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1、TheTopologicalNon-connectivityThresholdandmagneticphasetransitionsinclassicalanisotropiclong-rangeinteractingspinsystemsR.Trasarti-Battistoni,1F.Borgonovi,1,2andG.L.Celardo11DipartimentodiMatematicaeFisica,Universit`aCattolica,viaMusei41,25121Brescia,Italy2I.N.F.N.,SezionediPavia,Italy(Dated:Febr
2、uary2,2008)WeanalyzefromthedynamicalpointofviewtheclassicalcharacteristicsoftheTopologicalNon-connectivityThreshold(TNT),recentlyintroducedinF.Borgonovi,G.L.Celardo,M.Maianti,E.Pedersoli,J.Stat.Phys.,116,516(2004).ThisshowsinterestingconnectionsamongTopology,Dynamics,andThermo-Statisticsofferro/p
3、aramagneticphasetransitioninclassicalspinsystems,duetothecombinedeffectofanisotropyandlong-rangeinterections.PACSnumbers:05.45.-a,05.445.Pq,75.10.HkI.INTRODUCTIONanalytically,inaclassofspinmodelswhereimportantandrigorousresultshavebeenobtainedduringthelastcentury,thoughgenerallyonlyinthethermodyna
4、micalThemagneticpropertiesofmaterialsareusuallyde-limit(totalnumberofmicroscopicconstituentsofthescribedintheframeofsystemmodels,suchasHeisenbergdsystemN→∞keepingfixedthenumberdensityN/R,orIsingmodelswhererigorousresults,orsuitablemeanwhereRisthespatialsizeofthesystemanddisitsspa-fieldapproximation
5、sareavailableinthethermodynami-tialembeddingdimension).Nevertheless,tothebestofcallimit.Ontheotherside,modernapplicationsrequireourknowledge,apparentlynobodytookcareofthedy-todealwithnano-sizedmagneticmaterials,whoseintrin-namics,andconsequentlynobodyspokeofthissimplesicfeatureslead,fromonesideto
6、theemergenceofquan-butrelevantproperty.tumphenomena[1],andtotheothertothequestionofThisdynamicalpointofviewhasafewinterestingapplicabilityofstatisticalmechanics.Indeed,fewparti-classicalconsequences.Firstofallitexplains,fromtheclesystemsdonotusuallyfitintheclassofsystemswherepointofviewofmicroscop
7、icdynamics,thepossibilityofthepowerfultoolsofstatisticalmechanicscanbeappliedferromagneticbehaviorinsmallsystem.Indeed,inab-atglance.Inparticular,anexhaustivetheoryableto23senceofexternalfieldandexternalnoise(temperatur
