A framework for scalingandrenormalizationin thetriangularlattice
DominiqueDeserableLaboratoiredeMecaniqueAppliquee,Automatique& Geomecanique
INSA – 35043Rennescedex – Francehttp://[email protected]
Keywords: cellularautomata,triangularlattice,hierarchi-cal Cayley graphs,arrowhead,scalingandrenormalization.
Abstract
We exposethe suitability of a hierarchicalCayley networkunderlyingacellularautomatonto therenormalizationmeth-ods. Thenetwork is a hexagonalarrowheadtorusgeneratedon thetriangularlattice. We show how therecursive topolo-gy andthesymmetriesof thearrowheadarelikely to providea convenientframework for thesemethods.
1 Introduction
Therenormalizationmethodsappearedtheselastthirty yearsin theareaof statisticalphysicsanddynamicalsystems[8].They apply quite well to the study of large scalehomoge-neoussystems, illustratedby the imageof the “chessboard”whoseside of length
�representsthe macroscopicscale
while the squareof the chessboard,whosesideis of length��� �andwhere
�is the “correlation length”, represents
themesoscopicscalewhich definesa homogeneoussubsys-tem.Thesesystemsarethuscharacterizedby a translationalinvarianceby avectorof modulus
�, namely, they areperiod-
ic. Therenormalizationmethodshaveespeciallyprovedtheirsuperiorityin the studyof critical phenomenawhereclassi-calmethodshadshown to beunsuccessful.A critical systemis characterizedby a scaleinvariancewhoseeffect is a self-similar behavior anda divergenceof the correlationlengthat the critical point ������ of the control parameter. Usu-ally, the divergencefollows a power law (or “scaling law”)of the form
�� � ��� ����� � ��� where ����� is the criticalexponent. The samegoesfor the correlationlengthaswellas any macroscopicquantityexpressedin termsof the de-viation ����� � bearinga specificcritical exponent. It is upto therenormalizationmethodsto deducethevalueof theseexponents,andtherebythebehavior of thesystemin a criti-cal situation,from theonly propertyof scaleinvariance.Byrelying uponthe self-similarityof the system,the way is tosplit it into nestedblocksandsub-blocks...andto yield lo-
cal averagesin eachof them(Kadanoff construction).Thelong-rangecorrelationsremainunchangedafter a sequenceof transformationsonly if the systemlies in a critical state.Theoperatordefinesa“renormalizationgroup”– in thealge-braicmeaning– andthesequenceconvergesin thatcaseto-wardsa fixedpoint (Wilson & Kogut[12], Fisher[5], Ma[9],Stanley[11]).
We focushereon thestrictly topological aspectof theprob-lemof scalechangeunderlyingthesemethods.After a recallin Sect.2 of theKadanoff constructiondefinedonanorthog-onal lattice,we presentin Sect.3 andSect.4 a new schemeiteratedon thetriangularlattice.As anexample,thisprocesswill beappliedin Sect.5 to thedeterminationof a represen-tative elementaryvolume(or “REV”) in a globally homoge-neousmedium.
2 Kadanoff construction
The Kadanoff construction[6, 12, 11, 7, 8], definedon anorthogonallattice of spinsfor the Ising model,is displayedin Fig. 1 wherethelength � separatingtwo neighboringsitesis theminimal scale.Thevalidity of theconstructioncomesfrom the fact that the interactionsare short-range,that al-lows us to consideronly the nearest-neighborinteractions.Eachspin, associatedto a site, hasa properenergy ( � ) or( � ) dependingon the externalmagneticfield andeachpairof neighbors,associatedto a bond,hasan interactionener-gy ( �! ) or ( !� ) dependingon that they point towardsthe samedirectionor towardstwo oppositedirections. Thecalculationof theenergy of theglobalconfigurationis renor-malizedby forming squareblocks of "$#�" sites(Fig. 1a)andyieldinga localaverageaccordingto amajority rule(theblockhasanarbitrarysizebut shouldcontainasmallnumberof sitesandsize "%#&" is usuallychosenin theconstruction).Eachblock is thenlikenedto a singlespinat theupperscaleof length "'� andthe lengthsarecontractedby a factor " tomaintainthe initial densityof sites(Fig. 1b). Theconstruc-tion is thusiteratedby forming blocksof size ( , (*) , . . . , (*+until thecorrelationlength
�of thesystemin critical stateis
reached.
In the simplecaseof a globally homogeneoussystem,it is
a
a
- a - - b -
Figure 1: Kadanoff constructionon the orthogonallattice:dottedlines standfor inner interactionswithin a block, fulllinesfor interactionsbetweensitesof neighboringblocks(afterA. Lesne[8])
clear that a similar nestedblock constructionwould reachthemesoscopicscaleof a REV asit will beseenin Sect.5.
3 The arrowhead torus
The topologyunderlyingour cellular automatanetwork [4]is an“arrowhead”torus,a hierarchicalCayley graphdefinedfrom groupsandpresentationsin thetriangular(or “hexava-lent”) grid [3]. Another family derived from the triangulargrid andwhich oftenappearsin theliteraturewasinvestigat-ed by Yebraet al. [13, 1, 2]: the grid is also a hexagonaltorusbut composedof circular ringsarrangedarounda cen-tral node. The relevant featureof the arrowheadis that thesymmetriesaremaximizedin theconstruction.
So,let , thedimensionalityof thearrowhead,denoted-/.1032and 4��5( 0 thenumberof sites(or by duality, thenumberofhexagonalcells in the associatedautomaton).The sitesarenumberedin theset6 0 �87:9�2;�<2=">2@?;?@?A2B( 0 ���<C (1)
accordingtoaschemeresultingfromarecursiveconstructionof thefinite toruswithin theinfinite triangularlattice.
The connectionschemein -/. 0 is defined as follows.Let D be any direction in the orderedset
4E2GFIHJ2GFIK1�A2ML@N 2 L;O>P 2 L:O*Q � = (1,2,3) be the set of associatedincre-mentsas shown in Fig. 2 and D be the oppositedirectionof D . Then RTSVU 9*�I� RTSVU 9*�I�W9 (2)
and X3,$YZ� R[S 0 (<\3�]�^(*\`_ L R (3)R[S 0 (<\3�]��( RTS 0badc \3�e�f_ L R (4)
where g S'h ji � denotestheneighborin direction k of asitei
in -/. h .
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Figure2: Scalechangeby contractionof network -/.`0 in-to the network -1.10badc (for ,Z�ml ). The centersof blocksn 0bo c \p� aredisplayedin gray
As anillustration,knowing thewholeconfigurationof -1. ) ,let usexaminewhich the F , 4EK and 4$H neighborsof site(*9 arein -/.`q :O S q (*9*�r� ( O S ) �;9 �e�d_ L@N �5(ts;lu_W�v�w�:lNxQ S q (*9*�r� ( NyQ S ) �:9*�B�d_ L:ObP �^(ts;zv_�"T�Wl'(N{P S q (*9*�r� ( NuP S ) �:9*�e�|_ L O*Q �^(ts<�}_~lt���4 Construction on the triangular lattice
Starting from Kadanoff construction, Niemeijer & vanLeeuwengive an iteratedconstructionof spinblockson thetriangularlattice [10, 14]. Referingto the topologyof thearrowhead,we proposea new constructioninducedby thenumberingschemeof thecellsandwhich takesadvantageofthesymmetriesof thetorus.We definethesubsets6 0bo � �w7;\�� 6 0�� \��59 ����� ( � �AC 9��$���!,�� (5)
of thesitesof6 0 whoseindex is amultipleof ( � . For afixed� , a subset
6 0>o � definesa subdivision of thelatticeinto ( 0ba��blocksof size ( � denotedn 0bo � \3�]� 6 0bo �u_ 6 �1��7;\�_�� � \�� 6 0bo �/2|��� 6 �bC (6)
andwhosesites \ having their index in6 0bo � arecenters.For���w� in particular, weobtainacontractionof network -/.10
into a network -/.`0ba�c accordingto Fig. 2. Eachblock of( sitesnumberedin the set 7;(*\�2e(*\�_8�<2e(*\�_5">2e(*\�_5l>C isthuscontractedinto a singlesite \ where \$� 6 0ba�c'? Iterated� times, this schemeappliedto a triangularlattice inducesa renormalizationprocessanalogousto thatof theKadanoffconstruction. The advantageis that the symmetriesof thisframework seemto bebetterthanin themodelof Niemeijer& vanLeeuwen.
5 Application to the determination of a REVin a globally homogeneous medium
Let us considerthe trivial example of a scenerepresent-ing a compositematerialmadeup of two solid phases� c
Figure3: Determinationof a REV in a homogeneousmate-rial at the global scalewith concentrations�3cE���9 � and� ) �5l*9 � (phase��c appearsin light gray).Thepatterncon-tains (*9*l<9�� cells. Thehexagonalpolyhexe �[��o � 9 � contains"*�'� cellsanddefinesarepresentativevolume
and � ) , mixed accordingto concentrations�3c and � ) with�3c{_W� ) ���*? By applyingthe above constructionscheme,we planto determinatea mesoscopicscalereferredto a rep-resentative (elementary)volume. At this scale,the materialmaybeconsideredashomogeneous.
Let �;� betheoccurencefrequency of phase��c in a block ofsize ( � ( 9��!�J�Z, ). For a givensite,onehasin particular� U ��� if ��c is presentin this siteand � U �59 otherwise.Fora block of size ( �@ |c , the frequency �:�� fc follows from therecurrencerelation
�;�@ |c{�^¡ �;�<�]� �(q¢ £¤ U �|¥
£§¦� (7)
wherethe � ¥£¨¦� denotethefrequencieswithin eachof thefour
sub-blocks.Wheneverthemediumisgloballyhomogeneous,thereexistsa
�suchthat ©'� + |c ��� + ©ª��« where« is apos-
itive realarbitrarily small. It follows from this thatsequence7�;� C is a Cauchysequencewhich convergestowardsthe fi-nite limit �d¬{�5¡ �d¬@�}�5�3cV? (8)
This iteratedconstructionleadsto partitionthenetwork into( 0ba + blocks of size ( + , eachof theseblocks having thepropertyof defining a representative volume of the wholenetwork.
Fig. 3 displaysa pattern1 of the material,generatedby theautomatonandcoveringabout ( 9T9<9<9 cells,with concentra-tions �3c~��9b� and � ) �®l*9 � , as well as a hexagonal
1Thenetwork hasdimensionalityT°&±
� U �*? 9*9<9*9<9� c 9�? �<9<9*9<9� ) 9�? �*z �<�'9� q 9�?²�>�;zb�'��� 9�?²�'9<l��:"�'³ 9�? �*´<´ "<"
Table 1: Convergenceof the iteratedsequenceof frequen-cies,attainedafter ( iterationswith accuracy «/�w��block � ��o � 9 � centredin cell 9 anddefininga representativevolumeof "*�'� cells. Table1 detailstheconvergenceof theiteratedsequence,ensuredfor
� ��( with an accuracy of«t���V��?6 Conclusion
Our contribution in this paperwasto tacklea topologicalas-pectof the scalingandrenormalizationprocessin the trian-gularlatticeandto provideaconvenientframework for thesemethods.While theKadanoff constructionmakesuseof thesymmetrieswithin the orthogonalgrid, we show how it isalso possibleto exploit a scalability propertyfor the trian-gular case. The way lies in the hierarchicalstructureof aCayley network: we claim that the symmetriesof the “ar-rowhead” shouldprovide a self-similar scheme,comparedwith theskewedconstructionof Niemeijer& vanLeeuwen.The last sectionis just a topologicalillustration of the self-similarity of our framework.
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