Lattice Gas Hydrodynamics

by J.-P. Rivet , J. P. Boon

• ISBN13: 9780521419444
• ISBN10: 0521419441
• Format: Hardcover
• Copyright: 2001-01-04
• Publisher: Cambridge University Press

Summary

Lattice Gas Hydrodynamics describes the approach to fluid dynamics using a micro-world constructed as an automaton universe, where the microscopic dynamics is based not on a description of interacting particles, but on the laws of symmetry and invariance of macroscopic physics. We imagine point-like particles residing on a regular lattice, where they move from node to node and undergo collisions when their trajectories meet. If the collisions occur according to some simple logical rules, and if the lattice has the proper symmetry, then the automaton shows global behavior very similar to that of real fluids. This book carries two important messages. First, it shows how an automaton universe with simple microscopic dynamics - the lattice gas - can exhibit macroscopic behavior in accordance with the phenomenological laws of classical physics. Second, it demonstrates that lattice gases have spontaneous microscopic fluctuations which capture the essentials of actual fluctuations in real fluids.

Table of Contents

Preface	xiii
Basic ideas	1	(9)
The physicist's point of view	1	(1)
The mathematician's point of view	2	(3)
Finite automata	2	(1)
Cellular automata	3	(1)
Lattice gases	3	(2)
Comments	5	(5)
The velocity vectors	5	(1)
Space and time	6	(1)
The exclusion principle	6	(1)
Bravais lattices	7	(1)
Local versus non-local collisions	8	(1)
Collision--propagation versus propagation--collision	8	(1)
Mathematical versus physical	9	(1)
Microdynamics: general formalism	10	(24)
Basic concepts and notation	10	(5)
The lattice and the velocity vectors	10	(2)
The Boolean field	12	(1)
Observables	12	(2)
Generalized observables	14	(1)
The microdynamic equation	15	(5)
Formal expression	16	(1)
The propagation operator	16	(2)
The collision operator	18	(2)
Analytic expressions of the microdynamic equation	20	(1)
Microscopic properties of a lattice gas	20	(9)
Detailed and semi-detailed balance	20	(1)
Duality	21	(1)
Conservation laws	21	(2)
G-invariance	23	(3)
Crystallographic isotropy	26	(2)
Irreducibility	28	(1)
Special rules	29	(4)
Solid impermeable obstacles	30	(3)
Sources and sinks of observable quantities	33	(1)
Comments	33	(1)
Microdynamics: various examples	34	(34)
The HPP model	34	(5)
The microdynamical equation	36	(2)
Microscopic properties	38	(1)
The FHP-1 model	39	(5)
The microdynamical equation	41	(1)
Microscopic properties	42	(2)
The FHP-2 model	44	(4)
The microdynamical equation	44	(2)
Microscopic properties	46	(2)
The FHP-3 model	48	(2)
The microdynamical equation	48	(1)
Microscopic properties	48	(2)
The `colored' FHP model (CFHP)	50	(3)
The microdynamical equation	52	(1)
Microscopic properties	52	(1)
The GBL model	53	(4)
The microdynamical equation	55	(1)
Microscopic properties	56	(1)
Three-dimensional models	57	(11)
Models with multiple links	58	(1)
Models with biased collisions	59	(1)
The FCHC models	59	(4)
Collision rules	63	(1)
The microdynamical equation	64	(1)
Microscopic properties	65	(3)
Equilibrium statistical mechanics	68	(41)
The Liouville description	68	(3)
Macrostates	69	(1)
Ensemble-averages	69	(1)
The lattice Liouville equation	70	(1)
The Boltzmann description	71	(2)
The Boltzmann approximation	71	(1)
The lattice Boltzmann equation	72	(1)
The H-theorem	73	(7)
Some basics about communication and information	74	(3)
The H-theorem for lattice gases	77	(3)
Global equilibrium macrostates	80	(6)
The Liouville approach	81	(3)
The lattice Boltzmann approach	84	(1)
The variational approach	85	(1)
Natural parameterization of equilibria	86	(12)
Low-speed equilibria for single-species non-thermal models	89	(4)
Nearly equally distributed equilibria for thermal models	93	(5)
Statistical thermodynamics	98	(7)
Static correlation functions	105	(4)
Macrodynamics: Chapman-Enskog method	109	(28)
Local equilibria and the hydrodynamic limit	110	(1)
The multi-scale expansion for macrodynamics	111	(7)
The scale separation parameter	111	(1)
Perturbed local equilibrium	111	(1)
Macroscopic space and time scales	112	(2)
The averaged microdynamic equation	114	(2)
The expansion in powers of e	116	(2)
First order macrodynamics	118	(8)
Solvability conditions for the first order problem	118	(3)
Solution of the first order problem	121	(5)
Second order macrodynamics	126	(1)
Solvability conditions for the second order problem	126	(1)
The macrodynamic equations	127	(2)
Transport coefficients within the Boltzmann approximation	129	(3)
Non-thermal models	132	(3)
First order macrodynamics	132	(1)
Second order macrodynamics	133	(1)
The macrodynamic equation	134	(1)
The transport coefficients	135	(1)
Comments	135	(2)
Linearized hydrodynamics	137	(16)
The linearized Boltzmann equation	138	(2)
Slow and fast variables	140	(4)
The hydrodynamic limit	144	(6)
The coupling function	145	(1)
The memory function	145	(2)
The random force term	147	(1)
The long-wavelength, long-time limit	148	(2)
The transport matrix	150	(1)
Comments	151	(2)
Hydrodynamic fluctuations	153	(30)
The dynamic structure factor	154	(1)
Fluctuation correlations	155	(2)
The hydrodynamic modes	157	(6)
The spectral decomposition	157	(3)
The eigenvalues	160	(3)
The hydrodynamic spectrum	163	(1)
The eigenvalue spectrum	164	(4)
Hydrodynamic regime: klf << 1	165	(1)
Generalized hydrodynamic regime: klf < 1	166	(1)
Kinetic regime: klf ⋧ 1	167	(1)
Power spectrum	168	(6)
High density	168	(3)
Low density	171	(1)
Dispersion effects	172	(2)
Diffusion and correlations	174	(9)
The two-species lattice gas	175	(2)
The hydrodynamic limit	177	(1)
The power spectrum	178	(5)
Macrodynamics: projectors approach	183	(16)
Preliminaries	184	(4)
Multiple scales analysis	188	(2)
The hydrodynamic equations	190	(3)
Linear response and Green--Kubo coefficients	193	(3)
Long-time tails	196	(3)
Hydrodynamic regimes	199	(8)
The acoustic limit	200	(3)
The incompressible limit	203	(2)
Comments	205	(2)
Invariances	205	(1)
Four-dimensional models	205	(1)
Lattice gases to simulate real fluid dynamics	206	(1)
Lattice gas simulations	207	(26)
Lattice gas algorithms on dedicated machines	208	(2)
Lattice gas algorithms on general purpose computers	210	(5)
Channel-wise vs. node-wise storage	210	(3)
Collision strategies	213	(1)
Obstacles	214	(1)
Essential features of a lattice gas simulation code	215	(4)
Initialization	215	(1)
Raw physical data extraction	216	(1)
Post-processing	217	(2)
Measurement of basic lattice gas properties	219	(3)
Measuring g(ρ) and ν(ρ)	220	(1)
Measuring cs(ρ) and ν'(ρ)	220	(1)
An example: the FCHC-3 model	221	(1)
Examples of lattice gas simulations	222	(11)
The Kelvin--Helmholtz instability	222	(1)
Particle aggregation	223	(2)
Two-dimensional flow past an obstacle	225	(2)
Three-dimensional flow past an obstacle	227	(4)
Two-dimensional flow of a two-phase fluid in a porous medium	231	(2)
Guide for further reading	233	(17)
The historical `roots'	234	(2)
Discrete kinetic theory	234	(1)
The early days	235	(1)
Cellular automata	235	(1)
Three-dimensional models	236	(1)
Theoretical analyses	236	(5)
General lattice gas theory	237	(1)
Statistical physics and thermodynamics	237	(2)
Violation of semi-detailed balance	239	(1)
Invariants and conservation laws	240	(1)
Obstacles and Knudsen layers	240	(1)
Models with particular features	241	(3)
Fluid mixtures and colloids	241	(1)
Reaction-diffusion systems	241	(1)
Immiscible fluids and free interfaces	242	(1)
Flow in porous media	243	(1)
Thermo-hydrodynamics	243	(1)
Elastic waves	243	(1)
Other models	244	(1)
Lattice Boltzmann method	244	(2)
Lattice Bhatnagar--Gross--Krook model	246	(1)
Numerical simulations and implementations	246	(2)
Implementation on dedicated hardware	246	(1)
Simulations on general purpose computers	247	(1)
Books and review articles	248	(2)
Appendix Mathematical details	250	(25)
References	275	(6)
Author index	281	(4)
Subject index	285