Book contents
- Frontmatter
- Contents
- Preface
- Acknowledgements
- 1 A simple model of fluid mechanics
- 2 Two routes to hydrodynamics
- 3 Inviscid two-dimensional lattice-gas hydrodynamics
- 4 Viscous two-dimensional hydrodynamics
- 5 Some simple three-dimensional models
- 6 The lattice-Boltzmann method
- 7 Using the Boltzmann method
- 8 Miscible fluids
- 9 Immiscible lattice gases
- 10 Lattice-Boltzmann method for immiscible fluids
- 11 Immiscible lattice gases in three dimensions
- 12 Liquid-gas models
- 13 Flow through porous media
- 14 Equilibrium statistical mechanics
- 15 Hydrodynamics in the Boltzmann approximation
- 16 Phase separation
- 17 Interfaces
- 18 Complex fluids and patterns
- Appendix A Tensor symmetry
- Appendix B Polytopes and their symmetry group
- Appendix C Classical compressible flow modeling
- Appendix D Incompressible limit
- Appendix E Derivation of the Gibbs distribution
- Appendix F Hydrodynamic response to forces at fluid interfaces
- Appendix G Answers to exercises
- Author Index
- Subject Index
3 - Inviscid two-dimensional lattice-gas hydrodynamics
Published online by Cambridge University Press: 23 September 2009
- Frontmatter
- Contents
- Preface
- Acknowledgements
- 1 A simple model of fluid mechanics
- 2 Two routes to hydrodynamics
- 3 Inviscid two-dimensional lattice-gas hydrodynamics
- 4 Viscous two-dimensional hydrodynamics
- 5 Some simple three-dimensional models
- 6 The lattice-Boltzmann method
- 7 Using the Boltzmann method
- 8 Miscible fluids
- 9 Immiscible lattice gases
- 10 Lattice-Boltzmann method for immiscible fluids
- 11 Immiscible lattice gases in three dimensions
- 12 Liquid-gas models
- 13 Flow through porous media
- 14 Equilibrium statistical mechanics
- 15 Hydrodynamics in the Boltzmann approximation
- 16 Phase separation
- 17 Interfaces
- 18 Complex fluids and patterns
- Appendix A Tensor symmetry
- Appendix B Polytopes and their symmetry group
- Appendix C Classical compressible flow modeling
- Appendix D Incompressible limit
- Appendix E Derivation of the Gibbs distribution
- Appendix F Hydrodynamic response to forces at fluid interfaces
- Appendix G Answers to exercises
- Author Index
- Subject Index
Summary
We now provide our first detailed derivation of the hydrodynamics of lattice gases. To keep matters from becoming unnecessarily complicated, we mostly restrict the discussion in this chapter to two-dimensional (2D) models. We begin with the simplest possible 2D model on a square lattice. We then repeat the calculation for the hexagonal lattice model. The principal result of this chapter is the derivation of the Euler equation of both models. This equation has the form already indicated in Chapter 2 but here the unknown coefficients are explicitly calculated. For future use we also include a general calculation in arbitrary dimension D and with an arbitrary number of rest particles.
The hydrodynamic behavior that we thus find at the macroscopic scale is a consequence of the existence of a kind of thermodynamic equilibrium. This equilibrium state is described by the Fermi-Dirac distribution of statistical mechanics. How this distribution arises is described in detail in Chapters 14 and 15. In this chapter we give a simpler derivation of some properties of equilibrium, which are sufficient to obtain the Euler equation.
Homogeneous equilibrium distribution on the square lattice
Our first task is to calculate 〈ni〉, the average value of the Boolean variable ni introduced in Section 2.6. Repeated applications of the rules of propagation and collision in the lattice gas cause these average particle populations to quickly reach an equilibrium state regardless of initial conditions.
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- Lattice-Gas Cellular AutomataSimple Models of Complex Hydrodynamics, pp. 29 - 45Publisher: Cambridge University PressPrint publication year: 1997