representations of geometrical and topological quantities in cellular automata

Physica D 45 (1990) 271-277 North-Holland

R E P R E S E N T A T I O N S O F G E O M E T R I C A L A N D T O P O L O G I C A L Q U A N T I T I E S I N C E L L U L A R A U T O M A T A

Mark A. S M I T H Department of Physics, Massachusetts Institute of Technology, Laboratory for Computer Science, Cambridge, MA 02139, USA

Received 4 February 1990 Revised manuscript received 28 February 1990

This paper addresses the problem of finding novel representations of analogs of physical field variables in cellular automata. Several examples illustrate how this might be done: (1) a manifestly covariant vector law for diffusion, (2) a model for potential energy and exact one-forms, (3) a counting algorithm using a winding number density, and (4) fields which support topological charges. In order to combine the advantages of cellular automata and continuum analysis, it is necessary to have representations which involve more than one cell.

1. I n t r o d u c t i o n 2. V e c t o r s a n d g r o u p invar iances

Cel lu la r a u t o m a t a have proved useful for physical mode l ing [1]. Th is is because the i r phys ica l s t r u c t u r e and c o m p u t a t i o n a l power give t h e m the ab i l i t y to s imu la t e the complex , non l inea r behav- ior found in m a n y s p a t i a l l y e x t e n d e d sys tems . Ef- for ts are u n d e r way to ex t end the phys ica l mode l - ing power of ce l lu lar a u t o m a t a to ever more gen- eral domains . Toffoli [2] mo t iva t e s th is p r o g r a m wi th a r e p r e s e n t a t i o n of sca la r quan t i t i e s as well as a genera l d i scuss ion of how ce l lu lar a u t o m a t a can be viewed as an a l t e r n a t i v e m a t h e m a t i c a l s t ruc- tu re for physics .

The f u n d a m e n t a l laws of phys ics do not de- pend on a p a r t i c u l a r c o o r d i n a t e sy s t em nor on the choice of a basis. S imi lar ly , the p r o p e r t i e s of a ce l lu lar a u t o m a t a mode l of a field should not d e p e n d on the way the m o d e l is coo rd ina t i zed . The fol lowing sec t ions p resen t several e x a m p l e s of coo rd ina t e - f r ee ce l lu lar a u t o m a t a r ep r e sen t a t i ons of fields which occur in physics . T h e y were imple- m e n t e d us ing C A M - 6 [3], and the conf igura t ions below come from the r e su l t ing 256 × 256 images .

Th is sec t ion presen ts a re la t iv i s t i c diffusion m o d e l which o r i g ina t e d from a s t u d y of the sense in which ce l lu la r a u t o m a t a d i sp l a y Loventz invariance [4]. The m o d e l p rov ides a n a t u r a l r ep resen ta - t ion of two-d imens iona l ( spaee t ime) vector fields. Th i s mode l has in te res t ing p rope r t i e s and has a wide range of a pp l i c a t i ons [5]. Re la t iv i s t i c mode l s of diffusion have been deve loped before [6], and a s imi la r mode l has been used as an e x a m p l e of q u a n t u m a u t o m a t a [7].

T h e c o n t i n u u m mode l desc r ibes one-d imens ional p r o b a b i l i t y dens i t ies , p+(x , t) and p - ( x , t) , for f inding a pa r t i c l e moving in the pos i t ive and nega- t ive d i rec t ions respect ively . The l ight -cone coordi - na tes , x ± = (t ± x ) / x / 2 , will t u r n out to be useful. Cons ide r an ensemble of sys tems , each of which con ta ins a s ingle m a r k e d pa r t i c l e which bounces back and for th at uni t speed due to col l is ions wi th a b a c k g r o u n d of s imi la r pa r t i c l es as shown in fig. 1. T h e b a c k g r o u n d par t i c les give wel l -def ined mean free pa ths , A+(x +) and A _ ( x - ) , for the m a r k e d pa r t i c l e to reverse d i rec t ion . Th i s d y n a m i c s can be t hough t of as a r a n d o m walk wi th i ne r t i a or as the diffusion of a mass less par t i c le .

One can also define a conserved two-cur ren t , J~' = ( p , j ) , where p = p+ + p - and j = p+ - p - .

0167-2789/90/$ 03.50 (~ 1990 - Elsevier Science Publishers B.V. (North-Holland)

272 M.A. Smith / Geometrical and topological quantities in CA

× 1 I

Fig. 1. A spacetime diagram showing the paths of the particles in the relativistic diffusion model. A typical trajectory for a marked particle undergoing collisions with the background gas is emphasized.

The t r anspor t equat ions for the densities are then

Op + ap + p+ p- O--T + Oz ;~+ + ~_ ' (1)

Op- Op- p- p+ - - + - - ( 2 )

Ot Ox ~_ ~+

Adding these equat ions gives

Op oj 0 ~ + ~ = 0 ,

and sub t rac t ing them gives

~ / + ~ = - 3 ~ + +p ~+ •

(3)

(4)

The condi t ion tha t the background part icles s t ream at unit speed is

Ox + = 0 , and ~ = 0 . (5)

If one defines

1 1 1 1 c rt = - - + and c ~ - , (6)

~+ ~__' ~_ ~+

then eqs. (3)-(5) can be rewri t ten in manifest ly covariant form as

O,J" = 0, (7)

(o. + ~.)~'~Jv = 0, (s)

0,~" = 0, (9)

O,z'vcr~ = 0. (10)

The pa ramete r ~r~ is essentially the two-momen- t u m densi ty of the background particles and gives a propor t iona l cross section for collisions. Wri t ing the equat ions in this form proves tha t the model is Lorentz invariant .

This model is also conformally invariant. This is ment ioned because conformal invariance has pro- found consequences for physics. In 1+1 dimensions, conformal invariance means tha t the light- cone axes can be locally scaled by any amoun t (x + ~ f + ( x + ) ) , and the result ing evolution is still described by the original equations. The fact tha t the above model is conformal ly invariant is easy to see from fig. 1, since any change in the spacing of the diagonal lines yields a similar picture.

In the case of a uniform background with no net drift ()~+ = )~), eqs. (7)-(10) reduce to the te legrapher ' s equation, which has been s tudied ex- tensively [8]. For the initial condit ions j (x ,O) = p(x,O) = 5(x) the solution inside the light cone (i.e. the region t 2 - x 2 > 0) is

V + e - t / ~ ( t - x), (11)

p - ( x , t ) - ~ I0 . (12)

Outs ide the light cone, p+ = 0. The total density, p(x , t ) , for )~ = 32 is p lot ted in fig. 2. Note how the del ta funct ion decays exponent ia l ly as it moves along the x + axis and how the rest diffuses. Eqs. (11) and (12) can be used to find the solution for any other initial conditions. First , they can be reflected a round x = 0 to give the solution s ta r t ing from a lef t -moving pulse. Second, the l ight-cone axes can be scaled to change ~ ~ )~±(x +) corre- sponding to an a rb i t r a ry background gas. Finally, solutions s ta r t ing from different values of x can be

M,A. Smith / Geometrical and topological quantities in CA 273

0

I00

0.015

0.01

0.005

-i~o ~ iBo

"'%

Fig. 2. So lu t ion of the c o n t i n u u m equa t ions in the re la t iv i s -

t ic diffusion mode l s t a r t i n g f rom a loca l ized pulse mov ing to the r ight . The t -ax is runs from 0 to 128, and the x -ax i s runs from - 1 2 8 to 128.

Fig. 3. H i s t o g r a m of 17 500 pa r t i c l e s in the r e l a t iv i s t i c diffu-

sion ce l lu lar a u t o m a t a mode l for t = 128. Pa r t i c l e s execu te a r a n d o m walk in the upper ha l f and fall into one of the 256 bins in the lower half.

combined to give any p+(x, 0) and p (x, 0). Fig. 3 shows the result of a cellular au t om a t a

simulation with A ~- 32. In this case, a particle is restricted to a discrete set of paths, but for

)> 1, the continuum distribution of free path lengths (a decaying exponential) is well approx- imated by the actual discrete geometric progres- sion. The odd columns contain p - (x, 128), and the even columns contain p+(x, 128). The separation of these components is quite apparent in the fig- ure and is reflected by eqs. (11) and (12). The rightmost bin, p+(128,128), is full, so the delta function it represents has been truncated. There is a slight excess of particles in the neighboring bin, p+ (126, 128), because of spurious correlations in the noise source.

3. P o t e n t i a l s and o n e - f o r m s

One-forms are geometrical quantities very much like vectors. Like vectors, they are rigorously described in te rms of infinitesimals of space. Unlike vectors, they cannot always be pictured as arrows. A bet ter picture of a one-form is that of a sepa- rate contour map at each point. If the maps fit

together to make a complete contour map, then the one-form field is said to be exact. In this case, it can be writ ten as a gradient, dV. The scalar field of which it is the gradient is a potential, V. Potentials are impor tant in physics because they create forces; in this case, F ~- - d V .

Contours are unbroken lines (or hypersurfaces in higher dimensions) that do not cross and never end. They can be represented in any way which has these properties and consistently specifies which side of each contour is higher. The mag- nitude of the field is inversely proport ional to the "distance" between contours. The implementat ion used in fig. 4 encodes the potential very efficiently by digitizing it and then only storing the low- est bit. In this case, it is a harmonic potential, 1 2 2 ~mw r , in real space or alternatively, the classi- cal kinetic energy, p2/2m, in momen tum space. The entire potential can be recovered (up to a constant) by integration. Note that the overall slope of the potential is only discernable by look- ing at a relatively large region. This il lustrates the multiple-cell character of this representation.

An impor tant physical application of this representation is to make a reversible lattice gas spon- taneously become denser in a part icular region of


• ..::..'•::'.. :

Fig. 4. Contours representing the gradient of a harmonic potential well. The special encoding of the potential ac- counts for the width of the steps being nonmonotonic.

space by effect ively r e p r o d u c i n g an e x t e r n a l po- t en t i a l [9]. A n isolated s y s t e m con ta in ing a uni- fo rmly d i s t r i b u t e d l a t t i ce gas canno t revers ib ly se l f -organize in any way because i t is a l r e a d y in a s t a t e of m a x i m u m ent ropy. Th is would be tan- t a m o u n t to perfect compress ion of N b i t s of infor- m a t i o n into fewer t h a n N bi ts .

However , one can get a r o u n d th is r e s t r i c t i on by coupl ing the sy s t em to a hea t b a t h which serves to conserve the microscopic i n fo rma t ion of the ent i re sys t em whi le i ts macroscop ic en t ropy increases . The hea t b a t h consis ts of any revers ib le sys t em tha t has conserved tokens which can s t a n d for energy. The coupl ing mani fes t s i t se l f when a pa r t i c l e of the gas t r ies to cross a con tour of the po ten t i a l . The pa r t i c l e is a l lowed to go up (down) a con tour if the co r r e spond ing cell in the hea t b a t h is occu- p ied ( empty ) . In these cases, it m a y proceed by exchanging energy wi th the hea t ba th ; o therwise , it mus t bounce back.

A l a t t i ce gas in a p o t e n t i a l well d i s t r i b u t e s i t- self accord ing to the laws of s t a t i s t i c a l mechanics - it has a well defined ent ropy, t e m p e r a t u r e , free energy, chemica l po t en t i a l , etc. The force which causes it to fall into the well is en t i re ly s t a t i s t i - cal: the sy s t em mere ly wande r s freely and evenly t h r ough the space def ined by its conserved quan-

Fig. 5. The Fermi sea formed by a lattice gas in a harmonic potential well. This can be viewed as a spatial density or as a distribution in momentum space. The Fermi energy (compared to the contours above) is approximately 4, and the temperature is about 2.

t i t ies , bu t by far the bu lk of th is space lies where the dens i ty of the gas in the well is h igher . The resul t is shown in fig. 5.

A l a t t i ce gas has an exclusion pr inc ip le bui l t in because no two par t i c l es can be in the s ame s ta te ; therefore , t hey are cons ide red to be ]ermions, and they obey Fermi statistics. All m a t t e r is ul t i - m a t e l y m a d e out of fermions , and the exclus ion is phys ica l ly very i m p o r t a n t because i t keeps m a t t e r f rom col lapsing. Similar ly , the gas in fig. 5 cannot be compres sed beyond a b o u t the four th con- t ou r ( this l imi t is cal led the Fermi energy and is roughly the s ame as the chemical potential). The dens i ty falls f rom one to zero over a b o u t two contours ( th is is the temperature). A col lect ion of par- t ic les which follows such a Fermi distribution is cal led a Fermi sea. The Fermi d i s t r i b u t i o n is only a p p r o x i m a t e , s ince the sy s t em is not ful ly ergodic and has a f ini te size. Cor rec t ions to the d i s t r ibu - t ion have been ca lcu la t ed and m e a s u r e d and are desc r ibed e lsewhere [10].

4. W i n d i n g number

Winding number refers to any topo log ica l quan- t i t y which is given by the n u m b e r of comple t e t u r n s m a d e by m a p p i n g s be tween a circle and a

M.A. Smith / Geometrical and topological quantities in CA 275

plane. An archetype to keep in mind is provided by the function of a complex variable, w = f ( z ) = z ~. This maps the circle, z = r e io, 0 _< 0 < 2~r, around the point w -- 0 a total of n times. The terminol- ogy comes by analogy with the principle of the ar- gument in complex analysis, which gives the number of zeros minus the number of poles (of a mero- morphic function) surrounded by a contour. This number is the number of t imes the image of the contour winds around the origin.

4.1. Domain counting

A winding number can be associated with a set of domains in two dimensions (see fig. 6). It is the net number of turns made by following all the boundaries (taken with the same orientation) of all the domains. The orientation of a boundary is determined by whether the interior of a domain is to the right or to the left of the boundary as it is traversed. This number is the number of domains minus the number of holes in the domains and can be shown to be the same as the Euler number of the set. See ref. [11] for a more complete mathe- matical t rea tment of these topics.

,, - . - . • " ~ . ~ "~- ~ . , : . . . . a , . . . . - , ~ ' " ' , t h • . " | ,. ." ~ : ' . " ' ' . . _ _ •

, , . ~ " : ' ~ . . . " ' : - : - . '." " ' " " ~ . : e ' " " " " . , ' - '" ~-b . . . . . "~'~ : ~ . i / : . . . . . .. , ' . " . . . . . *~ : ." ~ ~ ~, ' . ~ . . . . ". . ~ . - , I

; . . : . ".,_.'.-" • , t ~ . . " • . ~ . ~ " - ~ , - , . ~ "~ ,

" ' " . ' - i ".. . - . . . . . :

, . . ~ . . . . . .. • ' . . . , . ,_ - . . . ~ P i q l . , - ' . • , , - ; . ~ I . . .,- '. .- , ".,, ~ * . .:'- . " "" . - . .

," ' . " " ' : , i i . , o " . : ,~ • .

i I ~ I q d " i ~ " , "..:" ~ I I ~ l * ~ l ~ l l I ' ~ d I ~ I I I , .~ . . . . . . . . . . - ~ . -

," . I ~ . i i ' ; ' " . ~ I , ' I ' : . ~ " I . : " , I~ ILP~I . " . ~ : i { ~ l i ~ . ~ l l ' ' ; ~ ' [ i ,

.~ " . , ~ " '~, '" , . '~ " ~J1~.'.,~'1~'1_ '-. n ~ r . , . . . , . ' : ,. " . : . . . " . " . : , . . . w - ~ i i . , ' : . : ' ~ . . .

• . : . . . . • . • . : . , .

Fig. 6. Magne t i c d o m a i n s (in black) t aken f rom an Is ing mode l wi th 64K spins. E ach black cell can be t h o u g h t of a s a par t ic le . T h e wind ing n u m b e r is 1357, which is some- wha t less t h a n t he ac tua l n u m b e r of doma i ns . T h e a v e r a g e

n u m b e r of par t ic les per d o m a i n is a p p r o x i m a t e l y 7.9.

In the discrete case (as for cellular au tomata) , the plane is tiled with cells. A connected group of occupied cells (or tiles) consti tutes a domain. I f the sides of the tiles are straight, the boundaries of the domains only turn where three or more cells meet at a point (the set of all such points form the vertex set of the lattice dual to the original lattice). The net number of turns is s imply the sum of these angles (xcel ls /360 °) taken over all the cells of the dual lattice. Hence it is possible to define a density whose integral gives the winding number. This connection between a topological number and a differential quanti ty is surprising and important .

The winding number density on the dual lattice is defined to be zero unless there is a boundary passing though the vertex. In this case, it is 180 ° minus the angle subtended by the cells constitut- ing the interior of the cluster. This is just the angle that the boundary bends through. On a square lattice, the winding number density is assigned as shown in fig. 7. Note that this quanti ty depends on correlations in a group of cells, and that the groups overlap.

The winding number density is related to certain many-body potentials in deterministic Ising models. These potentials have been used to bias the curvature of the phase boundary in a model of droplet growth [12]. The winding number then provides an est imate of the number of droplets. In the low-density limit, the domains become simply connected (no holes), and the correspondence between winding number and the number of domains becomes exact. In the case of droplet growth, it

(a) 0 ° (b) +90 ° (c) 0°+180 °

I ( d ) 0 ° ( e ) - 9 0 ° ( f ) 0 °

Fig. 7. W i n d i n g n u m b e r dens i ty (in degrees per cell) for a squa re lat t ice. R o t a t e d vers ions o f t h e s e blocks are equiv- a lent . Case (c) ha s two b o u n d a r y lines going t h r o u g h t h e

cente r and can be coun ted accord ing to h o w o n e w a n t s to define a d o m a i n .


can be used to find an est imate of the average droplet size as a function of time.

In order to find the average domain size, one needs to count the number of occurrences of cases (b), (c), and (e) in fig. 7 as well as the total number of occupied cells. Fig. 6 has 7380 90 ° angles, 2310 - 9 0 ° angles, 179 180 ° angles, and a total of 10 749 particles. The built-in counter in CAM-6 makes doing these counts over the 64K cells very fast. Finding and tracing out domains one by one requires a relatively slow, complicated procedure, whereas finding the average domain size on CAM- 6 only takes four steps or ~ of a second; hence, this is a practical method to use while doing real- t ime simulations.

4.2. Topological defects

Another form of winding number is i l lustrated by fig. 8. The cell states of this au tomaton have a definite cyclic order. A configuration is cont inuous if the states of adjacent cells differ by at most one with respect to this order. The winding number of a closed loop of cells is the net number of times the cell states go through the cycle as the loop

is traversed in a counterclockwise direction. Any loop with a nonzero winding number is said to contain a defect (see the illustration in ref. [13]). The defects have a topological character because the winding number of a loop cannot be changed by modifying the state of one cell at a t ime while still maintaining continuity. The centers of the spirals in fig. 8 are the topological defects.

Analogous topological objects are impor tan t in several disciplines. The mathemat ica l s tudy of these objects falls under the field of algebraic topology. Their physical relevance s tems from the fact that , depending on the dynamical laws and the overall topology of spacetime, most fields can form "knots" - continuity and energy conserva- tion forbid such configurations of fields from being untied. Two physical instances of topological charges which have been postulated to exist (but not definitely observed) are magnetic monopoles and cosmic strings. The dynamical effects of topological objects range from cardiac a r rhy thmias to extinction in certain p reda to r -p rey models.

Many cellular au toma ta models [3] have behav- ior in which "activity" in a cell spreads to nearby cells and is then suppressed. These include the game of life, neural models, p reda tor -prey models [14], and chemical systems [15]. These rules gener- ate spiral pat terns (in a generalized sense) similar to those in fig. 8. One model in part icular [16] has definite configurations which are the growth centers for defects. Again, these structures reflect organization over a relatively large region.

Fig. 8. Configuration from a cellular automata model of an oscillatory chemical reaction. The cells assume eight different values, and the values change gradually between neighboring cells. The centers of the spirals are topological defects.

5. Conc lus ions

The above examples show that it is possible to find representations of physical quantities in cellular au toma ta which respect the geometrical and topological foundations of physics. Some peo- ple have the misconception tha t cellular au tomata models of fields must contain discretized approx- imations of the variables in individual cells. But the representations given here are distr ibuted over several cells and need not be interpreted as aver- ages over those cells - they can also have an in- terpretat ion as a single correlated pat tern. These alternatives il lustrate how cellular au toma ta have a distinct character and should not be viewed as

M.A. Smith / Geometrical and topological quantities in CA 277

necessari ly inferior subs t i tu t e s for the con t inuum. This paper is a step in the development of a

catalog of techniques for physical model ing with cellular a u t o m a t a .

Besides yie lding pract ical a lgor i thms, bu i ld ing discrete models of physics will give us the oppor- t u n i t y to reevaluate our u n d e r s t a n d i n g of physics

and computa t ion .

A c k n o w l e d g e m e n t s

I would like to t h a n k Tommaso Toffoli and the In fo rma t ion Mechanics Group in the M I T Labora- tory for C o m p u t e r Science for insp i r ing this work.

Suppor t was provided in par t by the Nat iona l Science Founda t ion , G r a n t No. 8618002-IRI, and in par t by the Defense Advanced Research Proj-

ects Agency, Gran t No. N00014-89-J-1988.

R e f e r e n c e s

[1] T. Toffoli and N. Margolus, Cellular Automata Ma- chines: A New Environment for Modeling (MIT Press, Cambridge MA, 1986).

[2] T. Toffoli, Cellular automata as an alternative to (rather than an approximation of) differential equations in modeling physics, Physica D 10 (1984) 117- 127.

[3] Information Mechanics Group, MIT Laboratory for Computer Science, CAM-6 hardware and software (Systems Concepts, San Francisco, 1987).

[4] T. Toffoli, Four topics in lattice gases: ergodicity; rela- tivity; information flow; and rule compression for par- allel lattice-gas machines, in: Discrete Kinetic Theory, Lattice Gas Dynamics and Foundations of Hydrody- namics, ed. R. Monaco (World Scientific, Singapore, 1989) pp. 343-354.

[5] M. A. Smith, T. Toffoli, H. Hrgov6i6 and N.H. Mar- golus, A conformally invariant cellular automaton, J. Phys. A, to be submitted for publication.

[6] D.C. Kelly, Difusion: a relativistic appraisal, Am. J. Phys. 36 (1968) 585-591.

[7] G. 't Hooft, Equivalence relations between deterministic and quantum mechanical systems, J. Stat. Phys. 53 (1988) 323-344.

[8] S. Goldstein, On diffusion by discontinuous move- ments, and on the telegraph equation, Quart. J. Mech. Appl. Math. 4 (1951) 129-156.

[9] M.A. Smith, A Fermi lattice gas in a potential well, J. Stat. Phys., to be submitted for publication.

[10] M.A. Smith, Broken ergodicity and finite size effects in the Fermi lattice gas model, MIT Laboratory for Computer Science Technical Memorandum, in prepa- ration.

[11] S.B. Gray, Local properties of binary images in two dimensions, IEEE Trans. Computers C-20 (1971) 551- 561.

[12] T. Toffoli and N.H. Margolus, CAM-6 experiment, private communication (1988).

[13] A.K. Dewdney, Sci. Am. (August 1989) 103. [14] R. Fisch, Cyclic cellular automata and related pro-

cesses, Physica D 45 (1990) 19-25, these Proceedings. [15] P. Tomayo and H. Hartman, Cellular automata,

reaction-diffusion systems and the origin of life, in: Artificial Life, SFI Studies in the Sciences of Com- plexity, ed. C. Langton (Addison-Wesley, New York, 1988) pp. 105-124.

[16] J.M. Greenberg and S.P. Hastings, Spatial patterns for discrete models of diffusion in excitable media, SIAM J. Appl. Math. 34 (1978) 515-523.

representations of geometrical and topological quantities in cellular automata

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