Boris D. Lubaehevsky and Frank H. Stillinger- Geometric Properties of Random Disk Packings

8/3/2019 Boris D. Lubaehevsky and Frank H. Stillinger- Geometric Properties of Random Disk Packings

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Journal o f Statistica l Physics, VoL 60, Nos. 5/6, 1990

G e o m e t r i c P r o p e r t i e s o f R a n d o m D i s k P a c k i n g sB o r i s D . L u b a e h e v s k y 1 a n d F r a n k H . S t i ll in g e r 1

Received January 9, 1990; fin al M arch 27, 1990Random packings of N~


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562 L u b ach evsky an d St i l l in g era r r an g e m e n t s o f p a rt ic l es a d s o r b e d o n s m o o t h su r fa c e s , (6 '2 6) a n d t h e y m a yprov id e a he lp fu l wa y to in te rp re t expe r imen ta l o bse rva t ion s o f m ono laye rco l lo ida l suspens ions conf ined be tween na r rowly spaced g la s s p la te s /z~Unde r r ap id l a te ra l compres s ion o r i r r eve r s ib le adsorp t ion the se two-d i m e n s i o n a l s y s t e m s c a n a c h i e v e s t a t e s a p p r o x i m a t e d b y r a n d o m d i s kpackings . Bu t in spi te of the long scienti fic h is tory o f such con nec t ion s an da p p l i c a t i o n s , m a n y f u n d a m e n t a l q u e s t i o n s r e m a i n a b o u t t h e g e o m e t r i cna tu re of r ig id sphere an d disk packings . (25~

In the p re sen t pape r , w e s tudy som e nove l a spec t s o f r and om d iskpack ings . Th is p ro jec t ha s been fac i l i t a ted by recen t advances in p rogram-m ing techniqu es for even t-dr iven s imula t ions . (18~ The ap pro ach used isread i ly app l icab le to r andom sphe re pack ings in th ree d imens ions a s we l l ,bu t tha t i s r e se rved fo r l a te r s tudy .

T h e e a rl ie s t e x a m i n a t i o n s o f r a n d o m d i sk a n d s p h e re p a c k i n g sinvo lved o f ten c leve r , bu t imprec i se and unsys tema t ic , mechan ica l ana logsim ulation s. (5'19'24'27) M o re recently, co ns tru ctio n algo rithm s for ra nd ompacking s u ti lizing digi ta l com pu ters h ave b een im plemen ted. (4'1~ W hile thela t te r a re prec ise and sys temat ic , they use an in t r ins ica l ly s e qu en t i a l m o d e lin which sphe re s a re added one by one to an in i t i a l s eed s t ruc tu re and thef ina l aggrega te , tho ug h pos s ib ly la rge , possesses a f ree surface . T he p ackin gmodel inves t iga ted here in is in t r ins ica l ly c o n c u r r e n t and involves no f reesurface ; i t i s consequent ly c loser in spir i t to the quenching or compres-s iona l p rocedure s tha t a re norma l ly used expe r imen ta l ly to c rea teamorphous sol ids . At leas t for the r ig id d isk case cons idered a t lengthbe low, s eve ra l unu sua l p rop e r t i e s o f the r ando m pack ings gene ra ted havebeen iden t i f i ed , which we suspec t would no t r e ad i ly appea r w i th a s equen-t ia l cons t ruc t ion mode l . On e o f the se p rope r t i e s i s the p re sence o f" ra t tl e r s ," d i sks on ly loose ly imp r i soned by a r ing o f s even o r mo re t igh t lyjam m ed ne ighbors . Th is poss ib i l i ty had been p rev ious ly p os tu la ted , (29) bu t ,to the be s t o f ou r know ledge , had n o t been o bse rved d i rec tly p r io r to th iss tudy .S e c t i o n 2 p r o v i d e s s o m e b a s i c b a c k g r o u n d i n f o r m a t i o n a b o u t t h eaccess ib le con f igura t ion space for r ig id d isks and spheres. This is fo l low edby a de sc r ip t ion in Sec t ion 3 o f our dense pack ing mode l . Sec t ion 4pre sen ts the r e su lt s o f expe r imen t ing wi th ou r m ode l . Sec t ion 5 p rov idessom e d i scuss ion . Tw o append ice s p rov id e an ou t l ine o f the a lgor i thm ~18)used to r ea l iz e our mode l computa t iona l ly .

2 . S O M E B A S I C C O N C E P T SI n it s m o s t g e n e r a l v e r si o n , o u r i n q u i r y w o u l d c o n c e r n n o n o v e r l a p p i n g

a r r a n g e m e n t s o f D - d i m e n s i o n a l s p h e re s , D = 1 , 2 , 3 ..... c o n f i n e d t o a


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R a n d o m D i s k P a c k i n g s 563

" re c t a ng u la r" reg io n /2D o f siz e L 1 x L 2 x - . . X L D . W e s h a l l s u p p o s e t h a tp e r i o d i c b o u n d a r y c o n d i t i o n s a p p l y ; t h a t i s , /2 D a n d i t s c o n t e n t s a r epe r iod i ca l ly rep l i c a t ed i n a l l d i rec t i ons t o f i l l Euc l idean D -space .I f i t s d i am e te r i s a , t he n the co n te n t o f a D -d im ens ion a l sp he re is (14)

sD (a) = 7ZD/2aD/2DF(1 + D /2 )C o n s e q u e n t l y , if N n o n o v e r l a p p i n g s u c h s p h e re s i n h a b i t t h e i n t e r io r o f / 2 D ,t h e f r a c t i o n { o f t h a t r e g i o n c o v e r e d b y t h o s e s p h e r e s is

= N s o ( a ) / / 2 9I n t h e l im i t o f l a r g e /2 D , e x p l ic i t t i g h t u p p e r b o u n d s a r e k n o w n f o r 1 a n d2 d im ens io ns an d s t ron g ly co n jec tu red in 3 d im ens io ns (23'25)

rc~ < 2 ~ ( D = 2 )~ < - - ( D = 3 )

T h e b o u n d s a r e a t t a i n e d , r e s p e c t iv e l y , w i t h t h e g a p l e s s li n e a r a r r a y ( D = 1 ),t he t r i a ng u la r l a t t ic e (D = 2 ), an d the face -cen te red cub ic a r r ay (D = 3 ).

L e t t h e c e n t e r p o s i t i o n s o f t h e N s p h e r e s i n / 2 D b e d e n o t e d b y t h e s e to f v e c t o r s r l . . . r N - - R . T h e n o n o v e r l a p c o n d i t i o n o n a ll p a i r s o f D - s p h e r e sa n d t h e i r p e r i o d i c i m a g e s , w h e n t h e d i a m e t e r s a r e a , r e q u i r e s t h e D N -d i m e n s i o n a l c o n f i g u r a t io n v e c to r R t o b e l o n g t o s o m e s u b s e t ~ ( a ) o f t h es e t o f a l l p o s s i b le c e n t e r p o s i t i o n s , / 2 v x / 2 0 x . . . x / 2 D ~ 6 ( 0 ) . W e s h a l lr e f e r t o ~ ( a ) a s t h e " a v a i l a b l e c o n f i g u r a t i o n s p a c e " f o r t h e N n o n -o v e r l a p p i n g D - s p h e r e s i n r e g i o n / 2 D '

I t is o b v i o u s t h a t i n c r e a s i n g a d e c r e as e s t h e p o s i t i o n a l " f r e e d o m " o ft h e N s p h e re s , i.e ., t h e c o n t e n t C [ ~ ( a ) ] o f t h e a v a i l a b le c o n f i g u r a t i o ns p a ce i s a c o n t i n u o u s n o n i n c r e a s i n g f u n c t i o n o f a . I n d e e d , b e y o n d s o m eam ~x ( g e n e ra l ly d e p e n d e n t o n D , N , a n d t h e sh a p e o f / 2 D ) C [ ~ ( a ) ] w illv a n i s h :

C [ ~ ( a ) ] > 0 ( 0 -N


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5 6 4 L u b a e h e v s k y a n d S t i l li n g e r~(a) undergoes a dimensional reduction as a increases to amax. For all

a< areax the dimension is DN; otherwise C[~(a)] could not be positive.The dimension of ~(amax) is necessarily less than DN; this is a mathemati-cal consequence of close packing. The specific value of the reduced dimen-sion depends on D, N, and the shape of OD, but the presence of periodicboundary conditions and the implied free translation of any sphere packingassures that the reduced dimension must be at least D.

The connectivity of ~(a) and its variation with a 1, ~(a) will be connected, provided a is sufficiently small, for then anynonoverlap configuration could be continuously deformed into any otherone while easily avoiding overlaps. However, as a increases for D > 1 oneexpects ~(a) to undergo a sequence of disconnections as the nonoverlapconditions impose more and more severe constraints on available sphererearrangements. The presumption is that ~(a) should shed disconnectingportions that correspond to various nearly jammed and configurationallytrapped sphere packings, the majority of which for large N have irregularstructures. Each such disconnected portion would undergo its own dimen-sional reduction as increasing a caused it to reach its jamming limit, say atap for the packing denoted by p. Obviously

ama x = max (ap)P

Any jammed packing p belongs to a set of (N- 1)! equivalent packingsthat differ only by permutation of spheres (recall that free translations per-mitted by periodic boundary conditions automatically permit groups of Npermutations to be freely accessed from one another, even at jamming).Our primary interest, however, concerns geometrically distinguishablepackings that are not permutation-equivalent. When D > 1, the number Mof such distinguishable packings can be expected to rise exponentiallywith N, (29)

lnM(D,N, 12D)=tcN+o(N) , x>OThe objective of most random-sphere-packing inquiries concerns averagevalues of selected properties for the collection of packings, subject to someappropriate weighting. Let q be an index for the M geometrically dis-tinguishable packings, and let w(q) be a set of normalized weights:

Z w(q)= 1q


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R a n d o m Disk Packings 56 5

Then, if F ( q ) stands for the value of some property defined for thepackings, its mean value is

(F 5 = S w(q) F(q)q

In particular, mean values of powers of the jamming diameters ((aq) n) a r edefined this way, and the mean covering fraction for the random packingsbecomes

( ~ ) = [ Nz cD/2 /ZDF ( 1 + 0/2) D] ((aq) D ) (2.1)The set of weights w ( q ) will depend on the method of preparation of theD-sphere packings. The next section provides a specific family of suchmethods.

3 . M O D E LOur generation procedure begins by placing the desired number N of

points randomly and with a uniform distribution within the periodic unitcell g2 o. The N points are assigned initial velocities whose components areindependently distributed at random between - 1 and + 1. In the absenceof subsequent collisions each of these N points would continue to move atits initial velocity along a straight line that threads through an infinitesequence of image cells.

At the outset, time t = 0, the points are infinitesimal. However, theybegin to grow at a common rate into elastic D-spheres with diametersgiven for t ~> 0 by some function a ( t ) . We require that a(0) = 0 and that a ( t )be cont inuous nondecreasing function with a ( t ) ~ + oo for t ~ + oo. As aresult of the particle "growth," collisions become possible for positivetimes, and will increase in frequency as a ( t ) increases.

The intention is to sample initial configurations and velocities statisti-cally by generating many starting points and using a common a ( t ) . Wepermit time to progress in any realization until the system jams up, atwhich point the collision rate in principle must diverge. The final packingachieved obviously depends on the specific combination of initial con-figuration of the N points, their initial velocities, and the nature of thetime-dependent collision diameter a ( t ) . After averaging over the initialconditions, the weights w ( q ) with which the distinguishable packings q aresampled still depend on a ( t ) .

The conventional collision dynamics of elastic D-spheres with constantdiameters conserves kinetic energy. However, that is not the case whendiameters change with time. The collision dynamics now must be altered in


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5 6 6 L u b a c h e v s k y a n d S t i l li n g e ra w a y t h a t i s i n d i c a t e d f o r e a c h p a i r e n c o u n t e r b y F i g . 1. S u p p o s e p a r t i c l e sn u m b e r e d 1 a n d 2 f o r c o n v e n i e n c e h a v e r e s p e c t iv e v e lo c it ie s v l a n d v 2 j u s tb e f o r e c o l l i si o n . A s s h o w n i n F i g . 1 , t h e s e v e l o c i t ie s c a n b e r e s o l v e d i n t oc o m p o n e n t s p a r a ll e l ( p ) a n d t r a n s v e r s e ( t ) t o t h e li ne o f c e n te r s :

v ~ = v ~ ) + v ~'~V2 ~---v (2 P ) Ar V ( t )

w h e r e v l P ) . v l ~ ) - - 0 a n d v l t ) . ( r 2 - r l ) = 0 , i = 1, 2. T h e t r a n s v e r s e v e lo c it yc o m p o n e n t s a r e u n c h a n g e d b y co ll is io n , w h e r e a s t he p a ra l le l c o m p o n e n t sa r e e x c h a n g e d a n d m o d i f i e d i n m a g n i t u d e b y a n a d d i t i v e h. I f v * a n d v *a r e t h e v e l o c i t ie s j u s t a f t e r i m p a c t ( o c c u r r i n g a t t ~), t h e n

V ~ = I -V ( 2 ) " 4- h U 1 2 3 + V ~ ) (3 .1)v ~ = [ v ~ p ~ + h u 2 ~ ] + v ~ 'w h e r e u12 i s t he u n i t v e c t o r :

u12 = ( r l - r2 ) / i r l - r2t = - 1 1 2 1 (3 .2)I f 2 h e x c ee d s t h e d i a m e t e r g r o w t h r a t e a ' ( t c ), t h e n c o l li s io n s o c c u r a td i s c r e t e i s o l a t e d t i m e s . F o r r e s u l ts r e p o r t e d i n S e c t i o n 4 w e t a k e

h = a ' ( t c )

T h e d i f f e r e n c e i n k i n e t i c e n e r g i e s f o r t h e p a i r a f t e r a n d b e f o r e t h e c o l l i s i o ni s p r o p o r t i o n a l t o

8 9 + I v * l 2 - I v , I 2 - I v 2 1 2 )- h t , ~ p > - , ~ p ) ~ + h z (3 .3)- - " ~ - - - [ " 2 I "i 11 2 1

V 2

Fig. 1. Pair collision dynam ics for grow ing /)-spheres. Velocities are resolved in to com -ponents parallel and transverse to the line o f centers. Equations (3.1) and (3.2) specify howthese components change up on impact.


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R a n d o m D i s k P a c k i ng s 5 6 7

T h e e x i s t e n c e o f t h e c o l l i s io n r e q u i r e s(v(P )-- v(P)~ 9 > 0"1 "2 ] U21

C o n s e q u e n t l y , t h e d i f f e re n c e i n E q . ( 3 .3 ) is s t r i c tl y p o s i ti v e , s i n c e h > 0 .H e n c e , t o t a l k i n e t i c e n e r g y i n t h e s y s t e m i n c r e a s e s w i t h e a c h c o l l i s i o n .

T he w or k de s c r ib e d be low is r e s t r i c t e d t o th e c a se D = 2 , i.e ., r i g idd i s k s i n t h e p l a n e . F u r t h e r m o r e , t h e d i a m e t e r g r o w t h r a t e i s t a k e n t o b ec o n s t a n t :

a(t)=aot ( a o > 0 ) ( 3 . 4 )s o t h a t j a m m i n g a l w a y s o c c u r s a t a f in i te t im e . T h e s a m p l i n g w e i g h ts w(q)f o r t h e j a m m e d d i s k p a c k i n g s d e p e n d o n t h e a0 ch o ic e . A s a g e n e ra l r u le( a t l e a s t f o r l a r g e N ) , j a m m i n g o c c u r s i n a n i r r e g u l a r s t r u c t u r e i f a o i s l a r g ec o m p a r e d t o t h e m e a n i n it ia l p a r ti c l e s p ee d . F o r v e r y s m a l l a o , h o w e v e r ,t h e m o r e e x t e n d e d c o l l i s i o n d y n a m i c s i n p r i n c i p l e p e r m i t s t h e s y s t e m t or e a r r a n g e i n t o a m o r e n e a r l y r e g u l a r c r y s t a l l i n e p a c k i n g . I n d e e d , t h e l i m i ta o --* 0 + s h o u l d r e s u l t i n a c h i e v i n g t h e p a c k i n g w i t h a re ax w i t h h i g hp r o b a b i l i t y . T h e s a m p l i n g w e i g h t s w(q) t h u s w i ll d e p e n d o n a o in s u c h a

) ~ ~ i r i i ~ i i ~ / i8 4 ~ ~~ , ....i i i L ! i!!!!i!iiiiiii!i!iiiiii!i!i!ii!iiiii!iliiiiiii!iii!!iii!iiiii!i!i!! !iii!li!ii!ii!i!ii!lliii!iiiii!i!!iiiif!i!!!iiiiiiii!iii!iii!lliFig. 2. An initial rando m configu ration of 2000 points.


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5 6 8 L u b a c h e v s k y a n d S t i l l i n g e rw a y t h a t t h e m e a n c o v e r e d a r e a f r a c t i o n ( ~ ) , E q . (2 .1 ), w i ll b e a m o n o -t o n i c a l l y in c r e a s i n g f u n c t i o n o f a o .

F o r t h e r e s u l t s r e p o r t e d i n t h e n e x t s e c t i o n ,ao = 3.2

T h i s v a l u e o f a o h a s b e e n c h o s e n b y t h e f o l l o w i n g p r o c e d u r e . F o r2000 d i sks , w e do ub led ao f i ve t im es s t a r t i ng w i th a o = 0 .1 : w i th a o -3 . 2 ,i r r e g u l a r p a c k i n g s r e s u l t e d f r o m a n y i n i t i a l r a n d o m c o n f i g u r a t i o n w e t r i e d ,w hi l e w i th a o= 0 . 1 , 0 .2 , 0 .4 , 0.8 , an d 1 . 6 , i r r egu la r pack ings w e re no tg e n e r a t e d o r w e r e g e n e r a t e d s e l d o m .

T h e s er ie s o f Fi g s. 2 - 5 r e p r e se n t s f o u r s n a p s h o t s o f 2 0 00 d i s k s e x p a n d -ing w i th speed a0 = 3 .2 ; t he se a re qua l i t a t i ve ly t yp i ca l o f t he sys t em evo lu -t i o n u n d e r t h e s e g r o w t h c o n d i t io n s . I n a l l o f o u r r i g id - d i s k p a c k i n g c a l c u la -t i o n s w e h a v e t a k e n t h e p r i m i t i v e c e l l 0 2 t o b e a s q u a r e (Lx = L y ) . T h i sc h o i c e e x cl u d e s th e o c c u r r e n c e o f a p e r f e c t t r i a n g u l a r l a t ti c e, t h e m a x i m u md e n s i t y p a c k i n g a r r a n g e m e n t f o r d is k s. H o w e v e r f o r s o m e c h o i ce s o f th ein t ege r N , spec i f i c a l l y t hose o f t he fo rm

N = / ' l l / 7 2

F i g . 3 . D e p i c t i o n o f 2 0 0 0 d i s k s a t t = 2 .9 5 76 a f t e r 2 x 10 4 p a i r w i s e c o l l i s io n s ( 2 0 i m p a c t s p e rd i s k ) . T h e c o v e r i n g f r a c t i o n i s ~ = 0.5 62 8. T h e s t a r t i n g c o n f i g u r a t i o n i s s h o w n i n F i g u r e 2 .


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R a n d o m D i s k P a c k in g s 5 6 9

Fig. 4. D epi ctio nof 200 0 disks at t = 3.3901 after 2 x 105 pairwise collisions (200 imp actsper disk). The covering fraction is ~=0.7394. T h is shows a continuation initiated inFigs. 2 an d 3.

w h e r e t h e r a t i o o f in t e g e r f a c t o r s n l a n d n 2 c l o se l y a p p r o x i m a t e s t h ei r r a t i o n a l v a l u e

n l / n 2 ~ _ 31/2/2a s l i g h tl y s t r a i n e d v e r s i o n o f t h e t r i a n g u l a r l a t t i c e c a n b e f i t t e d in t o f2 2 w i t hi ts p r i n c i p a l d i r e c t i o n s a l i g n e d w i t h t h e s i d e s o f g 2 2 . S i n c e o u r p r i m a r yi n t e r e s t c o n c e r n s i r r e g u l a r d i sk p a c k i n g s a n d o u r c h o i c e o f a o w o u l dd i s c r i m i n a t e a g a i n s t t h e r e g u l a r l a t t i c e a n y w a y , t h e s q u a r e s h a p e c h o s e nf o r ~ 2 i s r e a s o n a b l e .

C o l l i s io n r a t e s i n c r e as e w i t h o u t b o u n d a s t h e j a m m e d p a c k i n g l i m i t isa p p r o a c h e d . T h i s r e s u l t s , f i r s t , f r o m t h e d i m i n i s h i n g m e a n d i s t a n c e t h a tp a r t i c l e s m u s t t r a v e l b e t w e e n s u c c e s s i v e c o l l i s i o n s . S e c o n d , i t h a s b e e np o i n t e d o u t t h a t k i n e t i c e n e r g y ( h e n c e t h e p a r t i c l e m e a n s p e e d ) i n c r e a s e sf o r e l a s t i c c o l l i s i o n s w i t h g r o w i n g d i s k s . F o r p r a c t i c a l r e a s o n s w e s e t a l lp a r t ic l e v e l o c it ie s t o z e r o r e p e a t e d l y d u r i n g l a t e s t ag e s o f th e d y n a m i c s t oa l l e v i at e ( b u t n o t e l i m i n a t e ) t h e c o l l is i o n r a t e d i v e r g e n c e p r o b l e m . T h i s


10/23

5 7 0 L u b a c h e v s k y a n d S t i l li n g e r

Fig. 5. De piction of 2000 disks at t = 3.6733 after 2 x 10 6 pairw ise collisions (2000 impactsper disk). The covering fraction is ~ = 0.8681. Th is shows a continuation nitiated in Figs. 2-4.a c t i o n i s u t i li z e d o n l y a f t e r t h e d is k s h a v e b e c o m e s o n e a r l y j a m m e d t h a tv i r t u a l l y n o i n f lu e n c e o n t h e f i n a l p a c k i n g d i s t r i b u t i o n i s e x p e c t e d t o o c c u r.

N u m e r i ca l ly , w e i d e n ti f y t h e j a m m e d s ta t e a s h a v i n g o c c u r r e d w h e nt h e s e v e n s i g n i f ic a n t d i g i ts o f t h e d i s k d i a m e t e r s t ab i li z e d e s p it e c o n t i n u i n gc o l l i s i o n s . N o t e t h a t t h e c o m p u t a t i o n s a r e c a r r i e d o u t w i t h d o u b l e p r e c i -s ion , i.e ., w i th p rec i s ion a t l e a s t 10 -~ 4. W hi l e fu r the r l en g th y co m pu ta t i onw i t h e v e n h i g h e r p r e c i s i o n c o u l d r e d u c e t h i s r e m a i n i n g l o o s e n e s s s o m e -w h a t , w e b e lie v e t h a t n o t h i n g n e w w o u l d b e l e a r n e d t h e re b y .

A s a n a l te r n a t iv e t o o u r m e t h o d o f p r o d u c i n g j a m m e d p a c k i n g s , o n em i g h t h a v e u t il i ze d p a r ti c le s o f f ix e d si ze a n d r e d u c e d t h e s y s t e m a r e a. T w ov a r i a n t s w o u l d b e p o s s i b l e : ( a ) i m p e n e t r a b l e b o u n d a r i e s , a n d ( b ) p e r i o d i cb o u n d a r y c o n d i ti o n s . T h e f o r m e r h a s t h e u n d e s i ra b l e p r o p e r t y o f p r o d u c -i n g a n a n o m a l o u s b o u n d a r y r e g i o n w i t h p r o p e r t i e s t h a t p r e s u m a b l y d i f f e rf r o m t h o s e o f b u l k p a c k i n gs . T h e l a t te r r e q u ir e s d e c i si o n a b o u t m o m e n t u md i s c o n t i n u i t y to b e r e q u i r e d w h e n a p e r i o d i c ce ll b o u n d a r y i s c r o ss e d . B o t hv a r i a n t s m i g h t b e i m p l e m e n t e d w i t h p a r t i c l e a c c e l e r a t i o n s b e t w e e n c o l l i -s io n s i n t h e s p i r i t o f " c o n s t a n t p r e s su r e m o l e c u l a r d y n a m i c s " as i n t r o d u c e db y A n d e r s e n . ~2) O u r s i m p l e p a r t i c le g r o w t h m e t h o d a v o i d s t h e s e i ss u es .


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R a n d o m D i s k P a c k i n g s 571

4 . D I S K P A C K I N G R E S U L T SAs a s imple i n t roduc t i on t o t he complex behav ior wi th l a rge r numbers

of disks , w e fi rst cons ider the spec if ic case N = 27. W i th a sq uare pr imi t ivece l l and pe r iod i c bounda ry condi t i ons , on ly two fundamenta l l y d i s t i nc tfami lies of packing s occur . T hese a re i l lus t ra ted in Figs . 6 and 7 . Aside f romthe poss ib i l it y o f ove ral l fr ee t r ans l a t ion and ro t a t i o ns by i n t ege r mul t ip l e sof 90 ~ t he se t ypes o f pa t t e rns have been rep ea t ed ly gen e ra t ed f rom therandom in i t i a l cond i t i ons .

The conf igura t i on p re sen t ed i n F ig . 6 is comple t e ly j amm ed. Eve rydisk i s t ight ly cons t ra ined by a t l eas t three ne ighbors wi th which i t i s incon t ac t . N o con t inuou s rea r rangem ent o f t he pa t t e rn is poss ib l e wi thou tv io l a t i ng t he d i sk nonove r l ap condi t i on . Consequen t ly , t he l imi t i ng reducedd i m e n s i o n o f th e d i s c o n n e c te d m a n i f o ld f o r th is c o m p l e t e ly j a m m e dpacking i s 2 .

Th e d i sk pac king exh ibi ted in Fig . 7 possesses a di s tinc t ive charac-te ri s ti c , nam ely the p resence o f a " ra t t l e r" di sk . Twen ty-s ix of the di sks a ret igh t l y j am m ed aga ins t one ano the r , and can on ly t r ans la t e a s a ri g idwhole , t hanks t o t he pe r iod i c bounda ry condi t i ons . The 27 th d i sk , a s t hef igure c learly show s, i s f ree to m ov e W ithin a r ig id cage of e ight j am m edneighb ors . As a resul t, the l imi t ing red uced d ime nsion of th is packing 'sd i sconnec t ed mani fo ld i s 4 .

. . . . . . . . . . . S: 5 7

6 9

Fig. 6. Completely jammed random packing formed from 27 disks in a square domain, withperiodic boundary conditions. The covering fraction is r = 0.83007.


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5 7 2 k u b a c h e v s k y a n d S t i l l i n g e r

Fig. 7. R an do m packing for 27 disks showing the occurrence of an unjamm ed rattler.Periodic boun dary conditions app ly to the squa re doma in used. The covering fractionis ~ = 0.82974.

N o t i c e t h a t t h e j a m m e d s u b s e t o f 2 6 d is k s i n F i g. 7 a p p e a r s t o p o s s es sa r e f le c t i o n s y m m e t r y p a r a l l e l t o t h e d i a g o n a l s o f t h e s q u a r e p r i m i t i v e c e ll~ 2 . N o s u c h s y m m e t r y a pp l ie s t o F i g. 6.

W h i l e n u m e r i c a l e x p e r i m e n t s w i t h r e l a t i v e ly s m a l l s y s t e m s s u f fi ce t os h o w t h e e x i s te n c e o f r a tt le r s, o t h e r a t t r i b u t e s o f r a n d o m d i s k p a c k i n g sb e c o m e c l e a r o n l y f o r la r g e N . S o m e o f t h e s e a t t r i b u t e s a r e i ll u s t r a t e d i nF i g . 8 , t h e f i n a l c o n f i g u r a t i o n o f t h e 2 0 0 0 - d i sk e x p e r i m e n t w h o s e e a r l ys t ag e s w e r e s h o w n i n F i g s. 2 - 5 . T o a i d v i s u a l i z a t i o n , a l l d i s k s h a v e b e e ns h a d e d e x c e p t f o r f o u r r a t t l e r s .

T h e p r e d o m i n a n t t e x t u r e o f t h e c o n f i g u r a t i o n s h o w n i n F ig . 8 c a n b ed e s c r i b e d a s p o l y c r y s t a l l i n e . I n d i v i d u a l c r y s t a l l i n e d o m a i n s a r e s e p a r a t e db y g r a i n b o u n d a r i e s a c r o s s w h i c h c r y s t a l o r i e n t a t i o n c h a n g e s . T h e s eg r a i n b o u n d a r i e s e x h i b i t a h i g h i n c i d e n c e o f p e n t a g o n a l h o l e s , i.e., g a p ss u r r o u n d e d b y f iv e d i sk s . N o t e t h a t t h e f o u r r a t t l e r s a ll re s i d e w i t h i ng r a i n b o u n d a r i e s .

F i g u r e 8 a l s o r e v e a ls t h e p r e s e n c e o f l i n e a r s h e a r f r a c t u r e s t h a t r u na c r o s s c r y s t a l g r a i n s . T h e s e r u n p a r a l l e l t o t h e p r i m a r y c r y s t a l l o g r a p h i cd i r e c t i o n s o f t h e a f f e c t ed g r a in s . T h e y s e e m t o a p p e a r o n l y i n t h e fi n a ls ta g e s o f t h e n u m e r i c a l e x p e r i m e n t , p r o d u c i n g s e q u e n c e s o f v ir t u a ll y i d en t i-c a l q u a d r i l a t e r a l h o l e s . I t i s i m p o r t a n t t o r e a l i z e t h a t i n t r a g r a i n t r a n s l a -


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Ra n d om Disk Packings 573

tional long-range order is disrupted by these shear fracture lines, whereasorientational long-range order (directions of nearest-neighbor pairs) ispreserved. This observation relates to the existence of "hexatic" phases oftwo-dimensional systems with just this property: long-range order in theorientational, but not translational, degrees of freedom. (2~) However, theaccepted description of hexatic phases involves unbonded dislocations, notlinear shear fractures. (21~ Whether or not linear shear fractures could be asignificant structural feature in real hexatic systems must remain an openquestion for now.

A monovacancy is obvious near the center of Fig. 8. That only onearises in a 2000-disk packing suggests that they have low occurrenceprobability under the packing formation conditions used.In order to analyze more deeply the local geometric properties of therandom packings, we have found it instructive to classify disks accordingto their values of a dimensionless "jamming" parameter ?:

7 = S/3a (4.1)

Fig . 8 . Dep ict ion of 2000 d isks a t t = 3 .6840 after 21 x 106 pai rwise col l i s ions (21 ,000 imp actspe r d i sk ) . Th e cover ing f rac t ion i s ~ =0 .8732 . S ign if i cant ra t t l e r s a re n o t shaded . T h i s i s t hef ina l con f igu ra t ion in t he se r i es whose beg inn ing i s i n F ig . 2 . No t i ce t he mo nov aca ncy n ea r t hecen te r o f t he pa t t e rn , who se p rec u rso r was c l ea r i n F ig. 5 .


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5 7 4 L u b a c h e v s k y a n d S t i l l i n g e rH e r e S is th e s u m o f t h e d i s t a n c e s t o t h e t h r e e c l o s e st n e i g h b o r s o f t h e d i s ku n d e r c o n s i d e r a t io n , a n d a i s t h e f in a l c o m m o n v a l u e o f t h e d i sk s 'd i a m e t e rs . F i g u r e 9 r e p r o d u c e s t h e c o n f i g u r a t io n s h o w n i n F i g u r e 8 , b u tw i t h f o u r 7 r a n g e s v i s u a l l y d i s t in g u i s h e d . T h e f o u r r a t t l e r s c o n t i n u e t os t an d o u t w i th l a rge va lues o f 7 /> 10 1. Th e su rp r i s i ng resu l t i s t ha t a sub -s t a n t i a l f r a c t i o n o f t h e d i s k s s e e m t o e n j o y a s m a l l, b u t d i s t i n g u i s h a b l yp o s it iv e , m o t i o n a l f r e e d o m , w i t h j a m m i n g p a r a m e t e r v a l u es in t h e r a n g e10-7~


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Fig . 10 . A m a g n i f i e d f r a g m e n t n e a r t h e c e n t e r o f F ig . 9 ( it is m a r k e d w i t h a s q u a r e i n F i g . 9 ).T h e m a g n i f i c a t i o n c l e a r l y r e v e a l s a r a t t l e r .

O Y < 10 7 (1823 disks) Q 10-7 < 7 < 10-4 (86 disks)10-4 -< 7 < 10 -I (91 disks) @ 10 -I -< y (0 disks)

F i g . 1 1. A n o t h e r j a m m e d c o n f i g u r a t i o n o f 2 00 0 d i s k s s ta r t e d w i t h a d if f e r en t r a n d o mc o n f i g u r a t i o n . I t j a m m e d a t t = 3.7 01 3 a f t e r 2 3 x 1 06 p a i r w i s e c o l l i s io n s ( 2 3 , 0 0 0 i m p a c t s p e rd i s k ) . T h e c o v e r i n g f r a c t i o n i s r = 0 . 8 8 1 4 .

822/60/5-6-4


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576 Lubachevsky and St i l l inge rtion of one of these other numerical experiments, with 7 classification ofdisks. Our results have consistently shown polycrystalline textures, withfrequent linear shear fractures. The relatively scarce monovacancies andloose (7/> 10-1) rattlers may or may not occur in any particular instance,as would be expected statistically, but in any case their concentrationsremain quite low.

5 . D I S C U S S I O NThe principal conclusion to be drawn from the present work is thatour concurrent construction procedure produces random disk packings

with several surprising characteristics that are unlikely to emerge fromstandard sequential construction procedures. These characteristics includethe presence of rattlers, vacancies, and linear shear fractures. In particular,we have found that obvious rattler disks tend to concentrate at grain boun-daries. Although it is possible for a sequential method to produce packingswith grain boundaries (specifically by starting with an irregular seed clusterof a few disks), it seems unlikely that an isotropic distribution of suchboundaries would arise in an extended packing created this way.Several nontrivial extensions of our study deserve future consideration.One would involve mistures of two or more species of disks with anarbitrary symmetric matrix of collision distances a i j ( t ) for species i and j.This would be particularly fascinating in light of recent advocacy of two-disk-size models for quasicrystalline order in two dimensions, u~'3~) Anotherdirection for generalization would be to consider more elaborate particleshapes, such as hard ellipses,/3~ for which random packings will doubtlessdisplay interesting patterns of local particle orientational order.We have carried out a few exploratory calculations designed toobserve the effect of changing ao in Eq. (3.4) on packing properties. Asmentioned earlier, we expect that the slower the growth rate, the less defec-tive should the packings tend to be. Indeed, this is the trend observed. Themost irregular packings were produced with the largest growth rate,ao = 1000, and have covering fraction ~ approximately 0.85; the mostregular packings were achieved with the slowest growth rate, ao=0.001,and have covering r just over 0.90 (maximum = 0.9069). Further computa-tion would be required to yield precise results for the way that ao affectsthe mean and variance of rThe most obvious direction for extension of the present study istoward packing of rigid spheres in three dimensions. A natural question iswhether "rattlers" would continue to be found inhabiting cages ofneighbors comprising no fewer than 13 spheres. Some preliminary calcula-tions have been performed, which seem to answer that question in the


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R a n d o m D i s k P a c k in g s 577

a f f ir m a t iv e : n o t o n l y d o t h e r a t tl e r s a p p e a r i n r a n d o m s p h e re p a c k i n g s , b u tt h e y o c c u r a p p a r e n t l y w i t h h i g h e r p r o b a b i l i t y t h a n i n t h e c o r r e s p o n d i n gd i sk case .F i n a l l y , w e e x p re s s t h e h o p e t h a t t h e r e su l ts o f o u r n u m e r i c a le x p e r i m e n t s w i ll s ti m u l a t e r i g o r o u s m a t h e m a t i c a l a n a l y si s o f t h e p r o p e r t ie so f r a n d o m d i s k a n d s p h e r e p a c ki n g s.

A P P E N D I X A . A L G O R I T H MLe t s t a t e o f a sph ere i a t t im e t be t he ve c to r s i ( t ) = ( r i ( t ) , v~( t) ), w here

r~( t) is the cen t e r pos i t i on vec to r o f sphe re i, an d v i ( t ) i s t he ve loc i t y vec to r .Th e g lob a l s t a t e S ( t ) o f the sys t em a t t ime t is t he s e t s l ( t ) , s2( t) ,..., S N ( t ) .A " n a i v e " se r i a l a l g o r i t h m a d v a n c e s t h e g l o b a l s t a te S ( t ) f r o m e v e n t t oe v e n t , w h e r e a n e v e n t is e i t h e r a c o l li s io n o f t w o s p h e r e s o r a b o u n d a r ycros s in g by a cen t e r o f a sphere . A l l N s t a t es s~ (t ), s2 ( t) . . . . SN ( t ) a reex am ine d an d up da t ed a t t imes to ~< t l ~< t2 ~< .. , wh ere to is t he i n i t i a l i za t i ont i m e a n d t~ +~ is t h e n e a r e s t n e x t e v e n t t i m e s e e n a t t i m e t i. T h e n a i v es c h e m e i s i n e ff ic i e n t f o r la r g e N b e c a u s e ( a ) t h e s a m e e v e n t i s r e p e a t e d l ys c h e d u l e d a n o r d e r o f N t i m e s u n t i l i t o c c u r s , a n d ( b ) a t a t y p i c a l c y c le ,m o s t s p h e r e s a r e n o t p a r t i c i p a t i n g i n e v e n t s ; s t i l l , t h e y a r e e x a m i n e d b y t h ea l g o r i t h m a s p o t e n t i a l p a r t i c i p a n t s .

A s i d e f r o m p r o b l e m s ( a ) a n d ( b ) , t h e r e e x is ts t h e p r o b l e m ( c ) o ff i n d i n g a n i n e x p e n s i v e m e t h o d o f d e t e r m i n i n g t h e n e a r e s t c o l li s io n fo r ac h o s e n s p h e r e . A s t r a i g h t f o r w a r d m e t h o d i s t o c o m p a r e t h e c h o s e n s p h e r ew i t h a ll N - 1 o t h e rs . T h e s t a n d a r d i m p r o v e m e n t i n t hi s m e t h o d is t h ed i v i s io n o f t h e r e g i o n O D i n t o a n o r d e r o f N r e c t a n g u l a r s e c t o rs co(k),

(2D = U CO(k), CO k) n CO(l)= ~ , i f k r lk

O n l y s p h e r e s i n t h e n e i g h b o r i n g s e c t o r s h a v e t o b e c h e c k e d t o d e t e r m i n et h e i m m e d i a t e n e x t c o l l i s i o n . F o r e a c h s e c t o r r ( k ) , a m e m b e r s h i p l i s t o fs p h e r e s w h o s e c e n t e r s b e l o n g t o c o(k ) i s m a i n t a i n e d . T h e o v e r h e a d o f t h em e t h o d r e s u l t s f r o m e x a m i n i n g a d d i t i o n a l b o u n d a r y c r o s s i n g e v e n t s w h e ns p h e re s c h a n g e s e c to r m e m b e r s h i p . D e s p i t e o f t h e o v e r h e a d , t h e m e t h o dr e d u c e s t h e w o r k f r o m O ( N ) t o O ( 1 ) p e r o n e c o l l i s i o n s c h e d u l e d .

A n a t u r a l i d e a f o r i m p r o v e m e n t i n ( a ) a n d ( b ) is t o p o s t p o n ee x a m i n i n g a n d u p d a t i n g a s p h e re s t a te u n t il t h e e v e n t a c tu a l l y oc c u rs .I m p l e m e n t i n g t h i s id e a d o e s n o t a p p e a r a s e a s y as i t m i g h t s ee m . A s th es i m u l a t i o n p r o g r e s s e s , a s c h e d u l e d c o l l is i o n o f a g i v e n s p h e r e m a y r e q u i r er e s c h e d u l i n g . T h e n e e d f o r s u c h r e s c h e d u l i n g a n d t h e d e s i r e n o t t o l o s ei n f o r m a t i o n a b o u t a l r e a d y p l a n n e d c o l l is i o n s l e d i n re f. 1 t o a c o m p l i c a t e dd a t a s t r u c t u r e a n d u p d a t e s c h e m e c a l l ed " t i m e - t a b l e " i n re f. 9 . O b s e r v e


18/23

5 7 8 L u b a c h e v s k y a n d S t i l l i n g e rt h a t , w i t h a l l i ts i n e f fi c ie n c y , t h e n a i v e s c h e m e h a s a v e r y a t t r a c t i v e d o u b l e -b u f f e r in g d a t a s t r u c t u r e . T h e s t r u c t u r e c o n s i s t s o f o n l y t w o c o p i e s o f t h eg l o b a l s t a t e v e c t o r S , t h e o l d a n d t h e n e w , s o t h a t t h e n e w v e c t o r i sc o m p u t e d o n t h e b a s is o f t h e o l d o n e a n d , i n t u r n , b e c o m e s t h e o l d o n ed u r i n g t h e n e x t c y c l e .

W e u s e a d i f f e re n t a l g o r i t h m p r o p o s e d i n re f. 18 . T h e a t t r a c t i o n o f t h isa l g o r i t h m i s t h a t i t u t i l i z e s a s i m p l e a n d e a s y t o h a n d l e d o u b l e - b u f f e r i n gd a t a s t r u c t u r e , w h i le a v o i d i n g p r o b l e m s ( a ) a n d ( b) . P r o b l e m ( c ) is h a n -d l e d u s i n g t h e s t a n d a r d t e c h n i q u e o f s e ct o ri n g. I n m o s t c a s e s th e a l g o r i t h me x a m i n e s a n d p r o c e s se s o n l y t h e e v e n t s w h o s e p r o c e s s i n g is u n a v o i d a b l e ,e . g . , s p h e r e c o l l i s i o n s a n d b o u n d a r y c r o s s i n g s . S o m e t i m e s , l i k e t h e n a i v ea l g o r i t h m , i t a l s o p r o c e s s e s e v e n t s w h o s e e x a m i n i n g i s n o t n e c e s s a r y .H o w e v e r , t h e f r a c t i o n o f s u c h o v e r h e a d e v e n t s is s m a l l a n d d o e s n o t g r o ww i t h N , w h i l e th e s p e e d u p d u e t o s i m p l i c i ty o f d a t a h a n d l i n g is s u b s t a n t ia l .

T h e f o l l o w i n g is a n o u t l in e o f t h e a l g o r i t h m . T h e b a s i c d a t a u n i t i n th ea l g o r i t h m i s c a l l e d even t a n d h a s t h e f o l lo w i n g f o r m a t

event = ( t ime, s ta te , pa r tner )w h e r e t ime i s t h e t i m e t o w h i c h s ta te o f a s p h e r e c o r r e sp o n d s . N o t e t h a ts ta te i s t h e n e w s t a t e o f t h e s p h e r e immed ia te ly a f t e r the eve nt , e .g . , if thes p h e r e h a s e x p e r i e n c e d a c o l l i s i o n a t time, t h e v e l o c i t y - c o o r d i n a t e o f t h es ta te i s t he new ve loc i t y vec to r a f t e r t he co l l i s i on ; par tner ident i f i es theo t h e r s p h e r e , i f a n y , i n v o l v e d i n t h e e v e n t. I f t h e r e is n o p a r t n e r i n t h ee v e n t , t h e p r o g r a m a s s ig n s a s p e c ia l " n o - v a l u e " s y m b o l A t o t h e par tnerc o o r d i n a t e . I f t i m e = + o % t h e n t h e o t h e r t h r e e c o o r d i n a t e s i n t h e eventhave no va lue , i . e . , s ta te = type =p ar tn er = A .

A t a n y s ta g e o f s i m u l a t i6 n , t h e a l g o r i t h m m a i n t a i n s t w o e v e n t s fo re a c h s p h e r e : a n o l d , a l r e a d y p r o c e s s e d i n t h e p a s t e v e n t a n d a n e w , n e x ts c h e d u l e d e v e n t . T h i s i n f o r m a t i o n is s to r e d i n a r r a y e v e n t [ l : N , 1:2] ,w h e r e , a s b e f o re , N i s t h e n u m b e r o f s p h e r e s o f t h e s i m u l a t e d s y s te m . L e tu s a g r e e t o u n d e r s t a n d a r e f e r e n c e l i k e t ime[3 , 1 ] as the t ime c o o r d i n a t e o fe l e m e n t even t [3 , 1 ] o f t h i s a r r ay .

T w o a r r a y s , n e w [ l : N ] a n d o l d [ l : N ] , w i t h e l e m e n t s e q u a l t o 1 o r 2 ,a r e m a i n t a i n e d . F o r s p h e r e i , new[ i] i s t h e p o i n t e r t o t h e n e w e v e n t a n do l d [ i ] i s t h e p o i n t e r t o t h e o l d e v e n t . T h u s , t h e n e w e v e n t f o r s p h e r e i i ss t o r e d a t even t [ i , new[ i ] ] a n d t h e o l d e v e n t i s s t o r e d a t even t [ i , o ld[ i ] ] .W h e n n e w [ i ] is u p d a t e d , o l d [ i ] is u p d a t e d i m m e d i a t e l y a f t e rw a r d , s o t h a tt h e r e l a t i o n n e w [ i ] + o l d [ i ] = 3 r e m a i n s i n v a r i a n t .

F i g u r e 1 2 r e p r e s e n t s t h e a l g o r i t h m p s e u d o c o d e g i v e n i n r ef. 1 8. T h ea l g o r i t h m is f o r m u l a t e d i n t e r m s o f b a si c f u n c ti o n s in teract ion- t ime, jump,a n d advance ( h e re " b a s i c " m e a n s t h a t t h e a c t u a l c o m p u t a t i o n s o f t h e sef u n c t i o n s a r e n o t r e p r e s e n t e d ) . T h e f u n c t i o n s h a v e t h e f o l lo w i n g s e m a n t i c s .


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R a n d o m D i sk P a c k in g s 5 7 9

Given s t a t e 1 of sphere 1 at t i m e l and s t a t e 2 of sphere 2 at t i m e 2 , func-tion i n t e r a c t i o n _ t i m e computes the t i m e of the next potential interaction ofsphere 1 with sphere 2 while ignoring the presence of other sphere andboundaries:

t i m e * - - i n t e r a c t io n _ t i m e ( s t a t e l , t i m e l , s t a t e 2 , t i m e 2 ) (A.1)where time >>.m a x ( t i m e l , t i m e 2 ) . If i n t e r a c t i o n _ t i m e cannot find such finitet i m e , e.g., when the spheres are moving away from each other, we assumethat + Go is returned. In the actual program, i n t e r a c t i o n _ t i m e is representedas a subroutine; the computations of this subroutine are specified inAppendix B.

i n i t i a l l y c u r r e n t _ t i m e ~-- 0 and fo r i = 1 , 2 , . . . N :n e w [ i ] ~ -- 1 , o l d [ i ] ~ 2 , t i m e [ i , 1 ] + --- 0 , p a r m e r [ i , 1 ] + - - A ,s t a t e [ i , 1 ] e - - i n i t i a l st a t e o f s p h e r e i, e v e n t [ i , 2 ] ~ -- e v e n t [ i , l ]1 . whi le c u r r e n t _ t i m e < e n d _ t i m e d o (2 . c u r r e n t t i m e ~ -- m i n ti m e [ i , n e w [ i l l ;

- - l- P c lo se " /} / " e n d R < + o o c l o s e " /

} / " e n d w h i l e l o o p ' 7

F i g . 1 2. T h e s i m u l a t i o n a l g o r i t h m .


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5 8 0 L u b a c h e v s k y a n d S t i l li n g e rT h e c h a n g e o f t h e s p h e r e v e l o c it ie s a t t h e m o m e n t o f c o l li si o n , di s-

c u s s e d i n S e c t i o n 3 , i s r e p r e s e n t e d i n t h e a l g o r i t h m b y f u n c t i o n j u m p , a l soi m p l e m e n t e d a s a s u b r o u ti n e . G i v e n s t a t e 1 a n d s t a t e 2 of co l l i d ing spheres 1a n d 2 , s u b r o u t i n e j u m p r e t u r n s n e w _ s t a t e 1 a n d n e w _ s t a t e 2 o f t h e se s p h e r e si m m e d i a t e l y a f te r t h e i n t e ra c t i o n :

( n e w _ s t a t e l , n e w - s t a t e 2 ) * - - ju m p ( s t a te 1 , s t a t e 2 )F u n c t i o n s i n t e r ac t i on_ t i m e a n d j u m p , a s d e s c r i b e d , d e p e n d o n t w o s p h e r ea r g u m e n t s . T h e a l g o r i t h m p s e u d o c o d e i n F i g. 12 al so e m p l o y s o n e - a r g u -m e n t v a r i a n t s o f i n t e r ac t i on_ t i m e a n d j u m p t o e x p r e s s b o u n d a r y c r o s s i n g .T h u s , i f k i s a n i n d e x f o r t h e s e t o f K b o u n d a r i e s ( w h i c h i n c l u d e s t h ee x t e r i o r b o u n d a r i e s o f (2 D a s w e l l a s i n t e r i o r b o u n d a r i e s w h i c h d e f in e t h esec to r s ) , t hen s equences

t i m e ~ - i n t e r ac t i on_ t i m e ( s t a t e 1 , t i m e l , k ) (A .2 )a n d

ne w _s t a t e , ,- - jum p ( s t a te , k ) (A .3 )s h o w t h e i n v o c a t i o n f o r m a t s o f t h e s e o n e - s p h e r e f u n c t i o n s . I n ( A .2 ), t i m eis t h e n e a r e s t t i m e o f c r o s s i n g b o u n d a r y k . T h e j u m p i n (A .3 ) i s t he i den t i t ym a p w h e n k i s a n i n t e r n a l b o u n d a r y , b e c a u s e n e i t h e r p o s i t i o n n o r v e l o c i t yo f a s p h e r e e x p e r i e n c e s a j u m p a t t h e i n s t a n c e o f c r o s s i n g i n t e r n a lb o u n d a r y . W h e n k is a n e x t e r n a l b o u n d a r y , o n e o f t h e c o o r d i n a t e s o f t h epos i t i on v e c t o r e x p e r ie n c e s a j u m p : t h e s p h e r e d i s a p p e a r s a t a f a c e o f (2 Da n d i m m e d i a t e l y r e a p p e a r s a t t h e o p p o s i t e f ac e.

G i v e n s tateO o f a s p h e r e a t t imeO a n d a v a l u e t i m e l >~ t imeO , f u n c t i o nadv anc e c o m p u t e s s t a t e 1 t h is s p h e r e w o u l d h a v e a t t im e l i g n o r i n g p o s s i b lec o l l i s i o n s w i t h t h e o t h e r s p h e r e s o r b o u n d a r y c r o s s i n g s o n t h e i n t e r v a l[ t imeO , t im e l ] :

s ta te l *--ad van ce (s tateO, t imeO , t im e l )I n o u r p r o b l e m , a d v a n c e h a s t h e o b v i o u s f o r m o f a d v a n c e m e n t a l o n g as t r a igh t l i ne pa ra l l e l t o veloci tyO s t a r t i n g f r o m pos i t ionO a t t imeO.

I n F i g. 12, / " a n d " / m a r k t h e be g i nn i n g a n d t h e en d o f a c o m m e n t ,a n d t h e m i n i m u m o v e r a n e m p t y s e t o f v a l u es is a s s u m e d t o b e + ~ . T h ef o l lo w i n g s h o r t - h a n d n o t a t i o n s a r e u s e d :

P ~ = - i n t e r ac t i on_ t i m e ( s t a t e [ i , o l d [ i ] ] , t i m e [ i , o l d [ i ] ] ,s t a te [ j , o l d [ j ] ] , t im e [ j , o l d [ j ] ] )


21/23

Ran dom Disk Pack ings 581

w h er e 1 ~< i, j ~< N a n dQ ~k - interaction_ time(state[i, old [ i ] ] , time[i, old[i] ], k)

w h e r e 1


22/23

5 8 2 L u b a c h e v s k y a n d S t i l l in g e r

T h e l a s t b u t n o t t h e l e a s t c o m m e n t i n t h i s s e c t i o n c o n c e r n s t h e a c t u a lc o m p u t a t i o n a l s p e e d a ch i e v ed b y th is a l g o ri th m . O n a V A X 8 5 5 0 c o m p u t e rw h o s e a c c e p t e d v a lu e o f s p e e d is a b o u t 1 M F L O P ( m i ll io n fl o a ti n g -p o i n t i n s t r u c ti o n s p e r s e c o n d ; f o r c o m p a r i s o n , t h e a c c e p t e d v a l u e o f s p e e do f C R A Y -1 is m o r e t h an 2 0 M F L O P s ) , th e F O R T R A N - i m p l e m e n t e da l g o r i t h m i n F i g. 1 2 p r o c e s s e s 1 5 0 - 4 5 0 p a i r w i s e c o l l i s i o n s p e r s e c o n d ,d e p e n d i n g o n t h e d i s k d e n s i t y . P r o c e s s i n g i s f a s t e r , t h e h i g h e r t h e d e n s i t y .I t ta k e s le ss t h a n 2 h r o f C P U t i m e o f V A X 8 5 5 0 t o r e a c h t h e st a tep r e s e n t e d i n F i g . 5 s t a r t i n g f r o m t h e i n i t i a l c o n f i g u r a t i o n i n F i g . 2 .

A P P E N D I X B .

W e h a v e

w h e r e

A C O N C R E T E E X P R E S S I O N F O RA S S I G N M E N T ( A . 1 )

t i m e ~ t , + t

t , = m a x ( t i m e l , t im e 2 )a n d

r E - B - - (B 2 - A C ) t / 2 ] / At = + oo ot he rw ise , i .e ., i fw i t h

if ( B < ~ O o r A < O ) a n d B 2 - A C > ~ O( B > 0 a nd A / > 0 ) o r B 2 - A C < 0

A = l v l 2 - a o 2 , B = r . v - a o a ( t , ) , C = l r l 2 - a ( t , ) 2r = r2o - - r io , v = V 2 - - V I

r l o = r l + v l ( t , - - t i m e l ) , r 2 o = r 2 + v 2 ( t , - - t i m e 2 )T h e v a l u e t i s t h e l e a s t p o s i t i v e r e a l s o l u t i o n o f t h e e q u a t i o nA t 2 + 2 B t + C = O , w h i c h i s d e r i v e d f r o m I r + v t l 2 = [ a ( t , ) + a o t ] 2. T h el a t t e r r e p r e s e n t s t h e c o n d i t i o n t h a t t h e d i s t a n c e b e t w e e n t h e c e n t e r s o fs p h e r e s 1 a n d 2 e q u a l s t h e c u r r e n t d i a m e t e r .

R E F E R E N C E S1. B. J. A lde r and T. E. W ainwright, J. Chem. Phys . 31(2):459466 (1959).2. H. C. And ersen, J. Chem. Phys . 72:2384 (1980).3. C. J. Bashe et aL, IBM 's Early Computers (M IT Press, Camb ridge, M assachusetts, 1986).4. C. H . Benn ett, J. Appl . Phys . 43:2727 (1972).


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Rand om Disk Pack ings 583

5. J . D. Bernal , Proc. R. Soc. A 280:299 (1964).6 . F. P . Buff an d F . H. S t i l linger , J . Chem. Phys. 39:1911 (1963).7 . S . C h a p m a n a n d T . G . C o w l i n g , The Ma thema tical Theory of Non-Uniform Gases( C a m b r i d g e U n i v e r s i ty P r e ss , C a m b r i d g e , 1 95 3) , C h a p t e r s 5 a n d 6 .81 K. E. Davis , W. B. Russel , and W. J . Glantschnig , Science 245:507 (1989).9 . J . J. E r p e n b e c k a n d W . W . W o o d , M o l e c u l a r d y n a m i c s t e c h n i q u e s f o r h a r d - c o r e s y s t em s ,in Statistical Mechanics. P a r t B : Time-Dependent Processes, B. J . Be rne , ed . (P l enum,

N ew Y ork , 1977 ) .10. J. L. Finney, Nature 266:309 (1977), a nd referen ces there in .11. H. L. Frisch, Adv. Chem. Phys. 6:229 (1964).12 . P. H. Gaskel l , in Glassy Metals, Vol . I I , H. Beck and H.-J . G fin th ero dt , eds . (Sprin ger ,

Berl in , 1983), pp . 5-49 .13 . K . H uang and C . N . Y ang , Phys. Rev. 105:767 (1957).14 . M. G . K enda l l , A Course in the Geometry o fn Dimensions (H afne r , N ew Y ork , 1961 ) ,p. 36.15 . D. E. Knuth , A rt o f Computer Programming, Vol. 3, Sorting and Searching ( A d d i s o n -

Wesley , 1973).16 . T . D . Lee , K . H uang , and C . N . Y ang , Phys. Rev. 106:1135 (1957).17 . P. W. Leung, C. L. Henley , and G. V. Chester , Phys. Rev. B 39:446 (1989).18 . B . D . Lubachevsky , J . Comp. Phys. (1990).1 9 . G . M a s o n a n d W . C l a r k , Nature 207:512 (1965).2 0 . C . A . M u r r a y a n d D . H . v a n W i n k l e , Phys. Rev. A 34:562 (1986).21 . D . R . N e l son and B . J . H a lpe r in , Phys. Rev. B 19:2457 (1979).22 . R . P indak , D . J . B i shop , and W . O . Sp renge r , Phys. Rev. Lett. 44:1461 (1980).23 . C. A. Rogers , Pack ing and C overing (Cambr idge U n ive rs i t y Pre ss , Cambr idge , 1964 ) , p . 3 .24 . G. D. Scot t , Nature 194:957 (1962).25 . N. J . A. Sloane, Sci . Am. 250(1):116 (1984).26. W. A. Steele, J. Phys. Chem. 69:3446 (1965).27 . F . H . S t i l li nge r , E . A . D iM arz io , and R . L . K orn egay , J . Chem. Phys. 40:1564 (1964).28 . F . H . S t i ll i nge r an d Z . W . S a l sbu rg , J . Stat. Phys. 1:179 (1969).29 . F. H. S t i l l inger an d T. A. W eber , J . Chem. Phys. 83:4767 (1985).30 . J . Vie i l la rd-Baron , J. Chem. Phys. 56:4729 (1972).31 . M. W idom, K . J . S t randbu rg , and R . H . Sw endsen , Phys. Rev. Lett. 58:706 (1987).

Boris D. Lubaehevsky and Frank H. Stillinger- Geometric Properties of Random Disk Packings

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