A conjecture about the size of a particular cellular automaton : the perfectly growing crystal

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A conjecture about the size of a particular cellular automaton : the perfectly growing crystal

H. Dreyssé, R. Riedinger

To cite this version:

H. Dreyssé, R. Riedinger. A conjecture about the size of a particular cellular automaton : the perfectly growing crystal. Journal de Physique, 1987, 48 (6), pp.915-920. �10.1051/jphys:01987004806091500�.

�jpa-00210521�

A CONJECTURE ABOUT THE SIZE OF A PARTICULAR CELLULAR AUTOMATON :

THE PERFECTLY GROWING CRYSTAL

H.

Dreyssé

^{and R.} Riedinger

Laboratoire de Physique ^du Solide, Faculté des Sciences et Techniques, ^{4 Rue} des Frères Lumière,

68093 Mulhouse Cedex, ^France

(Reçu

^{le 17 dicembre 1986, rivisi le} 25 fivrier 198?’, accepti ^{le 10 mars}

1987)

Résumé.- Pour minimiser l’occupation mémoire dans la mise en oeuvre de la méthode récursive, proposée

initialement _par

Heine,

Haydock ^et

Kelly,

nous avons été amenés à construire un amas à partir ^d’un

ensemble de

départ

par

applications

successives d’un ensemble de générateurs ^{sur un} réseau. Cet amas

est organisé en couronnes et nous donnons une conjecture ^{sur sa} taille. Cette conjecture ^{est testée sur}

des réseaux avec des sites de connectivité différente où elle semble rester _{exacte ;} cette conjecture ^devrait

être comprise ^à

partir

^de

l’aspect

auto-similaire de l’amas en croissance. Nous relions cet amas à d’autres aspects : ^automate

cellulaire,

animaux, croissance d’un cristal

parfait

^et analyse combinatoire. Nous illustrons

également

^la génération de certain joints ^de grains par ^ce

procédé.

Abstract.-In order to optimize ^a computer implementation ^{of the recursion} method,

(initially

^proposed

by Heine, Haydock ^and

Kelly),

^{we build a cluster from an} initial seed of points by adjoining ^{the new sites}

obtained from the actual cluster by translation of a given ^{set of vectors}

(the generators),

^{on a lattice. This}

cluster is

organized

in shells and a

conjecture

^is

given

^{about its size. This} conjecture ^{is checked on lattices}

with inequivalent ^{sites and seems to remain} valid this case. The conjecture about the size seems to be related to a

self-similarity

^{of the} growing cluster. We relate the construction of these clusters to other

topics : ^cellular automata,

animals,

growth ^{of a} crystal ^and combinatories. We show also how the growing

process may be used to generate cluster with

grain

boundaries.

Classification Physics ^Abstracts

05.50 - 61.50C - ^02-10

In this letter we present ^a conjecture ^{about the size}

of clusters built by ^a

geometrical

recursion law. This

problem ^can be related to the study of cellular au-

tomata, ^the

partition

^{of an} integer, animals, ^{and per-} fectly growing

crystals [1].

^{Let us first} define these

particular clusters, ^{which we call} "zebra", ^{due to their} special organization ^{in shells or} strips.

1. Definition

The recursion method

[2]

^{is a powerful method to cal-}

culate _many

physically interesting

quantities ^{in real}

space.

We implemented

^{it in our} previous ^{studies of}

the electronic structure of impurities ^{near surfaces}

[3].

In order to minimize the computer memory require- ments, ^{we built an} optimal cluster of sites which avoids any useless site for ^a given number of levels of the ^as-

sociated continued fraction. Let us

briefly

^{recall the}

essence of the recursion method :

starting

^{from an}

initial function

10) (atomic

^or

molecular),

^{we build a}

sequence of orthonormalized states

in)

^{given by the}

recursion relation :

where, ^at stage

n(n

>

1), n), [n - I)

^{and bn are}

known ; ^{H is} the Hamiltonian of the system, an =

(n IH I n) .

The initial conditions are to select a nor-

malized vector

10)

^{and to assume that bo = 0.}

In a tight-binding formalism, ^the natural basis is the whole set of atomic states located on all atomic sites,

and the action of H on an atomic state

li)

^{can be writ-}

ten

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01987004806091500

916

This algorithm may also be transposed ^to generate ^the cluster of sites ; ^{one has to} replace ^the Hamiltonian H

by ^the connectivity ^{matrix :}

Let us assume that the set of the sites is part ^{of a} per- fect ordered crystal. The direct

neighbourhood

^{of each}

atom is

exactly

^{the same. If Rs denotes the}

position

of the site i and

Tri

^the

position

^{of an} interacting ^site

(his 0 0), Rï - Ri

^{is a vector of a finite} set vectors

(g),

^called generators.

We can define

paths

^{and steps : one step}

corresponds

to a single

application

^{of the}

operation

^{T onto a site.}

(go

from the site Ri ^{to the site}

Tri

⁼

1ii Rï

^{= R, +}

ga, ga E

fg}).

^{It is similar to the presence of a} "time".

The optimal cluster, ^the "zebra", ^{needed for an exact}

determination of n stages of the continued fractions, ^is

built by ^the following recursion laws :

1. The starting ^{seeds is} given ; it forms the shell 0 and may be either a single ^{site or a finite cluster.}

2. From the shell, n ^one builds the shell

(n

⁺

1) ,

^which

is the set of all the sites connected in one step ^{to the} full cluster at stage ⁿ

(in

fact, ^{one needs} only ^{to con-}

sider the shell

n)

^sites which does not yet

belong

^{to the}

cluster

(in

fact, ^{one has} only ^{to check a} finite number of preexisting ^shells

(n, n -1, ...)).

3. The "zebra" at stage n ^is the reunion of all shells,

up to ^n.

In figure ^{1 we show a} growing cluster with a seed of two sites.

2. Conjecture ^over the size of the "zebra"

For a large variety ^of generators spanning ^a dspace

(d=2,3,4)

^we have checked that

s(n),

the number of sites in the shell _n, satisfies to

for n greater ^{than a} finite threshold value, nc. Here A is the finite difference operator.

Moreover, the value of the constant T ⁼

A’-’s(n)

^{for a}

d-dimensional crystal depends only ^{on the} generators.

We have listed the values of _nc and T for different _gen- erators and seeds in table I and figure ^2.

So it is possible ^to determine the number of sites in the full "zebra" at stage

n, S(n)

Fig.l.-Example ^{of 2D} "zebra". 0 denotes the seed ; and

A denote the points ^{in two} successive shells. Full lines connect

points

^{in the same} shell ; ^{these lines and} stripes

are drawn for a better visualization of the different shells. Note the

self-similarity

^of the "zebra".

It is clear from equation

(4)

^that

s(n) [S(n) resp.]

^is

a polynomial ^of highest ^order

(d - 1) [d resp.]

^{in n,}

fully determined from

(4)

^{and the knowledge of a finite}

number of

s(n)

^{as initial data. In a 3d-cubic lattice,}

S (n)

^verifies

with

S(0)

^{= 1}

(the

^{seed is a}

point),

^and the values of

T are given ⁱⁿ table I. Let us note that the simplicity

of formula

(7)

^{stems from the fact that, in the present}

examples, ^the asymptotic ^{threshold nc is} attained for

nc ⁼ 2. The values

S (0)

^and

S (1)

^{also comply with}

this formula, ^{due to the} empirical ^{law about r :}

where Nt} ^is the number of sites connected to a given

site. As mentioned above, this formula can be used as a check on the size of the minimal cluster for obtain-

ing n ^{exact levels in a} continued fraction built by ^the

recursion method on a lattice. We worked out similar relations for other lattices from

(4).

3. Relations of the "zebra" with other

topics

1. The "zebra" is a cellular automaton

[4,5].

^According

to Wolfram

[6],

the "zebra" is a discrete lattice of sites

Table 1.- Value of ^{the threshold} n,, and limit of

ð. d-1 s(n)

for different systems, for various dimensions d, generators, and seeds.

(s.c. :

simple cubic, ^{b.c.c. :} body ^centered cubic, f.c.c. : face ^centered cubic, ^{n.n. : nearest} neighbours,

n.n.n. : next nearest

neighbours).

which evolves in discrete time steps. ^{Each site takes}

two possible ^values

(occupied

^or

not),

according ^{to the}

same deterministic rules

(one

^{site is}

occupied

^{in the}

shell n if, ^without belonging ^{to the}

previous

shells, ^this

site is connected to a site in these previous

shells).

Thus the rules for the evolution of a site depend only

on a local neighbourhood ^{of sites around it.}

2. The "zebra" is connected to combznatorzes and to lattice animals. Suppose ^{we have} ng generators gj,

spanning ^a d-dimension _space and the seed is a unique point. Any ^{site x in shell n can be}

represented

^as spanning ^a d-dimension _space and the seed is a unique point. Any ^{site x in shell n can be} represented ^as

From the recursion law, ^we deduce the following prop-

erty : any point ^of the shell n _may be reached from the

source in a number of steps greater ^or

equal

^{to n.}

This implies that it is always possible ^{to find a set of} integers

{njl

^such that the site is reached in n steps from the seed, ^or

Equation

(8c)

^{defines a partition of the integer n into}

ng

(or less)

non-negative integers.

But even if the generators ^are linearly dependent, ^the

statement that the path ^{from the} origin ^to point ^{x is the}

shortest _one, measured in steps ^{holds. The} generators being linearly dependent, there exist _p constraints.

918

Fig.2.-Value ^{of r =}

A1s(n)

^{for different} generators.

where cij and

cIJ’,

^{are positive or null integers which}

satisfy ^the following ^relation

This relation implies ^that p >

(ng - d).

^{This entirely}

determines the

possible

^{sets of values of} nj in equation

(8c)

and thus the conditional

partition [7].

^Equation

(9)

^can also be used as a sieve, ^{as follows}

and if all

the n’.

^{are greater than or}

^equal

^{to a certain}

set ci j

^{it is}

^possible

^{to write x as}

and thus,

This point belongs ^{to the} shell q ^{of lower} index. The

partition resulting ^from

( l0e)

^{is not allowed. These}

conditional partitions may lead to a formal proof ^of

our conjecture. ^{In the case of d} linearly independent generators

applied

^{to a} punctual ^seed

(the

^{site at ori-}

gin),

^our conjecture about the number of sites in the shell

n, s(n),

^{allows us to} determine the number of di- rected

animals,

An, ^{of size n on} the lattice defined by

the d generators.

Thus, ^{even in the} general ^case

(linearly

dependent gen-

erators, which is close to the polyominos

[8]),

^any point

of the shell n can be attained in n steps ^{from the} origin

but not less and the shell n is the set of the extremities of the stretched animals of size _n, the other extrem-

ity being ^{fixed at the} origin

(this

^{animal is in fact a}

worm

!).

3. The "zebra" can be considered as a perfectly growing crystal, since it involves no randomness.

Conjecture (4)

is valid for _{any set} of generators ^{on a} lattice. Let us

note, ^that

generally,

^{our set of} generators ^is compat- ible with a lattice where the possible interaction be-

tween two sites is not necessarily ^{related to a} geomet- rical

proximity

^{but is} associated with the generators.

In the case of d linearly independent generators ^and

one point ^as seed, simple arguments ^of self-similarity

can be invoked to give ^a proof ^of conjecture

(4).

^We

can also

give

^a proof ^of conjecture

(4)

^{in the} following

case. Let _p linearly dependent generators ^{be with the} following property : all lattice sites inside the geomet- rical envelope ^{of the} generators

applied

^{to a} point ^seed

are occupied. ^{then we can} tile the whole _space into a

set of

pyramidal

cones, defined with the seed

point

^{as a}

summit and a base of adjacent vertices of the first shell.

In each _cone, the preceding

self-similarity

property ^re- mains valid and this leads to a

proof

^of conjecture

(4).

More

generally,

^{one seems to find such}

self-similarity

properties ^{which are related to} conjecture

(4).

^An-

other interesting property ^of the "zebra" is its

ability

to build clusters with some

restrictions ;

e.g. surfaces with steps ^or grain boundaries

(Fig.3).

^For instance,

with a

judicious

^{choice of} generators

operating

^{in half-}

spaces

only,

^{one can build a}

large family

of flexion

grain

boundaries. Let us mention that the generators ^used in this case to build the lattice no

longer

represent physical Hamiltonian

interactions,

^as in the recursion method.

Finally

^{one can} visualize the growing process by using

Fig.3.-Example

^{of 2D} crystal ^with periodical steps ^with the used generators ^{and one} point ^{as seed}

(·).

^{The first}

four shells are represented.

a computer.

One can then see that, ^when conjecture

(4)

^{is satisfied,}

all the defects in the crystal are "blocked" i.e. they ^are just moving ^{in a} translational _way, and no longer grow.

4. We can extend our conjecture ^to lattices with sites

having ^different connectivities. Conjecture

(4)

^remains

valid, ^{but now}

A-1 s(n)

^{also depends on the origin}

and the number of steps ^n. However, ^{in the cases re-} ported ⁱⁿ figure 4, ^{we recover the}

periodic

^behaviour

Fig.4a.-Rectangular-triangulax

^{lattice with nearest neig-}

bour interactions,

s(n)

^{is the number} of sites in the shell n.

Fig.4b.-Lattice ^{with nearest} neigbour interactions,

8 (n)

is the number of sites in the shell n.

with a period v independent ^{of the} starting point

(here,

in both _cases,

v=3).

Simple geometrical arguments ^can be used to transform this problem ^{into a linear chain} problem ^{and the values of}

Ad-1S (n)

^are easily ^deter-

mined. Moreover one can consider a "supershell" ^con- sisting of the reunion of v consecutive shells. If

E(n)

denotes the number of sites in the "supershell" n, ^we find again ^{that the} remarkable property

6.1 E (n)

^does

not depend ^{on the} starting point ^but only ^{on the lat-}

tice and thus on the generators.

(In

Fig.

4a, &’E(n) =

14=4+6+4=5+5+4, ⁱⁿ Fig. 4b,

A’E(n)

^=16=6+5+

5=4+6+6).

^{Thus it seems that} conjecture

(4)

^found

for lattices with sites having ^{all the same direct} neigh-

920

bourhood, ^{can be extended to} lattices with sites of different connectivities. Let us note that, ⁱⁿ two cases, the lattices can be built by applying v ^{different sets of} generators

periodically.

^{Let us note} general expression

of this conjecture would be : given ^a lattice built from

a initial set of states by periodical

application

^{of v sets}

of generators, following the recursion law

(each

point

of a shell is connected at least to a point ^{of the} previous

shells but is not present ^{in these} previous

shells) ;

^one

can consider a "supershell" ^{as the} reunion of v con-

secutive shells. Thus if

E ( n)

^{is the} number of sites in

"supershell" n

for n greater ^{than a} threshold value _nc, depending ^of

the sets of generators ^{and the} seed, ^r depending only

on the sets of.generators.

4. Conclusion

In spite ^{of our} efforts, ^{we were unable to find a} general proof ^{of the} conjecture ^{on the size of} the "zebra". How- ever, it was

possible

^to find connections of the "zebra"

with different topics ^of statistical physics. ^In any ^case this conjecture ^{is useful in the recursion method} itself, by giving ^an estimation of the _memory storage ^needed for a given-number ^{of levels of the continued fraction}

to be built buy the recursion method. The ability ^of

the "zebra" to generate ^some corrugated surfaces and

grain boundaries has also been emphasized.

Acknowledgments

We whish to thak Profs. G. Rauch and Foata and Dr. J. Vannimenus for stimulating discussions. We

are indebted to Dr. I. Bose and P. Ray

for

^suggesting

the study of lattices with sites having different ^con- nectivitiet. Computational facilities form Groupement

pour ^un Centre de Calcul Vectoriel en Recherche

(GC-

CVR,

Palaiseau),

^Centre

Interregional

^{de Recherche en}

Calcul Electronique

(CIRCE, Orsay)

^{and Institut de}

Recherches

Polytechniques (IRP, Mulhouse)

^are grate-

fully acknowledged.

References

[1]

^{STAT PHYS} XVI,

Proceeding

^{of the XVI IUAP}

Conférence on Thermodynamics and statistical Me-

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(North Holland,

^Amster-

dam),

^1986.

[2]

^R. HAYDOCK, ^{V. HEINE and M.J.} KELLY, ^J. Phys.

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(1972)

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Ibid in Solid State Physics, ^edited by H. Ehrenreich.

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(Academic

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New-York)

1980, ^{vol. 35.}

[3]

R. RIEDINGER and H. DREYSSE, Phys. ^{Rev. B27}

(1983)

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Phys. ^{Rev. B31}

(1985)

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H. DREYSSE and R. RIEDINGER, Phys. ^{Rev. B28}

(1983)

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and S. WOLFRAM

(North

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[6]

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[7]

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^G. VIENNOT, Séminaire Bourbaki n°626

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