34.3 Relativistic Boltzmann Equation
34.4 Relativistic Equilibria: the Juettner Distribution
34.5 Relativistic Hydrodynamics
34.6 Relativistic LBE
34.7 Top-Down Approach: RLB by Moment Matching
34.7.1 Dissipative Hydrodynamics
34.8 Early RLB Applications
34.8.1 Shock Wave in Quark-Gluon Plasmas
34.8.2 Astrophysics: Supernova Explosions
34.8.3 Electron Flows in Graphene
34.9 Strongly Interacting Fluids, AdS-CFT Duality and Super-Universality
34.9.1 Caveat
34.10 Summary
References
Exercises
35 Advanced RLB models
35.1 Relativistic LBE from Relativistic Kinetic Theory
35.1.1 Relativistic Basis Functions
35.1.2 Relativistic Lattice Polynomials
35.1.3 Third-Order KRLB with Improved Dissipation
35.1.4 KRLB Equilibria
35.1.5 Kinetic RLB schemes for non-zero mass (finite-temperature) particles
35.2 Applications of Kinetic RLB
35.3 RLB in Spherical Coordinates
35.4 Lattice Boltzmann in Curved Manifolds
35.4.1 Hydrodynamics in Curved Manifolds
35.4.2 Kinetic Theory in Curved Manifolds
35.4.3 Lattice Boltzmann in a Curved Manifold
35.5 Future Developments: Fluid-Structure Interactions
35.6 General Relativity
35.6.1 LB for Numerical Relativity
35.6.2 Hybrid Scenarios for Numerical Relativity
35.6.3 Kinetic Theory of spacetime
35.7 Summary
35.8 Appendix: Covariant and Controvariant Coordinates
References
Exercises
36 Coda
36.1 The Question
36.1.1 Biblio-Numbers
36.1.2 LB Cannot Be Too Wrong
36.1.3 Breakthroughs
36.1.4 Weakly Broken Universality and Chimaera Simulation
36.2 Boltzmann and Computers
References
37 Notation
37.1 Vectors and Tensors
37.2 Scalar Products and Tensor Contractions
37.3 Gradients, Divergence, Curl and All That

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