Renormalization group approach to an Abelian sandp

arXiv:cond-mat/0204243v2[cond-mat.stat-mech]13Apr2002RenormalizationgroupapproachtoanAbeliansandpilemodelonplanarlatticesChai-YuLin∗andChin-KunHu+InstituteofPhysics,AcademiaSinica,Nankang,Taipei11529,TaiwanAbstractOneimportantstepintherenormalizationgroup(RG)approachtoalatticesandpilemodelistheexactenumerationofallpossibletopplingprocessesofsandpiledynamicsinsideacellforRGtransformations.HereweproposeacomputeralgorithmtocarryoutsuchexactenumerationforcellsofplanarlatticesinRGapproachtoBak-Tang-Wiesenfeldsandpilemodel[Phys.Rev.Lett.59,381(1987)]andconsiderboththereduced-highRGequationspro-posedbyPietronero,Vespignani,andZapperi(PVZ)[Phys.Rev.Lett.72,1690(1994)]andthereal-heightRGequationsproposedbyIvashkevich[Phys.Rev.Lett.76,3368(1996)].UsingthisalgorithmweareabletocarryoutRGtransformationsmorequicklywithlargecellsize,e.g.3×3cellforthesquare(sq)latticeinPVZRGequations,whichisthelargestcellsizeatthepresent,andfindsomemistakesinapreviouspaper[Phys.Rev.E51,1711(1995)].Forsqandplanetriangular(pt)lattices,weobtaintheonlyattrac-tivefixedpointforeachlatticeandcalculatetheavalancheexponentτandthedynamicalexponentz.OurresultssuggestthattheincreaseofthecellsizeinthePVZRGtransformationdoesnotleadtomoreaccurateresults.Theimplicationofsuchresultisdiscussed.05.40-a,05.50,05.60Cd,89.75DaTypesetusingREVTEX1I.INTRODUCTIONIn1987,Bak,TangandWiesenfeld(BTW)[1]proposedtheconceptofself-organizedcriticality(SOC)inordertounderstandtheautomatic(i.e.withoutatuningparameter,suchastemperature)appearanceofabundantself-similarstructuresandscalingquantitiesinnature.BTWalsoproposedalatticesandpilemodelandusedMonteCarlosimulationstosimulatethismodelonsquareandsimplecubiclattices.Theydidobserveself-similarstructuresandscalingquantitiesinthesimulationdatawithouttuninganyparameter.Since1987,manynaturalphenomenahavebeenrelatedtoSOC,suchasearthquakes[2],forestfires[3],biologicalevolution[4],ricepiledynamics[5],turbulence[6],etc.ManylatticemodelshavealsobeenproposedtoillustratethebehaviorofSOCoravalancheprocesses[7,8].IthasbeenfoundthattheBTW’ssandpilemodelisAbelian[9]andsomequantitiesforthismodelcouldbecalculatedexactly[9–12].TheBTW’sAbeliansandpilemodel(ASM)[1]hasbeenconsideredtobeaprototypicalmodelforSOC.ManyideasaboutbehaviorofSOCmodels,suchasuniversalityandscaling[13],ornewmethodsforstudyingSOCmodels,suchasrenormalizationgrouptheory[14–17],areoftenfirsttestedintheBTW’sASM.Inthepresentpaper,weproposeacomputeralgorithmwhichisusefulforcarryingoutrenormalizationgrouptransformationsfortheBTW’sASM.TheBTW’sASMonalatticeℜofNsitesisdefinedasfollows.Eachsiteofℜisassignedaheightinteger;thei-thsiteisassignedzifor1≤i≤N.Inthebeginningofthesimulation,theheightintegerateachsiteisrandomlychosentobe0,1,...,orzc−1,wherethecriticalheightiszc−1andzcisthecoordinationnumberofthelattice.ForeachtimeintervalTa,oneparticlefallsonarandomlychosenlatticesite,saythei-thsite;theheightzithenbecomeszi+1.Ifthenewzizc,thenrandomlychooseasiteagain,saythek-thsite,toaddaparticle;ifzi≥zc,thenthei-thsitetopplesanditsheightzibecomeszi−zc.Atthesametime,eachofthenearestneighbor(nn)sitesofthei-thsitereceivesoneparticle,i.e.,zω(i,j)→zω(i,j)+1,∀jwhereω(i,j)isthelabelofthejthnnsiteofthei-thsite.Thisrelaxationproceduretakestimetwandweassumetw/Ta→0.Ifsomeof2thenewzω(i,j),∀j,areequalorlargerthanzcagain,thesesitesaredenotedbyω(i,j′).Then,thetopplingprocesscontinuesinparallelforeachj′withzω(i,j′)→zω(i,j′)−zcandzω(ω(i,j′),k)→zω(ω(i,j′),k)+1whereω(ω(i,j′),k)isthek-thnnsiteofthej′-thnnsiteofthei-thsite.Therelaxationtimefortheseparalleltopplingandreceivingprocessesbetweenω(i,j′)andω(ω(i,j′),k)takesanothertimetw.Usually,theopenboundaryconditionsareusedsothatwhenaboundarysitetopples,theparticlecanleavethesystem.Thedynamicalprocesscontinuesuntiltheheightsofallsitesarelessthanzc.Ingeneral,ifthelasttopplingsiteisω(ω(...(ω(ω(i,i1),i2),i3)...),in),thetotaltopplingprocesstakestimen×tw.Inthisway,aseriesoftopplingprocesseswithtopplingareas(i.e.,thetotalnumberoftoppledsites)andrelaxationtimentw≡tappearsandformsanavalanchewhichhasnocharacteristicsize.Afterrepeatingmanytimestheprocessofaddingoneparticleonarandomlychosensitewithsubsequentrelaxationwhentheheightofthesiteisequalorlargerthenzc,wecanobtainadistributionoftopplingareaP(s)andcalculatetheaveragerelaxationtimetforavalancheswithtopplingareas.IthasbeenfoundthatP(s)∼s−τwiththeavalancheexponentτandt∼sz/2withthedynamicalexponentz.Manna[18]usedMonteCarlosimulationstocalculateτ=1.22andz=1.21fortheBTWmodelonthesquare(sq)lattice.MajumdarandDharconjecturedthatz=5/4whichisconsistentwiththeirownnumericalsimulations[19],andPriezzhevetal.[20]proposedthatτ=5/4.Byscalingargument,Tebaldietal.[21]suggestedτ=6/5.Manyinvestigationshavebeenjustfocusedonnumericalsimulationsorexactresultsforheightprobabilitiesandexponentsforthesqlatticeandthereislittleattentiontootherkindsoflattices.Itisnotclearwhetherthesandpilemodelontwodimensionallatticeshavethesamesetofcriticalexponents.Therenormalizationgroup(RG)theoryhasbeenusedsuccessfullytocalculatecriticalexponents,orderparameters,e