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D. H. Rothma n and S. Zales ki: Lattice-gas mode ls of phase separation

1 . B o l t z m a n n a pp r o x im a t ion 1 4 4 9

2 .

Sur face tensi on 1449

a . The o r e t i c a l c a l c u l a t i on 1 4 4 9

b . C om pa r i so n w i th s im u la t i on 1 4 5 1

B . I n t e r f a c e s i n l i qu id - ga s m od e l s 1 4 5 2

C. Int erfa ce fluctuations 1453

V I I I . P h a s e T r a n s i t i o n s i n P h a s e - S e p a r a t i n g A u t o m a t a 1 4 54

A . L iqu id - ga s t r a ns i t i on i n t he l i qu id - ga s m od e l 1 4 5 5

B .

S p inoda l de c om pos i t i on i n im m isc ib l e l a t t i c e ga se s 1 4 5 5

1 . C ha pm a n- E nsko g e s t im a te o f t he d i iF us ion

coefficient 1456

2 . I m m isc ib l e l a t t i c e -ga s pha se d i a g r a m : The sp ino

da l curv e 1457

C. I so trop y and se l f - s imila r i ty 1457

I X . N u m e r i c a l S im u la t i on s 1 4 5 8

A. S im ula t i ons of s ing le -co mpo nen t f luids 1458

1. Tw o-d ime nsio na l f lu ids 1458

a . F lo ws in s imp le geom etr ies 1458

b .

S t a t i s t i c a l m e c h a n i c s a nd hyd r ody na m ic s 1 4 5 9

c . F low s i n c om ple x ge om e t r i e s 1 4 60

2 . Thr ee-d ime nsio na l f lu ids 1460

B . S im u la t i on s o f m u l t i c om pon e n t f lu id s 1 4 61

1 . P ha se s e pa r a t i o n a nd hyd r ody na m ic s 1 4 61

2 . M ul t i pha se f low th r o ugh po r o us m e d ia 1 4 62

3 .

Th r e e - p ha se f low , e m u l s ions , a nd s e d im e n ta t i on 1 4 63

4 . Th r e e - d im e ns iona l f low s 1 4 66

a . L iqu id - ga s m ode l 1 4 66

b .

I m m isc ib l e l a t t i c e ga s 1 4 66

X . C onc lus ions 1 4 67

A . S t a t i s t i c a l m e c h a n i c s 1 4 67

B . H y d r o d y n a m i c s 1 4 68

A c k n o w l e d g m e n t s 1 4 6 9

A pp e nd ix A : S ym m e t r y a nd R e l a t e d G e o m e t r i c a l P r ope r t i e s 1 4 69

1. Poly opes 1469

2 . Te ns o r sym m e t r i e s 1 4 69

a . I so trop ic tenso rs 1469

b . Te ns o r s i nva r i a n t und e r t he l a t t i c e po in t sym

metr ies 1470

i . Te nso r i nva r i a n t und e r t he w ho le g r ou p 1 4 7 0

ii . Te ns o r a t t a c h e d t o a g ive n l a t t i c e ve c to r 1 4 7 0

3 .

Te nso r s f o r m e d w i th ge ne r a t i ng ve c to r s 1 4 7 2

A p p e n d i x B : T h e L i n e a r i z e d B o l t z m a n n O p e r a t o r 1 4 72

A p p e n d i x C : T h e L a t t i c e - B o l t z m a n n M e t h o d 1 4 72

1.

Basic def ini t ions 1473

2 . Evo lu t i on e qua t i ons 1 4 7 3

3.

H y d r o d y n a m i c l i m i t 1 4 73

4 .

S tabi l i ty 1474

5 . M u l t i ph a se m od e l s a nd o the r a pp l i c a t i on s 1 4 7 4

References 1474

I . INTRODUCTION

Ma c rosc op ic c omple x i ty c a n ma sk mic rosc op ic s imp l i

c i ty . Fo r example , the swir ls and burs ts of a turbu lent

f lu id a r e just the col lec t ive dynam ics tha t emerg e from a

l a rge numbe r o f mo le c u le s i n t e ra c t ing w i th e a c h o the r

v i a N e w ton ' s e qua t ion o f mo t io n . W h e re a s t he mic ro

sc op ic dyna mic s i n suc h a sy s t e m a re s t r a igh t fo rw a rd in

p r inc ip l e , t he o rga n iz a t ion o f t he se mic rosc op ic mo t ions

to p roduc e tu rbu le nc e , o r e ve n hyd rodyna mic s itself,

r e ma ins sh roude d in mys te ry .

M uc h , how e ve r , is o f c ou rse know n . B o th the k ine t i c

the o ry o f ga se s a nd the N a v ie r -S toke s e qua t ions o f hy

d rodyna mic s da t e f rom the n ine t e e n th c e n tu ry , w h i l e i n

th i s c e n tu ry c ons ide ra b le p rog re ss ha s be e n ma de tow a rd

the unde rs t a nd ing o f t he c onne c t ions be tw e e n the mic ro

scopic or a tomist ic descr ip t ion of f lu ids and macroscopic

hyd r odyn a mic s (C ha pm a n a nd C ow l ing , 197 0 ). N e ve r

the less, some re la t ive ly s imple quest ions concerning the

re la t ion be tween these two leve ls of descr ip t ion a re only

ju s t be g inn ing to be a dd re sse d . Fo r e xa m ple , one ma y

ask prec ise ly how la rge a microscopic system of par t ic les

must be for i t to conta in enough degrees of f reedom to be

considered , a t a la rger sca le , as a cont inuously varying

ma c ro sc op ic me d iu m. To a nsw e r suc h que s t ions , t he a d

ve n t of mo de rn c om pu te r s ha s be e n e sse n t i al . A m ong

the many achievements in the f ie ld of molecular-

dyna mic s s imu la t ion ha s be e n the e xp l i c i t de mons t ra t i on

tha t hydrodynamic f lows can be obta ined (a lbe i t a t con

s ide ra b le c ompu ta t iona l e xpe nse ) f rom l a rge mo le c u la r

sys t e ms (R a pa p o r t a nd C le me n t i , 1 986 ; Ma r e sc ha l a nd

K e s te mon t , 1 987 ) .

W e now know , how e ve r , t ha t t he c omple x i ty o f hyd ro

dynamics not only may be descr ibed by an expl ic i t

" a ve ra g ing" o f t he iV -body p rob le m o f mo le c u la r dyna m

i cs , bu t t ha t v i r t ua lly t he sa me ma c rosc op ic hy d ro -

dyna mic e qua t ions ma y be ob ta ine d f rom a d ra s t i c a l ly

simpl i fied version of mo lecula r dynam ics. Spec ifica lly , in

1986 , F r i sc h , H a ss l a c he r , a nd Pom e a u show e d tha t one

ma y de r ive t he N a v ie r -S toke s e qua t ions f rom a mic ro -

dynamics consist ing of an a r t i f ic ia l se t of ru les for col

l i s ion and propaga t ion of ident ica l par t ic les , each of

which is const ra ined to move on a regular la t t ice in

discre te t ime wi th one of only a smal l , f in i te number of

possib le ve loc i t ies (Fr isch et al., 1 986 ) . Th i s r e ma rka b le

observa t ion has not only had impl ica t ions for s ta t i s t ica l

me c ha n ic s a nd k ine t i c t he o ry , bu t a l so fo r t he nume r i c a l

sim ula t ion of cer ta in hyd rod yna mi c f lows. Th is review is

therefore dedica ted to an expl ica t ion of the or ig ina l work

of Fr isch et al. and to a survey of some of the resul t ing

ramif ica t ions during the e ight years s ince i t s in t roduc

t ion .

B e c a use the mode l o f F r i sc h

et al.

i s c ons t ru c t e d f rom

discre te dynamica l var iables ( the ve loc i t ies) tha t evolve

on a d isc re te la t t ice in d isc re te t ime , i t i s an example of a

cellular automaton

( F a r m e r

et al.,

1984; Wo lfram , 1986b;

Toffbl i and Mar gol us, 1987). Th e idea of ce l lu la r auto

ma ta , w h ic h da t e s ba c k to t he w ork o f von N e uma nn a nd

Ul am in the 1940s (von Ne um an n, 1966), i s to f ind simple

rules of spa t ia l in te rac t ion and tempora l evolut ion , f rom

which col lec t ive , complex behavior emerges. The ear ly

mot iva t ions for th is work came from biology: the goa l ,

as descr ibed in the h istor ica l perspec t ive g iven by Dyson

(1979), was to provide a theory for how an artificial l ife

capable of reproducing i t se l f could be const ruc ted .

W hi l e a pp l i c a t ions o f c e l lu l a r a u toma ta t o b io logy

remain of in te rest [see , for example , the book by Weis-

bu ch (1991)] , in the last two to thr ee decade s mu ch of the

in te res t has sh i f ted to phys ics and com put a t io n . On e of

the ear l iest works in th is regard is tha t of Zuse (1970) ,

who was possib ly the f i rs t to perce ive the connec t ions be

tw e e n c e l lu l a r a u toma ta a nd the s imu la t ion o f pa r t i a l -

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D. H. Rothman and S. Zaleski: Lattice-gas models of phase separation

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diffe rent ia l equa t ions. Other examples inc lude studies of

t ime - re ve r s ib l e a u toma ta (Ma rgo lus , 1 984) , spe c u la t ions

on the s imu la t ion o f qua n tum-me c ha n ic a l phe nome na

(Fe ynm a n , 1 982) , c re a t ion o f a " s t a t i s t i c a l me c ha n ic s" o f

ce l lu la r au tom ata (Wo lfram, 1983) , and expl ic i t con

s ide ra t ions o f c e l lu l a r a u toma ta a s d i sc re t e dyna mic a l

systems (Vichniac , 1984) and as an a l te rna t ive to par t ia l -

differential equ atio ns (Toffoli, 1984). In dee d, by 1985

there was a surfe i t of specula t ion and expec ta t ion; but in

the absence of any wide ly known, concre te example of a

ce l lu la r-a utom aton mo del of a par t ia l -d i ffe rent ia l equ a

t i on , ma ny w e re l e f t w onde r ing w he the r suc h mode l s

c ou ld inde e d be c ons t ruc t e d .

In t h i s c on te x t t he mode l o f F r i sc h , H a ss l a c he r , a nd

Pom e a u (FH P ) w a s in t rodu c e d . The mode l , a n e x t e ns ion

o f e a r l i e r w ork by H a rdy , de Pa z z i s , a nd Pome a u (H a rdy

et aL, 1973, 1976), consis ted of ident ica l par t ic les th a t

hop from si te to s i te on a regular la t t ice , obeying simple

c o l l i s ion ru l e s t ha t c onse rve ma ss a nd mome n tum.

Fr i sc h , H a ss l a c he r , a nd Pome a u show e d tha t , a t a spa t i a l

sca le much la rger than a la t t ice uni t and a t a tempora l

sca le much slower than a d isc re te t ime step , the model

a sympto t i c a l ly s imu la t e s t he i nc ompre ss ib l e N a v ie r -

S toke s e qua t ions .

Th e FH P m ode l , the fi rs t of a wide c lass of mod els t ha t

soon be c a me know n a s lattice-gas automata, led to ma ny

inte res t ing ramif ica t ion s. Fi rs t , as a l ready ment io ned , i t

demonstra ted tha t the fu l l de ta i ls of rea l molecular dy

na mic s a re no t ne c e ssa ry to c re a t e a mic rosc op ic mode l

w i th ma c ros c op ic hyd rody na m ic be ha v io r (K a da no ff ,

1986; W ol f ra m, 1 986a ; F r i sc h et aL, 1987; Kadanoff

et aL, 1989; Za ne t t i , 1989) . Seco nd, la t t ice -gas au tom ata

w e re imme d ia t e ly c ons ide re d a s a n a l t e rna t ive me a ns fo r

the nume r i c a l s imu la t ion o f hyd r odyn a mic f low s

(d 'H umie re s , Pome a u , a nd La l l e ma nd , 1 985 ; d 'H umie re s

and Lal lemand, 1986, 1987) . Third , the method gave r ise

to some new ideas for const ruc t ing models of cer ta in

complex fluids, specifically, fluid mixtures including in

te rfaces, exhibi t ing phase t ransi t ions, and a l lowing for

mu l t ip ha se f low s (R o thm a n a nd K e l l e r , 1 988 ; A p pe r t a nd

Zaleski , 1990) . Thus la t t ice -gas automata have not only

be c ome " toy mode l s" fo r t he e xp lo ra t ion o f t he mic ro

sc op ic ba s i s o f hyd rodyna mic s , bu t a l so t oo l s fo r t he nu

merica l s tudy of cer ta in problems in f lu id mechanics.

Both aspec ts of the subjec t a re covered in th is review.

In what fo l lows, we f i rs t provide an overview of the

f ie ld. W e in t r odu ce the FH P mode l , descr ibe in genera l

te rms i t s hydrodynamic l imi t , and i l lust ra te i t s abi l i ty to

s imu la t e t he N a v ie r -S toke s e qua t ions . W e the n in t roduc e

la t t ice -gas models of mul t iphase f lu ids and br ie f ly de

scr ibe two exam ples. On e is a mo del of a b ina ry f lu id

mix tu re t ha t e xh ib i t s a pha se - se pa ra t ion t r a ns i t i on . Th e

oth er co nta in s jus t a s ingle spec ies of f lu id , but exhibi ts a

l i qu id -ga s t r a ns i t i on .

Fol lowing th is overview, we show how one may der ive

hyd rody na m ic e qua t ions f rom the se mic rosc op ic mode l s .

We f i rs t descr ibe the hydrodynamic l imi t of the s implest ,

s ing l e -c ompone n t , mode l s . W e the n re v i e w the s t a t e o f

the o re t i c a l unde r s t a nd ing o f t he more c omple x , mu l t i

pha se l a t t i c e ga se s . The hyd rodyna mic be ha v io r o f t he

mul t iphase models i s in one case prec ise ly the same as,

and in the o ther case very c lose to , tha t of the s implest

l a t t i c e ga se s . Thus ou r e mpha s i s on the mu l t i pha se mod

e ls i s concentra ted on aspec ts of the i r phase

t ra ns i t i ons — in o the r w ord s , t he fo rma t ion o f

in t e r fa c e s— a nd on the phys i c s of t he se i n t e r fa ce s t he m

se lves. W e descr ibe a ca ta log of resul ts , bot h th eore t ica l

a nd e mp i r i c a l , t ha t show tha t t he ma c rosc op ic be ha v io r

of the mul t iphase models i s qua l i ta t ive ly , i f not quant i ta

t ive ly , s imi la r to tha t obta ined from c lassica l models of

pha se t r a ns i t i ons a nd in te r fa c e s . W e a rgue tha t t h i s

agreement wi th c lassica l theory is important not only for

a pp l i c a t ions , bu t a l so fo r a be t t e r unde r s t a nd ing o f some

of t he founda t ions o f s t a t i s t i c a l me c ha n ic s . S t a t e d b lun t

ly , t he se mode l s b re a k ma ny c l a ss i c a l ru l e s— for e xa m

pl e ,

t he i r m ic ro dyna m ic s is t ime - i r r e ve r s ib l e — but a p

pare nt ly wi t hou t s ignif icant de le te r io us e ffec t. Un de r

s t a nd ing w hy th i s ma y be r e ma ins one o f t he more im

po r t a n t que s t ions t o be a dd re sse d .

In the remainder of the review we provide an overview

of t he va r i e ty o f nume r i c a l e xpe r ime n ta t ion tha t ha s be e n

pe r fo rme d w i th l a t t ic e ga se s. W e de sc r ibe p rob le ms o f

bot h two- and three- dim ensi ona l f low, and of bo th single

and mul t ip le f lu ids. While the la t t ice gas may, in pr inc i

pl e , be u se d fo r ne a r ly a ny p rob le m in hyd rodyna mic

s imu la t ion , w e e mpha s i z e t ha t ma ny o f t he mos t suc c e ss

fu l appl ica t ions have involved e i ther a complex f lu id , a

c omple x ge ome t ry , o r bo th . Suc h c omple x i ty i s pe rha ps

best exempl i f ied by the problem of mul t iphase f low

th rough po rous me d ia .

Having sta ted the content of th is review, we f ind i t a lso

worthwhi le to indica te some of the subjec ts we do not

cover . On e such topic is mu l t ispee d models in which

moving par t ic les a re no longer rest r ic ted to uni t speed,

thus a l lowing the def in i t ion of a tempera ture (Grosf i l s

et aL, 1992; Mo lvig et aL, 1992; Qian et aL, 1992) . Re

l a t e d to t he i n t e rna l e ne rgy t r a nspo r t i n t he rma l mode l s

are models of d i f fusion or passive-sca la r t ransport .

Diffusion is re la t ive ly s imple to s tud y wi th la t t ic e gases

and, indeed, has been the subjec t of considerable a t ten

t i on (B urge s a nd Za le sk i , 1 987 ; C h opa rd a n d D ro z , 1 988 ;

d ' H u m i e r e s et aL, 1 988 ; M c N a m a ra , 1 990 ; K o ng a n d

Co hen , 1991); i t i s , how ever , la rge ly neg lec ted by th is re

v iew. Likew ise , we do not d iscuss recent la t t ice -gas mo d

e ls of reac t ion-diffusion equa t ions (Dab et aL, 1990,

1991; K a p r a l et aL, 1 991 ; La w n ic z a k et aL, 1991) . Last

ly , we have chosen to devote only minimal a t tent ion to

the " l a t t i c e -B o l t z m a nn me th od , " a n imp or t a n t e x t e ns ion

of the la t t ice gas which is of both theore t ica l and prac t i

c a l i n t e re s t . W h e re a s w e de r ive t he l a t t i c e -B o l t z m a nn

equat ion from the Boolean dynamics of la t t ice gases in

S ec . IV, methods for so lv ing th is equa t ion a re considered

only in a brief, i n t roduc to ry d i sc uss ion in A ppe nd ix C .

Fu r the r de t a i l s o f t he l a t t i c e -B o l t z ma nn me thod c a n be

found in the recent review by Benzi et aL (1992).

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II .

LATTICE-GAS MODELS OF SIMPLE FLUIDS

In th is paper , the te rm " la t t ice gas" re fe rs to a system

of par t ic les tha t move wi th a d isc re te se t of ve loc i t ies

from si te to s i te on a regular la t t ice . This k ind of la t t ice

gas is in some ways a genera l iza t ion of the c lassica l

la t t ice -gas models tha t have been employed, for example ,

in theore t ica l models of the l iquid-gas t ransi t ion (Stanley ,

1971) . The major d i ffe rence be tween the "new" la t t ice -

ga s mode l s a nd the i r c l a ss i c a l c oun te rpa r t s i s dyna mic a l :

mome n tum i s e xp l i c i t l y c onse rve d in t he ne w mode l s ,

t he re by a l l ow ing one to ob ta in hyd rodyna mic e qua t ions

o f mo t io n . The se mom e n tum -c ons e rv ing l a t t i c e ga se s a re

thus o f i n t e re s t fo r bo th hyd rodyna mic s a nd s t a t i s t i c a l

me c ha n ic s .

In th is sec t ion , we f i rs t provide a br ie f h istor ica l over

v iew of some spec i f ic hydrodynamic la t t ice -gas models .

We then in t roduce in some de ta i l the la t t ice -gas model of

F r i s c h

et al.

(1986) and fo l low tha t d iscussion wi th some

examples of la t t ice -gas s imula t ions.

A. Historical overview

From the s t a ndpo in t o f hyd rodyna mic s , t he e sse n t i a l

i nnova t ion due to mome n tum-c onse rv ing l a t t i c e ga se s i s

the s imu l t a ne ous d i sc re t i z a t ion o f spa c e , t ime , ve loc i ty ,

and densi ty . Dis cre t iz a t ion of space and t ime is in

mode rn t ime s r e l a t i ve ly munda ne , be ing a n e ve ryda y oc

currence in the numerica l so lu t ion of par t ia l -d i ffe rent ia l

equa t ions by , for example , the method of f in i te

di ffe rences. Dis cre t iz a t ion of ve loc i t ies , how ever , i s a re l

a t ive ly unusu a l idea . I t seems to hav e fi rs t been con

sidered for hydrodynamic f lows by Broadwel l , who

c ons t ruc t e d a d i sc re t e -ve loc i ty , c on t inuous- t ime ,

c on t inuous- spa c e , a nd c on t inuous-de ns i ty mode l t o f i nd

e xa c t so lu t ions t o a B o l t z ma nn e qua t ion de sc r ib ing shoc k

waves (Broadwel l , 1964) . Further ramif ica t ions of th is

a pp roa c h a re de sc r ibe d in t he monogra ph by G a t igno l

(1975).

The f i rs t d isc re te -ve loc i ty model in s ta t i s t ica l mechan

ics appears to have been proposed by Kadanoff and Swif t

(1 968 ) . In a n a t t e mp t t o de mons t ra t e t he t he o re t i c a l pos

sib i l i ty of the d ivergence of t ransport coeff ic ients near the

cr i t ica l poin t , they c rea ted a version of a c lassica l Is ing

model in which posi t ive sp ins ac ted as par t ic les wi th

momentum in , say , one of four d i rec t ions on a square la t

t ice , whi le nega t ive sp ins ac ted as holes. Par t i c les we re

then a l lowed to col l ide wi th o ther par t ic les or to ex

change the i r posi t ions wi th holes, but only i f energy

(based on nearest -ne ighbor Is ing in te rac t ions) and

mome n tum w e re e xa c t ly c onse rve d . The mode l , pu re ly

ana lyt ic in the form of a maste r equa t ion , was d isc re te in

space , ve loc i ty , and densi ty , but not in t im e . On e of the

new resul ts was tha t , despi te i t s s impl ic i ty , the dynamics

led to hydrodynamics v ia the existence of sound waves.

A fu l ly d isc re te model of hydrodynamics was f i rs t in

t rodu c e d in t he 1 970s by H a rd y , de Pa z z i s , a nd Pom e a u

( H a r d y

et al.,

1973, 1976) . The ir model consisted of

ident ica l par t ic les moving from si te to s i te on a square

l a t t i c e i n d i sc re t e t ime , c onse rv ing pa r t i c l e numbe r a nd

mo me n tu m upo n c o ll i s ion . The i r ob j e c tive w a s no t t he

s imu la t ion o f hyd rodyna mic s i n t he b roa d se nse , bu t

ra ther the s tudy of i ssues in s ta t i s t ica l mechanics, such as

ergodic i ty and the d ivergence of t ransport coeff ic ients in

tw o d ime ns ions . The i r w ork u se d the s imp le s t poss ib l e

mo del of mo lecula r dyna mic s and is notab le not only for

the reasons c i ted , but a lso for the in te rest ing in te rp lay

p rov ide d by the c ompa r i sons be tw e e n the o re t i c a l p re d i c

t i ons o f t he mode l ' s t r a nspo r t p rope r t i e s a nd the e mp i r i

ca l resul ts obta ined from numerica l s imula t ions of i t .

A l t h o u g h t h e m o d e l o f H a r d y

et al.

led to a number of

in te rest ing resul ts , i t has had only l imi ted appl ica t ions

be c a use i t s hyd r odyn a mic l im i t is a n i so t rop ic . Th i s i s t he

d i r e c t — a n d r a t h e r u n s u r p r i s i n g — c o n s e q u e n c e o f t h e

c ons t ra in t s impose d by the unde r ly ing squa re l a t ti c e . I t

was not rea l ized unt i l 1986, in the a forem ent ion ed wo rk

o f F r i sc h , H a ss l a c he r , a nd Pome a u (F r i sc h

et aL>

1986),

tha t a s imple extension of the model to a

triangular

la t

t ice wou ld suffice for i so t ro pic hyd rod yna mic s. We th us

tu rn to a n in t roduc t ion to t he FH P mode l .

B. The Frisch-Hasslacher-Pomeau lattice gas

In the fo l lowing, we f i rs t in t roduce a microdynamical

de sc r ip t ion o f t he FH P mode l . W e the n p rov ide a n ou t

l ine of the der iva t ion of the macrodynamical, o r h y d r o -

dyna m ic , be ha v io r . Fu l l de t a i l s c onc e rn ing the hyd ro -

dynamic l imi t a re g iven in Sec . IV.

1 .

Microdynamics

The FH P ga s i s c ons t ruc t e d o f d i sc re t e , i de n t i c a l pa r t i

c les which move from si te to s i te on a t r iangular la t t ice ,

col l id ing when they meet , a lways conserving par t ic le

num be r a nd mo me n tum . Th e dyna m ic s e vo lve s i n

discre te t ime steps; an example of the evolut ion during

one t ime step is i l lust ra ted in Fig . 1 . T he in i t ia l

configura t ion is g iven in Fig . 1 (a ). E ach a rro w r epres ents

a par t ic le of uni t mass moving wi th uni t speed (one la t

t ice uni t per t ime step) in one of s ix possib le d i rec t ions

given by the la t t ice l inks. No more than one par t ic le may

reside a t a g iven si te and move wi th a g iven ve loc i ty ;

thus, in th is example , s ix b i ts of informat ion suff ice to fu l

ly descr ibe the configura t ion a t any si te .

Each d iscre te t ime step of the la t t ice gas i s composed

of two steps. In the f i rs t, each par t ic le hops to a ne igh

boring si te [Fig . Kb)] in the d i rec t ion g iven by i t s ve loc i

ty . In the second step [Fig . 1(c)], t he pa r t i c l e s ma y c o l

l ide . The prec ise col l i s ion ru les a re parameters of the

model ; a l l co l l i s ions, however , conserve mass and

mo me n tum . Tw o e xa mple s o f c o l l is ions t ha t r e su l t in a

change in the ve loc i ty of par t ic les a re evident by compar

ing the middle row in Figs. Kb) and 1(c) . The two-body

col l i s ion could jus t as easi ly have ro ta ted coun terc lo ck

w i se a s c loc kw ise . Typ ic a l imp le me n ta t ions pe r fo rm

Rev. Mo d. Phys . , Vo l . 66 , No. 4 , Oc to ber 1994

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D. H. Rothman and S. Zaleski:

FIG. 1. One time step in the evolution of the FHP lattice gas.

Each arrow represents a particle of unit mass moving in the

direction given by the arrow, (a) is the initial condition, (b)

represents the propagation, or free-streaming step: each parti

cle has moved one lattice un it in the direction of its velocity, (c)

shows the result of collisions. The only collisions that h ave

changed the configuration of particles are located in the middle

row.

bo th w i th e qua l p roba b i l i t y , e i t he r t h rough the u se o f

ra n dom n um be rs o r v i a a de t e rmin i s t i c sc he me . Exp l i c i t

examples of col l i s ions a re g iven in Fig . 2 .

Th e mic ro dyna m ic s i n F ig . 1 i s e xp re sse d by

/ i

|

. ( x + c

/

, r + l ) = «

J

. ( x , 0 + A

/

[ n ( x , r ) ] . ( 2. 1)

The B oo le a n va r i a b l e s n = (n

1

,n

2

, - - . ,n

6

) indica te the

prese nce (1) or absen ce (0) of par t ic les m ovin g from si te x

to s i te -x + Cj , where the par t ic les move wi th uni t speed in

the d i rec t ions g iven by

C/ = (cos7n / 3 , sinTri / 3 ) , / = 1,2, . . . , 6 . (2.2)

I

Ai

2)

= flw

/

+

1

»

f + 4

( l — Ji /K l — »

/ + 2

) ( l — »/+ 3) ( l — »/+ 5>

-I- (1 — a )«

f

+ 2

w

i +

5

< 1

—

» / X

—

» / + 1 X

— w

i + 3 X

— / , .«, .

+ 3

(1 —

n

i + l

)(l-n

i+2

)(l~n

i+ 4

)(l-n

i+ 5

)

Note tha t A*-

2)

a l lows for c lockwise ro ta t i ons w hen the

supp le me n ta ry B oo le a n va r i a b l e a (x>t) = 1 , and for cou n

t e rc loc kw ise ro t a t i ons w he n a(x,t)

—

0. Fo r t he s imp le s t

la t t ice gas, the fu l l co l l i s ion opera tor A, i s

A ^ A J . ^ + A}

3

* . (2.5)

More e l a bo ra t e c o l l i s ion ope ra to r s ma y be fo rme d by in

c luding, for example , four-body col l i s ions or by a l lowing

i-gas models, of phase separation 1421

Before After

<

o

>

FIG. 2. Explicit examples of some collisions that may occur in

the FHP model. The two-body head-on collision may result in

either a clockwise or a counterclockwise rotation; here we show

just one example. The two-body collision shown with nonzero

net momentum results in no change, since no other

configuration exists that conserves both the number of particles

and the net momentum.

The col l i s ion opera tor A, descr ibes the change in n

t

(x

y

t)

due to col l i s ions, and takes on the va lues ± 1 an d 0. It is

the sum of Boolean expressions, one for each possib le col

l i s ion . Fo r e xa m ple , t he ope ra to r fo r t he t h re e -bod y c o l

lision in Fig. 2 is given by

A[

3 )

=

«

I

.

+ 1

^

+ 3

r t

/ + 5

( l - / t

t

. ) ( l - w

I

.

+ 2

) ( l - n

/ + 4

)

- ^ • ^ • + 2 ^ + 4 ( 1 - « , . + ) ( l - / f

/ + 3

) ( l - / t

/ + 5

) ,

(2.3)

where the c i rcula r sh i f t

i

+ 3

=j

suc h tha t

Cj• = — C/, j = 1 , . . . ,6 . Th e o per a to r for th e two -bod y

collision in Fig. 2 is

n

i+A

)

(2.4)

for col l i s ions wi th s ta t ionary , or " rest ," par t ic les

(d 'H umie re s a nd La l l e ma nd , 1 987 ) . In t he u sua l fo rmu la

t i ons , the only rest r ic t ions on A,- a re tha t i t conserve

mass ,

2 A , ( n ) = 0 , (2.6)

a nd tha t i t c onse rve m om e n tu m,


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1 4 2 2


I c

/

A

/

( n ) = 0 . (2 .7 )

Using the f i rs t of these re la t ions, one may sum the micro-

dyn am ica l equa t ion (2 .1) over each d i rec t io n i to obt a in

a n e qua t ion fo r t he c onse rva t ion o f ma ss ,

i i

a nd , a f t e r mu l t i p ly ing the sa me e qua t ion by c , , summing

again over i , and using the second re la t ion , one obta ins

a n e qua t ion fo r t he c onse rva t ion o f mome n tum,

2 c

/

n

/

( x + c ^ + l ) = 2 c

/

n

I

. ( x , ? ) . ( 2.9)

i i

Equat ions (2 .8) and (2 .9) descr ibe the evolut ion of mass

a nd mome n tum in t he B oo le a n f i e ld a nd ma y be c on

s i d e re d t h e m i c r o s c o p i c m a s s - b a la n c e a n d m o m e n t u m -

balance equa t ions, respec t ive ly , of the la t t ice gas.

2.

Macrodynamics

C onse rva t ion o f ma ss a nd mome n tum a t t he mic ro

scopic or molecular sca le of a f lu id impl ies the same con

se rva t ion a t a ma c rosc op ic , o r c on t inuum, sc a l e . I t i s a t

th is sca le , and par t ly f rom these conserva t ion laws, tha t

the N a v ie r -S toke s e qua t ions a re de r ive d (La nda u a nd

Lifshi tz , 1959a; Batc he lo r , 1967) . One expec ts tha t mu ch

the same ana lysis should apply to the la t t ice gas.

To obta in an overview of why th is i s possib le , consider

a cont iguous, enc losed se t of la t t ice s i tes . Note tha t the

change in mass wi th in th is se t i s prec ise ly ba lanced by

the f lux of ma ss out of i t . Con sider , then , the e volu t ion

o f t he a ve ra ge qua n t i t y

{n

t

),

in wh ich the averag e is tak

en over an ensemble of systems prepared wi th d i ffe rent

in i t ia l cond i t ion s. W e ident i fy ^i(n

t

) w i th t he ma ss a nd

2 / ( f t / ) c / w i t h t h e m a s s f lu x. W h e r e a s t h e u n a v e r a g e d

Boolean f ie ld i s necessar i ly noisy a t the smal lest sca les,

w e ma y a ssume tha t

(n

l

-)(x,t)

is s lowly varyin g in bo th

spa c e a nd t ime . W e c a n thus i n fe r t ha t t e mpora l a nd

spa t ia l sca les much la rger than one t ime step and one la t

t ice uni t , but s t i l l smal l enough such tha t

{n

t

)

var ies

slowly, may be def ined in the l imi t of long t imes and la rge

l a t t i c e s . S inc e the sa me ba l a nc e be tw e e n ma ss c ha nge

and mass f lux tha t appl ies to n

i

a lso appl ies to (n

(

), w e

ma y c onc lude , v i a t he d ive rge nc e the o re m a nd the u sua l

a r g u m e n t s o f c o n t i n u u m m e c h a n i c s , t h a t

3 , 2 < « /> = - 3 « : S < * / > * / « , ( 2. 10 )

w he re t he

a

c ompone n t o f t he i t h ve loc i ty ve c to r c

z

is

g iven by

c

ia

,

a nd the E ins t e in summa t ion c onve n t ion i s

a ssum e d ove r i nd i c e s g ive n by G re e k l e t t e r s .

O ne ma y re a c h a s imi l a r c onc lu s ion fo r t he mome n

tum . Th e c ha nge in t he

a

c o m p o n e n t o f m o m e n t u m i n

any region of the la t t ice i s i t se l f prec ise ly ba lanced by the

flux of

a

m o m e n t u m i n th e /? d i r e c t io n , 2 /

(

n

i)

c

ia

c

ip>

ou t o f t h i s re g ion . Thu s , by the sa me a rgum e n t , one ob

ta ins

i i

Fina l ly , by def in ing the m ass de nsi ty p = X;

^

n

t^

a n c

*

t h e m o m e n t u m d e n s i t y

pu

a

= X / (

n

t ^

c

ia

a n

d subs t i t u t i ng

in to Eqs. (2 .10) and (2 .11) , we obta in the famil ia r con

t i nu i ty e qua t ion ,

d , p = - d a ( p "

a

) f (2.12)

a n d t h e m a c r o s c o p i c m o m e n t u m - b a l a n c e e q u a t i o n ,

d

t

(pu

a

)=-dpIl

a/3

, (2.13)

o f hyd rodyna mic s , w he re i n t he l a t t e r w e ha ve in t ro

duc e d the mome n tum f lux de ns i ty t e nso r (La nda u a nd

Lifshi tz , 1959a)

nap=2,Cicfiie-

(2.14)

While Eq. (2 .12) i s fu l ly expl ic i t , the wri t ing of momen

tum conserva t ion as an expl ic i t equa t ion in te rms of

p

a nd u re qu i re s some w ork . N o t su rp r i s ing ly , t he p re s

ence of an underly ing la t t ice makes th is der iva t ion of hy

dro dy nam ics som ew hat d i ffe rent f rom the usu a l f lu id

c a se . Th e se mina l c on t r ibu t ion o f F H P w a s to no t i c e

tha t in a low-ve loc i ty expansion, a second-order tensor

such as

H

a

p

is wri t ten (Fr isch

et al.

y

1986)

n

a

j

8

=

J

P o ( p ) S a

i

8 + ^ a

i

8

r

5 ( p )

w

r

W

S

+ 0 ( l / 4 )

>

( 2

'

1 5

)

w h e r e / ?

0

a n d

h

a

p

yb

mu st be obta i ned from Eq. (2 .14) and

the expressions for (n

t

) , t he a ve ra ge popu la t ions . In o r

d ina ry c on t inuous me d ia t he t e nso r A ,

a/3r

§ is readi ly found

to be i so t rop ic a nd to p re se rve G a l i l e a n inva r i a nc e .

H ow e ve r , be c a use w e w ork w i th a n unde r ly ing l a t t i c e , i t

i s not th e case for la t t ice gases. In fac t ,

k>

a

p

r

&

ac ts instead

a s a n e l a s t i c i t y t e n so r a nd inhe r i t s t he symme t ry p rope r

t ies of the la t t ice just as e last ic i ty tensors share the sym

me t ry p rope r t i e s o f a c ry s t a l l a t t i c e . Th i s " me mory" o f

the la t t ice would , in fac t , doom our e ffor t to s imula te a

f lu id , except for a remarkable property of hexagonal la t

t ices wel l not iced by Landau and Lifshi tz , who wrote

(Landau and Lifshi tz , 1959b) , " I t should be not iced tha t

de fo rm a t ion in t he xy -p l a ne ( . . . ) i s de t e rm ine d by only

two modul i of e last ic i ty , as for an iso t ropic body; tha t i s ,

the e last ic propert ies of a hexagonal c rysta l a re i so t ropic

in t he p l a ne pe rpe n d ic u la r t o t he [he xa gona l ] a x i s . " In

e qua t ions , i so t ropy imp l i e s t he ge ne ra l fo rm

w he re 8

a / ?

is t he K ro ne c ke r d e l t a a nd

A

a n d

B

a re tw o in

de pe nde n t " e l a s t i c " modu l i , w h ic h mus t be de t e rmine d

f rom the a ve ra ge popu la t ions

{n

t

),

as is do ne in Sec. IV .

O nc e th i s i s done , t he mom e n tum -c on se rva t io n e qua t ion

take s the fo l lowing form , f rom (2 .13) , (2 .15) , and (2 .16):

d

t

pu

a

+ 2d

p

B(p)u

a

u

p

=- d

a

[p

0

(p)+A(p)u

2

] . (2.17)


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1423

T hi s equ a t i on i s c lose , bu t no t qu i t e i den t i ca l , t o t he usu

a l E u l e r equa t i on fo r com pr es s i b l e f low . M or eo ve r , we

have not given the express ion for the coeff ic ients A and

B, nor f o r p

0

(p). Ho wev er , a s we de t a i l i n S ec . I V , i n t he

l i mi t o f van i sh i ng u, t h e l a t t i c e - g a s h y d r o d y n a m i c a l

equa t i on i s equ i va l en t t o t he usua l incompressible E u l e r

e q u a t i o n .

T o ob t a i n t he v i scous t e r m, and t he r e f o r e t he Nav i e r -

S t okes equa t i on o f t he l a t t i ce gas , one need a l so cons i de r

g r ad i en t s o f t he mom en t um f i el d a t s econd o r de r . T h e

f o u r t h - r a n k viscous stress tensor, i t se l f obeying the sym

met r i e s o f E q . ( 2 .16) , i s t hen i n t r od uce d t o r e l a t e v i s cous

s t r e s s t o ve l oc i t y g r ad i en t s i n t he l a t t i ce gas . T he t wo

f r ee pa r ame t e r s o f t h i s t ensor t hen de t e r mi ne t he shea r

and bu l k v i scos i t i e s ( j us t a s t hey woul d g i ve t he L ame pa

r ame t e r s i n e l a s t i c i t y t heor y) . T ha t t he l a t t i ce gas does

i ndeed have a v i scos i t y i s concep t ua l l y deduced by ob

se r v i ng t ha t co l l i s i ons and p r opaga t i on con t r o l t he r a t e a t

wh i ch m om en t um d i ff uses . T h e r e l evan t d i ff usi on

coe ff ic i en t , o r k i ne ma t i c v i s cos i t y , may t hen be ca l cu l a t ed

v i a a B o l t zm ann ap pr o x i m a t i on , i. e ., by ignor i n g cor r e l a

t i ons be t we en pa r t i c l e s ( Hen on , 1987b) . E ach o f these is

sues i s addressed in deta i l in Sec. IV.

F i na l l y , we no t e t ha t t he de r i va t i on o f hydr odynami cs

i n t h i s s ec t i on , c r ude a s i t i s , r evea l s one ve r y i mpor t an t

point : the precise deta i l s of the col l i s ion rules (as ide f rom

cer t a i n pa t ho l og i ca l cho i ces t o be d i scussed l a t e r ) do not

a ff ec t t he f o r m of t he cons t i t u t i ve , hydr od yna mi c equa

t i ons . R a t he r , t hey de t e r m i ne t he va l ues o f t he t r an spo r t

coefficients.

poss i b l e w i t h t he F HP mode l .

F i gu r e 3 show s one of t he f ir st hyd r od yn am i c f lows

s i m u l a t e d b y t h e l a t t i c e - g a s m e t h o d ( d ' H u m i e r e s ,

P om eau , and L a l l em and , 1985 ). T h i s two- d i men s i ona l

f low past a f la t pla te i s forced by inject ing par t ic les a t the

l e f t boundar y o f t he l a t t i ce and r emovi ng pa r t i c l e s a t t he

r i g h t b o u n d a r y , t h u s c r e a t i n g a p r e s s u r e g r a d i e n t . T h e

f l ow , a t a R eyno l ds number o f appr ox i ma t e l y 70 , c r ea t e s

v o r t i c e s , k n o w n a s von Karman streets, beh i nd t he p l a t e .

T h i s f l ow f i e l d qua l i t a t i ve l y ma t ches t hose t ha t woul d be

o b t a i n e d f r o m q u a s i - t w o - d i m e n s i o n a l e x p e r i m e n t s o r o t h

e r me t hods o f numer i ca l s i mul a t i on .

As we sha l l d i s cus s l a t e r , one i n t e r e s t i ng a spec t o f t he

l a t t i ce - gas me t ho d i s t he ease w i t h wh i ch one may s i mu

l a t e fl ows in compl ex geom et r i e s . An ap p l i ca t i on o f t h i s

capab i l i t y is t he s t udy o f fl ows t h r ou gh mi c r osc op i ca l l y

d i so r d e r ed por ou s med i a . A n exam pl e o f a s i mu l a t i on o f

f l o w t h r o u g h a t w o - d i m e n s i o n a l p o r o u s m e d i u m i s s h o w n

i n F i g . 4 ( R ot hman , 1988) . F l ows such a s t hese obey a

l i nea r fo r ce -f lux r e l a t i on kno wn as Dar cy ' s l aw ; t he s i m u

l a t i ons a l l ow es t i ma t i on o f t he conduc t i v i t y , o r pe r mea

bil i ty, coefficient of the bulk flow.

T hese and o t he r s i mul a t i ona l s t ud i e s a r e desc r i bed i n

mo r e de t a i l i n S ec . I X . No w, howe ver , we t u r n t o a con

s i de r a t i on o f how t he s i mpl e F H P mod e l ma y be m odi f ied

t o s i mul a t e t he dyn am i cs o f ce r t a i n m ul t i ph ase f lu ids and

inter faces .

I I I.

LATTICE-GAS MODELS

OF PHASE-SEPARATING MIXTURES

C. Simulations

B ef or e i n t r oduc i ng mode l s w i t h i n t e r f aces , i t i s use f u l

t o i l l us t r a t e t he k i nd o f hydr odynami c s i mul a t i on t ha t i s

A s i n d i c a t e d i n t h e I n t r o d u c t i o n , o n e o f t h e i m p o r t a n t

gene r a l i za t i on s o f t he F H P l a t t i ce gas has been t he i n t r o

duc t i on o f d i s c r e t e mode l s f o r t he s i mul a t i on o f hydr o-

dyn am i c mi x t u r es . T h e ea r l i e s t mo de l s o f mi x t u r es wer e

/ J S ~» *»

i 1

r /

u \ U t r ^ -

;-w^

_ "N.*^

\ \ -

SSS

>

r r r,

1

I I I

7 r

t

£

T t/T '/ ^ •*

» • * « - » - » - « • •» \ •

•

N N N N

\ \ i

I- v i , . . , \

\\XX\\

J * • > I l i f *

rJ

iX

7 T

T T *

-*~*~*Sl*SS/' T

« ? • • > > ) > > - » ^ » »

- » - r - y - » r - * - » » » > > » » » ^ » >

FIG. 3. Two-dimensional f low past a f lat plate, simulated using the FHP lat t ice gas (d'Humieres, Pomeau, and Lallemand, 1985).

The Reyn olds number is approximately 70.

Rev. Mod. Phys., Vol. 66, No. 4, October 1 994

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1424


conce i ved s i mpl y by add i ng a s econd spec i e s o f pa r t i c l e s .

In the case of a

passive

s c a l a r ( C h e n a n d M a t t h a e u s ,

1987;

B a u d e t et al., 1989), t he on l y new dyn ami c s o f i n

teres t i s di f fus ion of one species into the oth er . Th e

second spec i e s , however , can a l so be active. T h us , f o r ex

ampl e , B ur ges and Z a l e sk i ( 1987) c r ea t ed a mode l o f a

mi x t u r e t ha t was no t on l y d i ff usi ve bu t a lso bu oya n t . A

f ur t he r gene r a l i za t i on o f t h i s so r t is t he i n t r od uc t i o n o f

reactive fluids (Clavin et al, 1986 , 1988 ; d 'H um i e r es

et al, 1987 ; Da b et al, 1990 , 1991 ; Ka pr a l et a/ . , 1991).

I n t h i s ca se co l l i s i ons i nvo l v i ng mor e t han one spec i e s

need no t conse r ve t he number o f pa r t i c l e s o f each spec i e s

enter ing a col l i s ion.

I n each o f t he mi x t u r e mode l s c i t ed above , t he dynam

ics of col l i s ions and propagat ion of the (unforced) f luid

mi x t u r e a r e i ndependen t o f t he pa r t i cu l a r spec i e s t ha t a

pa r t i c l e may r epr esen t ; t he new behav i o r comes i ns t ead

from the red is t r ib ut io n of species ( i.e . , mass) af ter per

f o r mi ng t h e F H P co l l i s ions desc r i bed in t he p r ev i ous s ec

t i on . T h us a s econd , qua l i t a t i ve l y d i ff e ren t mi x t u r e mo d

e l r e su l t s f r om c r ea t i ng a dynami cs i n whi ch t he r ed i s t r i

b u t i o n o f momentum depe nds on t he d i s t r i bu t i o n o f mass

( o r poss i b l y a l so moment um) p r i o r t o co l l i s i on . W e i n t r o

du ce two such mod els below . In the fi rs t case , two

spec i e s i n t e r ac t w i t h each o t he r t o c r ea t e i n t e r f aces w i t h

surface tens ion. In the second , a s ingle species of pa r t i

c les interacts wi th i t se l f and a l so forms inter faces ; but

ra ther than separat ing two species of f luids , the inter face

separates a dense ( l iquid) phase f rom a less dense (vapor)

phase .

9>x

FI G . 4. Lattice-gas simulation of flow thro ugh a two-dim ensional poro us medium (R othm an, 1988). Th e fluid is forced from left to

right .

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A. Immiscible lattice gas

T h e

imm iscible lattice gas

( ILG) is a two-spec ies var i

a n t o f t he FH P mode l (R o thm a n a nd K e l l e r , 1 988 ). A t a

me cha nist i c leve l, the d i ffe rences f rom an d simi la r i t ies to

the F H P m ode l a re be s t r e al i z e d from a c omp a r i son o f

the mic rodyna mic s o f t he tw o mode l s .

F igu re 5 i l l u s t r a t e s t he ILG m ic rodyn a mic s . Th e in i

tial state [Fig. 5(a)] of the lattice is the same as in Fig. 1,

bu t now some o f t he pa r t i c l e s a re c o lo re d " re d , " w h i l e

the o the r s a re c o lo re d " b lue . " The hopp ing s t e p , F ig .

5(b), is prec ise ly as before : par t ic le s propa ga t e to the

ne ig hbo ring si te in the d i rec t io n of the i r ve loc i ty . Th e

col l i s ion step in Fig . 5(c) , how ever , i s d i f fe rent . Ro ugh ly

spe a k ing , t he ILG c o l l i s ion ru l e c ha nge s t he

configura t ion of par t ic les so tha t , as much as possib le ,

r e d pa r t i c l e s a re d i r e c t e d tow a rd ne ighbo rs c on ta in ing

red par t ic les , and b lue par t ic les a re d i rec ted toward

ne ighbo rs c on ta in ing b lue pa r t i c l e s . The to t a l ma ss , t he

to t a l mome n tum, a nd the numbe r o f r e d (o r b lue ) pa r t i

c les a re conserved . Tw o examples of th is col li s ion ru le

are seen by comparing the middle row of Fig . 5(b) wi th

that of Fig. 5(c).

The ILG mic rodyna mic s ma y be de sc r ibe d a s fo l low s .

Each la t t ice s i te may conta in red par t ic les , b lue par t ic les ,

FIG. 5. Microdynamics of the immiscible lattice gas, in which

the initial condition (a), the propagation step (b), and the col

lision step (c) are displayed as in Fig. 1. The initial condition

and propagation step are the same as before, except that now

some particles are red (bold arrows) while others are blue (dou

ble arrows). In the collision step, the particles are rearrang ed so

that, as much as possible, the flux of color is in the direction of

the local gradient of color. A compa rison of the middle row

here with Fig. 1 shows how IL G collisions can create a "color

blind" microdynamics different from that created by FHP col

lisions.

or both , but a t most one par t ic le ( red or b lue) may move

in each of the s ix d i rec t ions c

1 ?

. . . , c

6

. In the usua l im

p le me n ta t ion , e a c h s i t e ma y a l so ha ve a se ve n th s t a t i on

ary , or rest , par t ic le moving wi th ve loc i ty c

0

—

0 , and sub

jec t to the same exc lusion ru le . The configura t ion a t a

si te x i s thu s descr ibed by the Bo olean va r iables r = {

r]}

an d b = {

b

{

}

, w he re t he ro ma n inde x / a ga in i nd i c a t e s t h e

ve loc i ty , and

r

{

a n d

b

{

cannot s imul taneously equa l 1 .

At a site x, a color flux q is defined to be the difference

b e t w e e n t h e r ed m o m e n t u m a n d t h e b l u e m o m e n t u m :

q [ r( x ) , b (x ) ] = 2 c ^ U J - ^ t x ) ] . (3 .1)

i = i

A ve c to r p ropo r t iona l t o t he l oc a l c o lo r g ra d i e n t (o r

"field") is also defined,

« x ) = 2 c

f

2 [ o

( x + c

^ ~

6

7

( x + c

« - ) ] •

(3

-

2)

i

J

Th e ILG co l l i s ion ru le is ant id i ffusive : i t max imize s th e

f lux of color in the d i rec t ion of the loca l color gradient .

The resul t of a col l i s ion , r—•r ' , b - ^ b ' , is the

c on f igu ra t ion tha t ma x imiz e s

q ( r ' , b ' ) . f , ( 3. 3)

suc h tha t t he numbe r o f r e d pa r t i c l e s a nd the numbe r o f

blue par t ic les i s conserved,

2 ' / = 2 ' / . 2 * / = 2 * /

>

«.4)

/'

i i i

a nd so i s t he t o t a l mome n tum:

2 c

I

- ( r / + 6 / ) = 2 c

/

( r

/

+ 6

/

) . (3 .5)

i i

I f more tha n one c ho ic e fo r r ' , b ' ma x imiz e s (3 .3 ) , t he n

the outcome of the col l i s ion is chosen wi th equa l proba

bi l i ty f rom among these opt imal configura t ions.

A na logous to t he d i sc re t e mic rodyna mic a l e qua t ion

(2.1) fo r t he F H P mod e l a re tw o c oup le d mic r odyn a mic a l

e qua t ions for t he IL G — on e fo r t he r e d pa r t i c l e s ,

r

i

(x-hc

i

,t+l)

=

r

l

(x,t)

, (3.6)

and another for the b lue par t ic les ,

^•(x + c,.,*+ l) = &/(x,f) , (3.7)

w he re

r / =

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D. H. Rothm an and S. Zales ki: Lattice-gas mode ls of phase separa tion

t=500

t = 1 0 0 0

FIG. 6. Phase separation in the immiscible

lattice gas. The initial condition was a homo

geneous random mixture, with 50% red (black)

particles, and 50% blue (gray). Time t is given

in time steps. Boundaries are periodic in both

directions. From Rothman (1992).

f / | f | obta in ed from a simpl i f ica t ion (descr ibed in Sec .

V.C.2) of Eq. (3 .2) . Al though here we only d iscuss ILG

mode l s t ha t ob t a in /

#

f rom in fo rma t ion a t ne ighbo r ing

s i t e s , ILG mode l s w i thou t e xp l i c i t de pe nde nc e on ne igh

bo r ing s i t e s ha ve a l so be e n p ropose d (C he n , D oo le n

et aL

9

1991; Som ers and Re m , 1991) . Th ese models use

c o lo re d " ho le s" i n a dd i t i on to c o lo re d pa r t i c l e s t o ob ta in

f

m

using only the loca l s ta te .

A sa l ient fea ture of the ILG is i t s abi l i ty to s imula te

pha se se pa ra t ion in a b ina ry mix tu re ; a n e xa mple i s

show n in Fig . 6 . He re a 25 6X 25 6 la t t ice i s in i t ia l ized as

a r a nd om mix tu re w i th a ve ra ge de ns i ty p — 4.9 pa r t i c l e s

pe r s i te , w i th 5 0% o f t he pa r t i c l e s r e d , a nd 5 0% b lue . A s

t ime p rog re sse s , t he doma ins o f r e d a nd b lue g row l a rge r ,

eventua l ly resul t ing in a s teady sta te in which one th ick

blue st r ipe is para l le l to an equa l ly th ick red st r ipe . M ea

su re me n t s show tha t t he r e d r i c h pha se i s v i r t ua l ly

( » 9 9 % ) pu re r e d , a nd l ike w ise fo r b lue .

La te r w e sha l l d i sc uss s imu la t ions o f pha se se pa ra t ion

in g re a t e r de t a i l , bo th f rom a phe nome no log ic a l a nd

the o re t i c a l po in t o f v i e w . W e no te now , how e ve r , t ha t

the collision rule defined by Eqs. (3.1)—(3.3) can differ

from t he p la in FH P col l i s ions only i f ther e is mo re t ha n

one color present a t the s i te loca ted a t x ; i t i s only in th is

case tha t d i f fe rent combina t ions of r

t

a n d b

(

can c rea te

different values of the color flux q t ha t c a n c on t r ibu t e

diffe rent ly to the maximiza t ion of (3 .3) . Thus, a f te r we

have establ ished ( in a la te r sec t ion) tha t there i s indeed

surface tension a t the in te rfaces, we sha l l see tha t , in ad

di t ion to be ing a model of phase separa t ion in a b inary

f lu id , the ILG is a lso a model of the hydrodynamics of

two-phase f low.

B. Liquid-gas model

In our second model of a mul t iphase f lu id , a

liquid-gas

(LG) model , there i s only one spec ies of par t ic le , but two

" the rmodyna mic " pha se s (A ppe r t a nd Za le sk i , 1 990 ) .

O ne pha se — l iqu id— ha s a h igh de ns i ty o f pa r t i c l e s ,

w h i l e t he o the r pha s e — ga s — is r e l a ti ve ly r a ref i ed . Th e

tw o pha se s r e su l t f rom a ru l e t ha t e xc ha nge s mome n tum

between si tes separa ted by one or more la t t ice uni ts ,

which , as we sha l l show la te r , modif ies the re la t ionship

be tween pressure and densi ty ( the "equa t ion of s ta te" ) in

(a)

(b)

(c)

(d)

FI G. 7. Micrody namics of the liquid-gas model, in which the

initial condition (a), the propagation step (b), and the collision

step (c) are precisely the same as that shown in Fig. 1. T he new,

interaction step is shown in (d). In this example, the interaction

distance r=2.

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1427

such a way as to a l low coexistence of the dense and

raref ied phases.

F igu r e 7 i l l u s t r a t e s t he dyna m ic s . Th e in it i a l s t a t e a nd

the hopping step a re prec ise ly the same as in Fig . 1 .

Th ere a re , how ever , now two col l i s ion steps. Th e f i rs t

col l i s ion step , Fig . 7 (c ), i s the sa me as tha t in the F H P

mod e l . W e w r i t e t he ou tc om e o f t h i s " c l a ss i c a l " c o l l is ion

as

»/ = « / + A,.(n) . (3.10)

In the second, in te rac t ing col l i s ion step [Fig .

7(d)],

sites

a t loc a t ions x and x +

rc

f

-

(where here we choose

r=2)

t r a de pa r t i c l e s mov ing in d i r e c t ions —

c

t

a n d

c

i9

r e spe c

t ive ly , i f and only i f both par t ic les exist pr ior to the ex

c h a n g e

an d

t he e xc ha nge c a n be pe r fo rme d w i tho u t

vio la t ing the exc lusion ru le . Fig ure 8 i l lust ra tes the ru le

in de ta i l . In te rm s of Bool ean var iables , we def ine

y

/

= n / ( x ) n /

+ 3

( x ) n / ( x + r c

/

) n /

+ 3

( x 4 - r c

l

-) , (3.11)

w he re ove rba r s i nd i c a t e t he B oo le a n ope ra to r " no t " a nd

th e circu lar shift / + 3 is defined as in Eq s. (2.3) an d (2.4).

The abi l i ty to perform the in te rac t ion is thus g iven by the

B oo le a n va r i a b l e

y

i9

a nd the mic ro dyna m ic a l e qua t ion

de sc r ib ing the c omple t e se que nc e o f p ropa ga t ion fo l

lowed by c lassica l and in te rac t ing col l i s ions becomes

n

/

(x + c

/

, r + l ) = n / ( x , f) + r

J

. - y

/ + 3

• <

3

-

12

)

The l i qu id -ga s mode l s t ha t w e re o r ig ina l ly p ropose d

c on ta ine d more in t e ra c t ions , r e qu i r ing a more c ompl i c a t

e d a na ly s i s a nd imp le me n ta t ion (A ppe r t a nd Za le sk i ,

1 990, 1 993 ; A ppe r t et al, 1991). In th is review, we dis

c uss on ly the s imp le r mode l o f A ppe r t , d 'H umie re s , a nd

Zales ki (1993) , g iven by Eq s. (3 .11) and (3 .12) . Th ou gh

the o ld and new models d i f fe r quant i ta t ive ly (e .g . , the

va lues of the t ransport coeff ic ients) , the qua l i ta t ive

be ha v io r r e ma ins t he sa me .

As in the ILG, the sa l ient fea ture of the l iquid-gas

mod e l i s pha se se p a ra t ion , w h ic h w i l l oc c u r fo r c e r t a in

choice s of the in i t ia l densi ty of par t ic les . Th is beh avio r i s

i l l u s t r a t e d in F ig . 9 . U n l ike t he ILG , pha se se pa ra t ion in

the l iquid-gas model manifests i t se l f not as the segrega

t ion of two spec ies of par t ic les in to separa te regions, but

Before

< o > < o >

After

* o > < o >

FIG. 8. Interacting collision in the simplest liquid-gas model.

Solid arrows represent particles; broken arrows represent the

absence of a particle. The sites on which the interactio n o ccurs

are situated

r

lattice units apart.

as the segrega t ion of a s ingle spec ies of par t ic le in to re

g ions of h igh an d low densi ty . As we sha l l show, the re l

a t ive volume of each region depends on the in i t ia l to ta l

densi ty of the s ingle spec ies, ra ther than the re la t ive con

c e n t ra t i o n o f tw o spe c ie s a s i n t he IL G .

La s t ly , w e no te t ha t , un l ike t he ILG , t he hyd rodyna m-

ic behavior of the bulk phases of the l iquid-gas model

doe s no t a u tom a t i c a l ly r e duc e to t ha t o f t he FH P m ode l .

O ne mus t i n s t e a d pe r fo rm the sa me mu l t i sc a l e e xpa ns ion

used to ana lyze the p la in la t t ice gas; the resul ts (de ta i led

in Sec . VI) a re then seen to d i ffe r only in such te rms as

the v iscosi ty .

IV. THEORY OF SIMPLE LATTICE-GAS AUTOMATA

In th is sec t ion we review the theory tha t leads f rom the

mic rosc op ic de f in it i on o f s imp le , s i ng l e -c ompon e n t l a t t i c e

ga se s g ive n in Se c . IV .A to t he l a rge - sc a l e hyd rodyna mic

equ at ion s. A grea t s impl i f ica t ion is achiev ed i f one uses

the B o l t z ma nn mo le c u la r -c ha os a ssumpt ion , w h ic h i s

e qu iva l e n t t o c ons ide r ing tha t t he pa r t i c l e s e n t e r ing a

c o l l is ion a re no t c o r re l a t e d . F r om th i s a ssum pt ion , one

ob ta in s t he Fe rmi -D i ra c e qu i l i b r ium d i s t r i bu t ion fo r t he

l a t t i c e-ga s a u to ma ton . Th i s e qu i li b r ium d i s t r i bu t ion a l

lows one to f ind the hy dro dyn am ica l equ a t ion s. Th e f irs t

resul t i s the Euler equa t ion for the la t t ice gas. Th e

Fe rmi -D i ra c e qu i l i b r ium a nd the Eu le r e qua t ion a ppe a r

a t t he l ow e s t o rde r o f a mu l t i p l e sc a l e o r C ha pma n-

Enskog e xpa ns ion fo r t he l a t t i c e ga s . A t se c ond o rde r ,

t h i s e xpa ns ion y i e ld s t he N a v ie r -S toke s e qua t ions a nd e x

pl ic i t expressions for the v iscosi ty of the model , as we

show in Sec . IV.B.

A more ge ne ra l s t a t i s t i c a l -me c ha n ic a l a pp roa c h a ba n

dons the mo le c u la r -c ha os a ssumpt ion a nd c ou ld y i e ld a

more r i go rous t he o ry fo r t he l a t t i c e ga s . H ow e ve r , t h i s

app roa ch is only par t ia l ly deve lo ped, and we review i t

only br ie fly in Sec . IV .C . Th ere we show, for exam ple ,

tha t the equi l ibr ium sta te may be obta ined d i rec t ly as a

solu t ion of a Liouvi l le equa t ion , instead of a Bol tzmann

equa t ion . Th e d iscussion of mo re subt le e ffec ts of space

d i sc re t i z a t ion , suc h a s t he a ppe a ra nc e o f spu r ious i nva r i

a n t s , is a lso cover ed in Sec . IV .C. Th e reader in te res ted

on ly in ge ne ra l i de a s a bou t t he de r iva t ion o f t he N a v ie r -

Stokes equa t ions may skip Sec . IV.C and lose l i t t le of im

me d ia t e r e l e va nc e to t he more c omple x l a t t i c e -ga s mode l s

descr ibed in the remainder of the review.

A. Som e typical lattice-gas automata

We sha l l a t tempt a more prec ise def in i t ion for a num

ber of la t t ice -gas mod els . The se models a re def ined o n

one -

to four-d imensiona l la t t ices. We sha l l need a few

geom etr ica l prope rt ies of these la t t ices. Only a f rac t ion

of the theory of c rysta l lographic la t t ices wi l l be useful

here , namely , the theory of regular Brava is la t t ices, for

which we reca l l the most important fac ts and def in i t ions.


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1 4 2 8


t= 2 0

t= 1 8 0 t= 3 2 0

1=440 t=600

FIG. 9. Phase separation with particle removal in the 2D liquid-gas model (Appert and Zaleski, 1993). The pixels are black for more

than two particles per site and white otherwise. Thu s the liquid phase is mostly black, while the gas phase is mostly white. The lat

tice is initialized with a uniform particle distribution . As time progresses, particles are slowly removed at rand om. After an initial

transient, the density of the liquid and the gas remain constant, but the fraction of space covered by the dense phase is decreasing.

This leads to the formation of a 2D soap froth.

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1 4 2 9

1. Regular Bravais lattices

T h e t e r m

lattice

deno tes a se t of i so la ted poi nts X in

D -d ime ns iona l spa c e

R

D

.

In a

Bravais lattice,

each point

ha s i de n t i c a l su r ro und in gs . Ma the m a t i c a l ly , th i s me a n s

tha t the la t t ice i s invar iant by a t ransla t ion tha t br ings

any point of

JL

on any o ther point . In equa t io ns, we le t

T

u

be a t ran sla t ion of space by a vec to r u and wr i te the

definition

T

x

_

y

X=X

for any pa i r (x , y) of vec to rs of X . A la t t ice is periodic if

i t i s inva r iant by a gro up of t ran sla t ions . I t ma y be

p rove d tha t i n D -d ime ns iona l spa c e , B ra va i s l a t t i c e s a re

those pe r iod ic l a t t i c e s ge ne ra t e d by l i ne a r c ombina t ions

with integer (positive or negative) coefficients n

t

of D in

de pe nde n t ve c to r s u , , fo rming the se t o f po in t s

WJUJ+ • • * +n

D

u

D

.

B ra va i s l a t t i c e s a re d i s t i ngu i she d by the i r symme t ry

p r o p e r t i e s . T h e point symmetry group 9 of a latt ice is the

g r o u p of congruent transformations (o r isometries) leaving

a la t t ice point f ixed and the la t t ice g loba l ly invar iant .

There is a lways a smal lest se t

S

o f ne ighbo rs i nva r i a n t

by the symme t ry g roup

9

and conta in ing a se t of genera t

ing vec to rs . Th is se t of ne ig hbo rs forms a polyg on in 2D

f rom w h ic h the l a t t i c e d ra w s i t s na me (he nc e

monohe d ra l , squa re , r e c t a ngu la r , a nd he xa gona l l a t t i c e s ) .

In 3D the se t of ne ighbors i s a polyhedron, and, in any

dim ensio n, a poly tope (Coxeter , 1977). Ob viously 9 is

a l so t he symme t ry g roup o f a po ly tope a ssoc i a t e d w i th

the la t t ice . A regular B ravais lattice is a Brava is la t t ice in

w h ic h the se t S is a regular poly tope .

The re gu la r i t y o f t he l a t t i c e i s ve ry impor t a n t i n ob

ta in ing the desi red propert ies of i so t ropy for f lu id-

me c ha n ic a l be ha v io r . Th e c onne c t ion f rom l a t t i c e sym

metry to la rge-sca le i so t ropy goes a long the l ines d is

cussed in Sec . I I .B and is d iscussed in more de ta i l in Ap

pend ix A. Th e desi re for i so t ro py mot iva tes us to rest r ic t

a l l de ve lopm e n t s t o r e gu la r B ra va i s l a t t i c e s . A mo ng the

f ive Brava is la t t ices in 2D are only two regular ones, the

square and hexagonal la t t ices. In 3D we f ind three cubic

l a t t i c e s— the s imp le c ub ic , t he c e n te re d c ub ic , a nd the

face-cente red cub ic . In 4D the ( regular) face-cente red

hyp ercu bic la t t ice a lso tur ns out to be useful . In s ingle -

ve loc i ty models , where there i s a s ingle ve loc i ty modulus

for a l l par t ic les , we sha l l denote by c

t

the se t of vec tors

jo in ing a s i te wi th i t s se t of nearest ne ighbors S. The ve c

t o r s Cf ha ve symm e t ry p rope r t i e s t ha t a re imp or t a n t i n

the ca lcula t ion of var ious quant i t ies , which a re der ived in

A ppe nd ix A us ing some ge ome t r i c a l f a c t s a bou t r e gu la r

poly topes.

2. Models on the hexagonal lattice

a. Six-velocity model

Th e six-ve loc i ty mo del was descr ib ed in Sec . I I .B . Th e

col l i s ion ru les g iven in Eqs. (2 .3) and (2 .4) correspond to

the or ig ina l FH P -I mod el . I t i s useful whe n discuss ing

th i s a nd subse que n t mode l s t o ha ve in mind a

c lassi f ica t ion of a l l configura t ions by c lasses of equa l

mo me n tu m a nd ma ss . T he n e a c h c on f igu ra t ion is

c ha ra c t e r i z e d by th re e numbe rs (n,g*,g*), w he re t he i n -

n=1

9 * y

= 1

g*y = o

g* x = o

g*x = i

/

g *

x

= 2

—

^=g

g*x = o g *

x

= i

g *

y

= 2 V

g* y

= 1

\ ^

g *

y

= 0 ̂ •

(3)

9*x = 2 g *

x

= 3

z,

<

n=3

I

g*x = o

g *

y

= 2

g *

y

=

1

(2)

g* x= i g*x = 2 g*

x

= 3

v.

(2)

(2)

g * x = 4

J

F IG. 10 . Conf igu ra t ions fo r the s ix -ve loc i ty F HP la t t i c e gas .

Each en t ry co r re sponds to a g iven c la s s

in,g*,g*).

O t h e r

con f igu ra t ions in the same c la s s may some t imes be ob ta ined by

ro ta t ion . Th e numb er o f con f igu ra t ions in the c la s s i s then

show n in pa ren t heses . Conf igu ra t io ns fo r o the r va lues of

(g?,g*)

may be ob ta in ed by re f lec t ions . Con f igu ra t ions fo r

n >

3 a re a l l ob ta in ed by exchang in g pa r t i c le s wi th ho le s .

Rev. Mo d. Phys . , Vo l . 66 , No. 4 , Oc tober 1994

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1430

D. H. Rothman an d S. Zaleski: Lattice-gas models of phase separation

t e ge r s n

9

g*,g* l a bel t he pa r t i c l e num be r a nd mo me n tu m.

The y a re de nne d by

n(s) =

^s

t

,

gf(s) = 2g

x

(s) ,

n=1

g * y = 1

9*y = 0

g*x = o

•

g*x =

1

/

g*x

=

2

—-**

(4.1)

w h e r e g(s) = ^

ii

s

t

c

ia

i s t he mo me n tu m o f c on f igu ra t ion s

and the fac tors 2 and 2/V3 a re add ed to obta in in teger

va lues for

n,g*

y

g*.

The possib le configura t ions for a s ix-ve loc i ty la t t ice gas

are shown in Fig . 10 . I t i s imm edia te ly seen tha t th e

FHP-I model does not perform a l l possib le col l i s ions.

There a re , for instance , two members in c lass (3 ,1 ,1)

which could be t ransformed in to each o ther by col l i s ion .

n=2

g *

y

=

2

g *

y

= i

g*y = o

g *x=o

V

M

•

(3)

g * x = i

/

g * x = 2

<

g * x = 3

z.

r^i

FIG. 11. Configurations for the seven-velocity

FHP lattice gas, constructed using the same

scheme as in Fig. 10. The solid circle

represents a rest particle. Notice that for

(n,g?,gf) = (3,0,0) there are two subclasses A

an d B. In each subclass the configurations

may be deduced from each other by rotations

and reflections.

g*x=o g*x=i g*x=2 g*x = 3 g*

x

=4

•>» V

VL

9*y

=

1

\ ^

4

(2) «

'

>

g *

y

= 0

»y~^ „/*

( 3 ) - . • ; \ ^


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D. H. Rothman and S. Zaleski: Lattice-gas mode ls of phase separation

1431

Inc luding a l l such col l i s ions leads to a s ix-ve loc i ty

c o l l i s ion - sa tu ra t e d mode l (d 'H umie re s a nd La l l e ma nd ,

1987) . We re turn to the def in i t ion of these models be low.

b. Seven-velocity model

A l though the FH P- I mode l i s ve ry s imp le , i t ha s c e r

t a in unw e lc om e fe atu re s . Fo r i n s t a nc e , t he tr i p l e c ol

l i s ion is re la t ive ly infrequent compared to the pa i r col

l i s ions. Th is i s un for tun a te , s ince pa i r col l i s ions cons erve

no t on ly ma ss a nd mome n tum bu t a n a dd i t i ona l i nva r i

ant . As we sha l l see , th is v i t ia tes the der iva t ion of hydr o

dyna m ic s . A n o th e r d if f ic u lty i s t ha t t he c om pre ss ion

v i sc os i ty i s z e ro fo r t he FH P- I mode l (F r i sc h

et aL,

1987). A simple imp rov em ent i s to add one or severa l

rest par t ic les . W e sha l l rest r ic t ourse lves to only one rest

pa r t i c l e .

A t t h i s po in t i t i s u se fu l t o i n t roduc e the s t a nda rd no

t a t i on (d 'H umie re s a nd La l l e ma n d , 1 987) fo r mu l t i p l e -

spe e d mode l s . The pa r t i c l e ve loc i t i e s now op t iona l ly c a r

ry a double index

i =(kj).

T he spee d inde x A; is 0 for

re s t pa r t i c l e s a nd 1 fo r mov ing pa r t i c l e s . Th e inde x

j

varies f rom 1 to 6 and in dica te s the d i rec t io n of the ve

loc i ty vec to r . Veloc i t ies a re not ed

c

kj

-

or c , , where , in the

la t te r c ase , the s ingle index / impl ies the tw o indice s.

L ike w ise , pa r t i c l e B oo le a n va r i a b l e s a re no te d

n

kj

o r

n

i9

e t c .

T h e c

l y

a re the s ix uni t vec tors para l le l to the axes

on the he xa gona l l a t t i c e , a nd c

0 1

= 0.

The possib le configura t ions for 7 -bi t models a re shown

in Fig . 11 . In th is f igure , we descr ibe the configura t ions

only up to a ro t a t io n or re flec t ion . I t is seen th a t t her e

are two subgroups of c lass (3 ,0 ,0) , which we ca l l (3 ,0 ,0)^

a nd (3 ,0 ,0 )

5

, c on ta in ing tw o o r t h re e c on f igu ra t ions

w h ic h c a n be t r a ns fo rme d in to e a c h o the r by ro t a t i ons .

T h e

collision-saturated seven-velocity mo del

(also

know n a s FH P- I I I ) ha s t he fo l low ing c o l l i s ion ru l e s .

C onf igu ra t ions a re t r a ns fo rm e d in to a ny o f t he

other

configura t ions of the same c lass

(n,g*,g*).

H ow e ve r , i n

some cases, such as c lass (3 ,0 ,0) , the col l i s ion output i s

c hose n to be a no the r me mbe r o f t he sa me subc la ss , e i t he r

(3,0 ,0)^

o r ( 3 ,0,0 )^ . T he re a re a t mos t / ? = 3 me m be rs o f a

c lass or subc lass in th is schem e. Th us there a re a t mo st

tw o ou tpu t s f rom w h ic h to c hoose . The c ho ic e i s

achieved wi th a random bi t as descr ibed in Sec . I I .B .

Another model tha t wi l l be useful in what fo l lows is

th e

random-co llision seven-velocity m odel.

A configura

t ion in a c lass

(n,g*

f

g* )

i s t r a ns fo rm e d in to a ny

configu ra t ion in the sam e c lass , inc lud ing the or ig ina l

o n e .

The re a re a t mos t

p = 5

c on f igu ra t ions f rom w h ic h

to choose (Fig . 11) . The col l i s ion ra te for s ta te

s

going

in to s t a t e

s'

is defined to be

A(s,s')=l/p.

M o r e t h a n

one ra n dom b i t is now ne e de d . In p ra c t i c e a r a nd om

num be r ge n e ra to r i s u se d to c hoose f rom the p

c on f igu ra t ions .

3. A three-dimensional m odel: The face-centered

hypercubic lattice

Thre e -d ime ns iona l l a t t i c e -ga s a u toma ta w e re f i r s t i n

t r o d u c e d b y d ' H u m i e r e s

et al.

(1 986 ) . Th re e -d im e ns ion a l

mode l s a re c ons t ruc t e d on the f a c e -c e n te re d hype rc ub ic

(fchc) la t t ice , a genera l iza t ion to 4D of the face-

cente red-cubic la t t ice ( fee) . The fee la t t ice i s inadequate ,

as a re a l l o ther 3D Brava is la t t ices, because i t fa i l s to en

su re t he symme t ry o f fou r th -o rde r t e n so r s suc h a s

h.

a

p

r

$

defined in Sec . I I . Th e fchc la t t ice is gen era te d by the se t

o f 24 ve loc i ty ve c to r s c , of t he fo rm ( ± 1 , ± 1 ,0 ,0 ) t o ge th

e r w i th a l l pe rm u ta t io ns o f t he fou r c ompo ne n t s . I t is

a lso the se t of poin ts x = (a,b,c,d) w i th i n t e ge r c oo rd i

na t e s a nd a -hb -he -hd e ve n . The " v i sua l i z a t ion" o f

such a la t t ice is d i f ficult , i f not imp ossib le . Ho wev er , a

good grasp of the na ture of the fchc la t t ice may be ob

ta ined from an ana logy wi th s taggered la t t ices in 2D or

3 D . In par t icu la r , the face-cen te red-cu bic fee la t t ice is

made of a l l the points

x

—

(a,b,c)

w i t h

a -hb -he

even.

No t ice in Fig . 12 tha t the fee la t t ice i s not th e ent i re cu

bic la t t ice wi th a l l poin ts of in teger coordina tes

(a,b,c),

but jus t ha l f of i t . In a s imi la r way , the fchc la t t ice form s

a s t a gge re d subse t o f t he hype rc ub ic l a t t i c e , j u s t a s t he

fee la t t ice i s ha l f of the cubic la t t ice . When projec ted

o n t o t h e 3 D h y p e r p l a n e

d =0,

the ve loc i ty vec tors in the

fchc la t t ice fa ll in the tw o se ts dep ic ted in Fig . 13: 12 d i

agonal vec tors l ie in the p lane

d

= 0 , w h i l e t he 1 2 o the r

vec tors fa l l on the Cartesian axes wi th d = ± 1. It is i n

te rest ing to note tha t the vec tors c , a re a lso ver t ices of

the four-d imensiona l poly tope def ined by the Schlaf i sym

bo l { 3 , 4 , 3 ] . (See Ap pen dix A for a def in i t ion of the

Schlaf i symbol .) A projec t ion of the {3 ,4 ,3 } poly tope is

shown in Fig . 14 .

A l though the f c hc c o l l i s ions mus t be pe r fo rme d in 4D ,

s imu la t ions o f 3D f low ma y be pe r fo rme d on l a t t i c e s t ha t

a re only a few layers wide in the fourth d im ensio n. Be

c a use o f t he s t a gge re d na tu re o f t he l a t t i c e , t he t h inne s t

possib le s lab is two la t t ice spac ings wide in the fourth d i

me ns ion . A l th oug h suc h s l a bs a re c om mo nly u se d , B r i to

FIG. 12. Face-centered-cubic lattice. The solid circles belong

to the face-centered lattice and correspond to coordinates

(a

y

b,c) of even sum. The open circles have integer coordinates

with odd sum . This lattice is the analog in three dimensions of

the fchc lattice. A 2D layer of the lattice contains points of

coordinates c— 0 or 1. Similarly, a 3D layer of the fchc lattice

spans two values of the coordinate in the fourth dimension.


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1432 D. H. Rothma n and S. Zale ski: Lattice-gas mod els of phase separation

FIG. 13. A perspective view of the fchc primitive cell, project

ed into 3 D space. Instead of explicitly showing all 24 velocities,

only two of the 12 velocities which extend into the fourth di

mension are shown, along with just one of the velocities with no

component in the fourth dimension.

and Ernst (1991b) pointed out tha t excess corre la t ions

may appear in such th in s labs.

Fina l ly , we note tha t the def in i t ion of a col l i s ion opera

tor for the fchc la t t ice wi th 24 ve loc i t ies requires the

spec if ica t ion of the possib ly ran do m ou tpu t for each of

2

2 4

poss ib l e c on f igu ra t ions . M a ny p rop osa l s , some o f

w h ic h a re r e v i e w e d in Se c . IX .A .2 , ha ve be e n ma de fo r

the def in i t ion of such opera tors and the a lgori thms to ca l

c u l a t e t he m on c ompu te r s .

B. A derivation of hydrodynamics

from the Boltzmann equation

In t h i s se c t ion w e de r ive t he Eu le r e qua t ions a nd the

Navier-Stokes equa t ions for the s imple la t t ice gas wi th a

single mass and a t most one rest par t ic le .

FIG. 14. Projection of the polytope of Schlafi symbol (3,4,3}

(Coxeter, 1977). Vertices are shown as small circles. As indi

cated by the Schlafi symbol, each face has three edges. The

eight edges attached to each vertex join it to a cube. This cube

is the E t P polytope con nected to each vertex and its Schlafi

symbol {4,3}. A definition of Et P and the Schlafi symbol may

be found in Appendix A.


1. The microdynamjcal equation

We consider a s ingle - or mul t ip le -speed mo del . Th e

ve loc i t ies , equa l to the d istance be tween nearest ne igh

bors in the single-speed case, will sti l l be denoted by c,-.

Th e index / ma y denote a mu l t ip le index in the m ul t ip le -

ve loc i ty case . We denote by b the number of par t ic les of

a l l ve loc i t ies . Th e loca l configura t ion w i l l be a Boolean

6-vec tor

n(x,t)

depe ndin g on space and t im e. I n the col

l i s ion step of the dynamics, the configura t ion n on a

given si te i s t ransformed in to a postcol l i s ion

configu ra t ion n ' . Th e configu ra t ion n ' i s se lec ted ran

domly a mong a l l c on f igu ra t ions ha v ing the sa me va lue o f

the i nva r i a n t s w i th p roba b i l i t y o r t r a ns i t i on r a t e

A ( n , n ' ) .

I t i s useful to express th is s imple a lgori thm in te rms of

a se que nc e o f B oo le a n c a l c u l a t i ons , suc h a s t he mic ro -

dyna m ic a l e qua t ion o f Se c . I I .B . Fo r t h i s pu rpose w e

define a f ie ld of " ra te b i ts ," denoted by a

ss

>, whic h a re

equa l to 1 wi th pro babi l i t y A(s>s'):

(a

ss

,) = A(s,s

f

) .

Ra te b i ts should y ie ld a s ingle output for any input s ta te s

a nd spa c e - t ime loc a t ion x , t. Th is i s expressed by

J

t

a

ss

.(x

9

t)=l . (4.2)

s'

The n the mic rodyna mic a l e qua t ion ma y be ge ne ra l i z e d to

read

rijix + Cjyt + 1)

= n

i

{x,t) + *2ta

ss

'(x,t)(sj

—

s

i

)

J J

nj(x,t)

J

h~j(x,t)

j

,

s,s' j

(4.3)

w he re w e u se a ga in t he no ta t ion

x = l~x.

In the above

e qua t ion the B oo le a n p roduc t P

=

Hj nj(x

9

t)

j

rij(x,t)

j

is

a genera l iza t ion of the products g iven in Eqs. (2 .3) and

(2.4).

To establ ish the equiva len ce of Eq. (4 .3) and the

ra ndom a lgo r i t hm, i t i s u se fu l t o r e ma rk tha t P is in fact

a logica l opera tor tha t tests the equa l i ty of the Boolean

ve c to r s s and n . Th us the r igh t-ha nd side of (4 .3) rea l ly

t ransforms n in to the postcol l i s ion configura t ion n ' .

2. The lattice-Boltzmann equation

In B o l t z ma nn ' s mo le c u la r -c ha os a ssumpt ion , pa r t i c l e s

ente r ing a col l i s ion a re not corre la ted before they col l ide .

Fo r a ny c ombina t ion o f pa r t i c l e s a,b, . . . , x ente r in g a

col l i s ion , one assumes

<

n

a

n

b

• • •

n

x

> = <

n

a

> <

n

b

> • • • <

n

x

> . (4.4)

The ra te of col l i s ion can then be de te rmined by averaging

Eq. (4 .3) . The resul t ing lattice-Boltzman n equation has

the fo rm

N

(

(x

+

c

iy

t

+ \

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