Redefinition of site percolation in light of entropy and the second law of thermodynamics
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] D ec Redefinition of site percolation in light of entropy and the second law ofthermodynamics
M. S. Rahman and M. K. Hassan
University of Dhaka, Department of Physics,Theoretical Physics Group, Dhaka 1000, Bangladesh (Dated: December 10, 2019)In this article, we revisit random site and bond percolation in square lattice focusing primarilyon the behavior of entropy and order parameter. In the case of traditional site percolation, we findthat both the quantities are zero at p = 0 revealing that the system is in the perfectly ordered andin the disordered state at the same time. Moreover, we find that entropy with 1 − p , which is theequivalent counterpart of temperature, first increases and then decreases again but we know thatentropy with temperature cannot decrease. However, bond percolation does not suffer from eitherof these two problems. To overcome this we propose a new definition for site percolation where weoccupy sites to connect bonds and we measure cluster size by the number of bonds connected byoccupied sites. This resolves all the problems without affecting any of the existing known results. PACS numbers: 61.43.Hv, 64.60.Ht, 68.03.Fg, 82.70.Dd
I. INTRODUCTION
Percolation has been studied extensively in statisticalphysics due to the simplicity of its definition and theversatility of its application in seemingly disparate com-plex systems. The reason for its simplicity is that it re-quires neither quantum nor many particle interaction ef-fects and yet it can describe phase transition and criticalphenomena [1, 2]. To define percolation, we first needto choose a skeleton. It can either be a graph embeddedin an infinite dimensional abstract space or it can be alattice embedded in an Euclidean space. In either casethey always consist of nodes or sites and links or bonds.Percolation is known as site or bond type depending onwhether we occupy sites or bonds respectively. In thecase of random bond percolation, we assume that all thelabelled bonds are initially frozen. The rule is then tochoose one frozen bond at each step randomly with uni-form probability and occupy it. We continue the pro-cess one by one till the occupation probability p , fractionof the total bonds being occupied, reaches to unity. At p = 0 each site is a cluster of its own size and as we tune p we observe that clusters, a group of sites connected byoccupied bonds, are continuously formed and grown onthe average. In the process there comes a critical point p c above which a spanning cluster s span that spans acrossthe entire lattice emerges. Interestingly, its relative size P = s span /N varies with p such that P > p > p c and P = 0 at p ≤ p c . Moreover, it grows above p c fol-lowing a power-law P ∼ ( p − p c ) β . This is reminiscent ofthe order parameter of continuous phase transition andhence P is regarded as the order parameter for percola-tion [3, 4].Despite more than five decades long history of exten-sive studies we still have many unresolved issues in per-colation. For instance, we know that the order param-eter in general measures the extent of order. However,what order really means here is not known in percola- tion. Finding a way to associate order with the relativesize of spanning cluster is still an issue. It has been shownthat P = 0 in the entire regime 0 < p ≤ p c at least inthe thermodynamic limit and hence we must find a wayto regard it as the disordered phase. We therefore needanother quantity that can quantify the degree of disor-der where P = 0. The obvious choice is entropy H . Infact, no model for phase transition is complete withoutit since, like order parameter, entropy is also used as alitmus test to define the order of transition. Despite be-ing such an important quantity, its definition remainedelusive for five decades as the first article on entropy ap-peared in 1999 [5]. After that we find only two more arti-cles [6, 7] prior to our recent works since 2017 [8, 9]. Oneof the physically acceptable requirements of H is thatwhen H is minimally low P cannot be minimally low tooas this would imply that the system is in the perfectlyordered and disordered at the same time. On the otherhand, at and near the critical point both the quantitiesundergo an abrupt change revealing that the transitionis accompanied by symmetry breaking, as it also meansan order-disorder transition. However, surprisingly thishas never been an issue in percolation theory. Besides itsrelevance to phase transition, the percolation model hasalso been applied to a wide variety of natural and socialphenomena such as the spread of disease in a population[10], flow of fluid through porous media [11], conductor-insulator composite materials [12], resilience of systems[13, 14], dilute magnets [15], the formation of public opin-ion [16–18] and spread of biological and computer virusesleading to epidemic [19, 20].In this article, we revisit the random bond and sitepercolation in the square lattice focusing primarily onentropy and order parameter. For bond percolation, wefind that entropy is maximum where order parameter isminimum and vice versa as expected. However, this isnot the case for the traditional site percolation as wefind that initially both entropy and order parameter areequal to zero which cannot be the case since it meansthat the system is in ordered and disordered state atthe same time. Moreover, we have identified that 1 − p ,the fraction of the unoccupied site/bond, as the equiv-alent counterpart of temperature. Thus, entropy mustalways increase with 1 − p and at most remain the samebut must not decrease at any stage. This is, however,not the case for traditional site percolation instead wefind that entropy first increases from zero to its maxi-mum value and then decreases to zero again. It violatesthe second law of thermodynamics which states that en-tropy of an isolated system should always increase andcan never decrease again. It warrants redefinition of sitepercolation. We therefore propose a new definition forsite percolation where it is assumed that bonds are al-ready present in the system and we occupy sites one byone to connect the bonds. We then measure the clustersize in terms of the number of bonds connected by occu-pied sites. We then find that entropy for redefined sitepercolation is always increases with 1 − p and it is consis-tent with the corresponding order parameter. We thenargue that the opposing nature of order parameter andentropy suggest that percolation transition is accompa-nied by symmetry breaking like ferromagnetic transition.Besides, we reproduce all the known results for redefinedsite percolation which confirms that random bond andre-defined site percolation belong to the same universal-ity class. Furthermore, despite the difference betweenthe old and new definition of site percolation they stillgive the same critical point p c . However, the same isnot true for site and bond percolation as their p c valuesare different albeit they belong to the same universalityclass.The rest of the articles is organized as follows: In Sec.II, we briefly discuss the Newman-Ziff (NZ) algorithmfor simulating the percolation model is briefly described.Alongside, we also discussed the idea of convolution. isintroduced and the algorithm . Sec. III, contains generaldiscussions about entropy and order parameter. Inconsis-tencies of entropy and order parameter according to olddefinition of site percolation are presented in Sec. IV.The site percolation is re-defined in Sec. V to resolve theinconsistencies. In Sec. VI we have shown that muchknown site-bond universality is still valid. The resultsare discussed and conclusions drawn in Sec. VII. II. NEWMAN-ZIFF (NZ) ALGORITHM
To study random percolation, we use Newman-Ziff(NZ) algorithm as it helps calculating various observ-able quantities over the entire range of p in every realiza-tion instead of measuring them for a fixed probability p in each realization [21]. On the other hand, in classicalHoshen-Kopelman (HK) we can only measure an observ-able quantity for a given p in every realization and thisis why NZ is more efficient than HK [22]. To illustratethe idea, we consider the case of bond percolation first. According to the NZ algorithm, all the labelled bonds i = 1 , , , ..., M are first randomized and then arrangedin an order in which they will be occupied. Note thatthe number of bonds with periodic boundary conditionis M = 2 L . In this way we can create percolation statesconsisting of n + 1 occupied bonds simply by occupyingone more bond to its immediate past state consisting of n occupied bonds. Initially, there are N = L clustersof size one. Occupying the first bond means formationof a cluster of size two. However, as we keep occupy-ing thereafter, average cluster size keeps growing at theexpense of decreasing cluster number. Interestingly, allthe observables in percolation, this way or another, arerelated to cluster size and hence a proper definition ofcluster becomes crucial. One of the advantages of theNZ algorithm is that we calculate an observable, say X n ,as a function of the number of occupied bonds (sites) n and use the resulting data in the convolution relation X ( p ) = N X n =1 (cid:18) Nn (cid:19) p n (1 − p ) N − n X n , (1)to obtain X ( p ) for any value of p . The appropriate weightfactor for each n at a given p is P Nn =1 p n (1 − p ) N − n [21].The convolution relation takes care of that weight factorand hence helps obtaining a smooth curve for X ( p ). III. ENTROPY AND ORDER PARAMETER
The two most important quantities of interests in thetheory of phase transition and critical phenomena are theentropy H and the order parameter P . The reason is thatthey are the ones which define the order of transition. Inthe first order or discontinuous phase transition, entropymust suffer a jump or discontinuity at the critical pointwhich is why first order transition requires latent heat.Similarly, the order parameter too must suffer a jump ordiscontinuity at the critical point and that is why newand old phase can coexist at the same time in the firstorder transition. Besides, they are also used as a lit-mus test to check whether the transition is accompaniedby symmetry breaking or not. In the case of symmetrybreaking, the system undergoes a transition from the dis-ordered state, which is characterized by maximally high H and P = 0, to the ordered state, which is character-ized by non-zero P and minimally low H . Such transitionhappens with an abrupt or sudden change in P and H but without gap or discontinuity at p c . Percolation beinga probabilistic model for phase transition, there is abso-lutely no room for considering thermal entropy. To thisend, the best candidate is definitely the Shannon entropy H ( p ) = − K m X i µ i log µ i , (2)provided we use a suitable probability for µ i that givesentropy consistent with the order parameter and the Occupation Probability, p E n t r o p y , H ( p , L ) L=200L=250L=300L=350L=400 (a)
Occupation Probability, p
Entropy, H(t)/H max
Order Parameter, P(t)/P max
Disordered Phasep
p c (b) FIG. 1: (a) Entropy H versus p for bond percolation. (b)Entropy H/H max (dashed blue) and order parameter
P/P max (purple) for traditional site percolation. second law of thermodynamics [23]. Recently, we haveshown that the cluster picking probability (CPP) for µ i ( p ), that a site picked at random belongs to labelledcluster i at occupation probability p , is the appropriatequantity for measuring entropy for percolation [8, 9].On the other hand, order parameter for bond perco-lation is the strength P ( p, L ) of the spanning cluster forsystem size L is defined as P = Number of sites in the spanning clusterTotal number of sites . (3)Essentially, it describes the probability that a site pickedat random belongs to the spanning cluster at occupationprobability p for system size L . It has been found thatin the limit L → ∞ the probability P ( p, L ) = 0 for p ≤ p c and it reaches to its maximum value P ( p, L ) = 1 follow-ing a power-law P ∼ ( p − p c ) β near but above p c . This isindeed reminiscent of the order parameter of the contin-uous thermal phase transition like magnetization duringthe ferromagnetic transition. It is noteworthy to mentionthat for finite system size P ( p, L ) has a non-zero valueeven at p < p c . However, as we increase the linear size L of the system it always shows a clear sign of becom-ing zero at a higher value towards p c . There exists yetanother definition of P where we can use the size of thelargest cluster instead of the spanning cluster. However,both definitions give exactly the same qualitative resultsin the thermodynamic limit. IV. PROBLEM WITH EXISTING SITEPERCOLATION
We first measure entropy for random bond percolationwhere initially every site is a cluster of its own size. As wekeep occupying or reactivating frozen bonds, clusters arecontinuously formed and their sizes on average are grown.Consider that at an arbitrary step of the process thereare m distinct, disjoint, and indivisible labelled clusters i = 1 , , ..., m whose sizes are s , s , ...., s m respectively.We can therefore define CPP as µ i = s i / P i s i , thata site picked at random belongs to the labelled cluster i , which is naturally normalized P j s j = N [8]. Notethat for convenience we choose K = 1 in Eq. (2) sinceit merely amounts to a choice of a unit of measure ofentropy. Clearly, at p = 0 we have µ i = 1 /N for allthe sites i = 1 , , ..., N = L which is exactly like thestate of the isolated ideal gas where all the allowed mi-crostates are equally likely. Clearly, the entropy is max-imum H = log N at p = 0 revealing that we are in astate of maximum uncertainty just like the state of theisolated ideal gas. On the other hand, as we go to theother extreme at p = 1, we find that all the sites belongto one cluster that makes µ = 1. It implies that entropyis zero at p = 1 and hence we are in a state of zero uncer-tainty just like the perfectly ordered crystal structure. Inorder to see how entropy interpolates between p = 0 and p = 1, we use CPP in Eq. (2) and the resulting entropy isshown in Fig. (1a) as a function of p for different systemsizes.To see how entropy for random site percolation differsfrom that of the bond type, we now measure entropy fortraditional definition of site percolation. In this case, itis assumed that initially all the sites are frozen or empty.The process starts with the occupation of sites one byone at random and at the same time measure the clustersize exactly like in the bond percolation. It means thatinitially CPP does not exist and after the occupation ofthe first site we have µ = 1 and hence entropy H = 0 at p = 1 /N which is essentially zero in the limit N → ∞ .As we further occupy sites, we observe a sharp rise inthe entropy to its maximum value, see Fig. (1b), whichhappens near p = 0 .
2. Thereafter it decreases with p qualitatively in the same way as in the case of randombond type percolation. To check whether the behaviorof entropy and order parameter are consistent or not wehave plotted both in the same figure as shown in Fig. (1b)distinguished by blue and orange colors respectively. Itsuggests that at p ≈ − p instead of p for bondpercolation we find that the entropy and order parame-ter both behaves exactly like they should do in thermalphase transition and the behaviour is consistent with thesecond law of thermodynamics. The same behavior isalso expected for traditional site percolation but clearlyit is not the case. These two features thus warrant re-definition of site percolation.It is noteworthy to mention that phase transition al-ways involves change in entropy. No model or theory ofphase transition is complete without the idea of entropysince the nature of change in entropy defines the orderof phase transition like order parameter. Percolation has (a) (b) FIG. 2: Schematic illustration of (a) old and (b) new defi-nition of site percolation on square lattice. We assume thatprocess starts with isolated bonds (thick black lines) but sitesare empty (white circles). the history of extensive studies for more than 60 yearsas a paradigmatic model for continuous or second orderphase transition. Yet till 2014 only two groups namelyTsang et al. and Vieira et al. investigated entropy forpercolation. Both the groups used Eq. (2) but not thesame expression for µ i . For instance, Tsang et al. used w s , the probability that a site picked at random belongsto a cluster which contain exactly s sites, to measure en-tropy for percolation. On the other hand, Vieira et al. used n s , number of cluster of size s per site, to measureentropy H . In either case, they found entropy is equalto zero at p = 0 where order parameter is also equal tozero. Thus, their entropy too suffers the same problemas our entropy for site percolation suffers. V. SITE PERCOLATION RE-DEFINED
The questions is: How can we resolve the problem withthe existing definition of site percolation? Recall thedefinition of bond percolation where we occupy bondsto connect the already existing sites in the system andmeasure cluster size by the number of contiguous sitesconnected by occupying bonds. We already know that itis consistent with the second law of thermodynamics andthe nature of the order parameter. Using the spirit ofthe bond percolation we can define the site percolationas follows. We assume that the bonds are already therein the system as an isolated entity and the occupationof sites connect these isolated bonds to form clusters ofbond. How do the old and new definitions differ? Ac-cording to the old definition, occupation of one, two andthree isolated empty sites forms a cluster of size one,two and three as shown in Fig. (2a) by the green color.On the other hand, according to the redefined site per-colation occupation of one, two and three empty sitesforms a cluster of size four, seven and ten respectivelywhich are shown in Fig. (2b). In the case of bond per-colation, on the other hand, occupation of one isolatedbond forms a cluster of size two, two consecutive isolatedbonds forms cluster of size three, three consecutive iso-
Occupation Probability, p E n t r o p y , H ( p , L ) L=200L=250L=300L=350L=400 (a)
Occupation Probability, p
Entropy, H(t)/H max
Order Parameter, P(t)/P max
Disordered Phasep
p c (b) FIG. 3: (a) Plots of entropy H versus p for redefined sitepercolation. Its qualitative behavior is now the same as forits bond counterpart. (b) Here we plot entropy H ( p ) /H max and order parameter P ( p ) /P max in the same graph to seethe contrast. It can be easily seen that P = 0 where en-tropy is maximally high and order parameter is maximallyhigh where entropy is minimally low which is reminiscent oforder-disorder transition in the ferromagnetic transition. lated bonds form cluster of size four and so on. Thus allthree definitions are clearly different.Using the redefined site percolation, we again measureentropy and find that it behaves just like its bond coun-terpart. That is, entropy is maximum at p = 0 and as p approaches p c it starts dropping sharply where the oc-cupation probability p is now defined as the fraction ofthe bonds being occupied. At above p c it then decreasesslowly to zero as p → H ( p ) /H (0) and order parameter P ( p ) /P (1) as a func-tion of p are shown in Fig. (3b). It clearly shows that H is maximally high where P = 0 and the order parameteris maximally high where H is minimally low. Moreover,we find entropy never decreases with 1 − p rather it eitherincreases or remain constant and hence both the issuesare resolved. VI. IS SITE-BOND UNIVERSALITY STILLVALID?
We now check if the re-defined site percolation stillgives the same critical point p c = 0 . ν of the correlation length or not. Thebest quantity for finding them is the spanning probabil-ity W ( p ). It describes the likelihood of finding a clusterthat spans across the entire system either horizontally orvertically at a given occupation probability p . In Fig.(4a), we show W ( p ) as a function of p for different sys-tem sizes L . One of the significant features of such plots Occupation Probabilit , p Sp a nn i n g P r o b a b ili t , w ( p , L ) L=200L=250L=300L=350L=400L=∞ (a) −2.0 −1.5 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 (p−p c )L Sp a nn i n g P r o b a b ili t y , w ( p , L ) L=200L=250L=300L=350L=400 (b)
FIG. 4: (a) Spanning probability W ( p, L ) vs p for differentlattice sizes using new definition of site percolation. In (b) weplot dimensionless quantities W vs ( p − p c ) L /ν using knownvalue of ν = 4 / Occupation Probability, p P e r c o l a t i o n S t r e n g t h , P ( p , L ) L=200L=250L=300L=350L=400 (a) −4 −3 −2 −1 0 1 2 3 4 (p−p c )L P L β / ν L=200L=250L=300L=350L=400 (b)
FIG. 5: (a) Order parameter P ( p, L ) vs p for re-defined sitepercolation in the square lattice. (b) We plot P ( p, L ) L β/ν versus ( p − p c ) L /ν using know value of ν = 4 / β = 5 / is that they all meet at one particular value. It is actu-ally the critical point p c = 0 . p c does not depend onwhether we measure the cluster size in terms of the num-ber of sites or the number of bond it contains. Note thatfinding the p c value for different skeletons is one of thecentral problems in percolation theory [24, 25]. To checkwhether the ν value is still the same we use its standardknown value ν = 4 / W ( p, L ) ∼ L a φ w (( p c − p ) L ν ) , (4)where φ w is the universal scaling function for spanningprobability [26]. We then plot of W ( p ) L − a vs ( p c − p ) L ν and find that all the distinct curves of Fig. (4a) collapseinto a universal scaling curve as shown in Fig. (4b) for ν = 4 / a = 0. This is a clear testament that thecritical point ν is also the same as that of the traditionalsite percolation and that W ( p ) behaves like a step func-tion in the thermodynamic limit.Next we attempt to find the critical exponent β of the Occupation Probability, p Sp e c i f i c H e a t , C ( p , L ) L=200L=250L=300L=350L=400 (a) −4 −2 0 2 4 (p−p c )L C L − α / ν L=200L=250L=300L=350L=400 (b)
FIG. 6: Specific heat C ( p, L ) vs p in square lattice for re-defined site percolation. In (b) we plot dimensionless quan-tities CL − α/ν vs ( p − p c ) L /ν and we find an excellent data-collapse. order parameter P using the new definition of site perco-lation. First, we plot order parameter P ( p ) in Fig. (5a)as a function of p for different lattice size L . We nowapply the finite-size scaling P ( p, L ) ∼ L − β/ν φ p (( p − p c ) L /ν ) , (5)where φ p is the universal scaling function for order pa-rameter. We now use the standard known values for ν = 4 / β/ν = 0 .
104 to check if the plot of P ( p, L ) L β/ν versus ( p − p c ) L /ν give data collapse ornot. Indeed, we find that all the distinct plots of Fig.(5a) collapse into a universal scaling curve as shown inFig. (5b). It implies that the new definition of site per-colation reproduces the same known value of β = 0 . d random percolation. Note that P ( p, L ) L β/ν and( p − p c ) L /ν are dimensionless quantities and hence wehave P ( p, L ) ∼ L − β/ν and ( p − p c ) ∼ L − /ν . Eliminating L from these relations we find P ∼ ( p − p c ) β . (6)This is exactly how the order parameter, such as magne-tization in the paramagnetic to ferromagnetic transition,behaves near the critical point.Knowing entropy paves the way for obtaining the spe-cific heat since we know that it is proportional to thefirst derivative of entropy i.e. C = T dS/dT where S isthe thermal entropy. If we now know the exact equivalentcounterpart of temperature then we can immediately ob-tain the specific heat for percolation. In our recent workwe argued that 1 − p is the equivalent counterpart oftemperature and hence the specific heat for percolationis C ( p ) = (1 − p ) dHd (1 − p ) . (7)The plots of C ( p ) as a function of p for different systemsizes L is shown in Fig. (6a). Let us assume that it obeysthe finite-size scaling C ( p, L ) ∼ L α/ν φ c (( p − p c ) L /ν ) , (8) Occupation Probability, p S u s c e p t i b ili t y , χ ( p , L ) L=200L=250L=300L=350L=400 (a) −3 −2 −1 0 1 2 3 (p−p c )L χ L − γ / ν L=200L=250L=300L=350L=400 (b)
FIG. 7: (a) Plots of susceptibility χ ( p ) for redefined site per-colation as a function of p in square lattice of different sizes.In (b) we plot dimensionless quantities χL − γ/ν vs ( p − p c ) L /ν and we find an excellent data-collapse with γ = 0 .
853 whichis the same as for bond type. where φ C is the scaling function. We already know thevalue of α = 0 .
906 from our recent work on bond perco-lation in the square lattice [8]. Using the same values for α and ν we now plot CL − α/ν vs ( p − p c ) L /ν and find anexcellent data collapse as shown in Fig. (6b). It confirmsthat α = 0 .
906 is indeed the same for both bond andredefined site percolation. According to Eq. (7) we have C ∼ L α/ν at p = p c . Once again eliminating L from itusing L ∼ ( p − p c ) − ν we find C ∼ ( p − p c ) − α , (9)which implies that the specific heat diverges at the criti-cal point. This is one of the important attributes of thesecond order phase transitions. Divergence means thatthe thermodynamic property suffers a discontinuity andchanges drastically from its value in the disordered state.It is qualitatively related to the emergence of long rangeorder. The specific heat has so far been defined as thesecond derivative of the number of clusters n ( p ) with re-spect to p . Using this definition one find α = − / α = 0 . n ( p ) is the free energy and hence its firstderivative should have been the entropy. However, onedo not get expected entropy from this.In percolation, yet another quantity of interest is thesusceptibility. Traditionally, mean cluster size has beenregarded as the equivalent counterpart of susceptibility.Sometimes variance of the order parameter h P i − h P i too is regarded as susceptibility. Neither of the twoactually gives respectable value for γ to obey the Rus-brooke inequality. Recently, we have proposed suscepti-bility χ ( p, L ) for percolation as the ratio of the change inthe order parameter ∆ P and the magnitude of the timeinterval ∆ p during which the change ∆ P occurs i.e., χ ( p, L ) = ∆ P ∆ p . (10). Essentially it becomes the derivative of the order pa- rameter P since ∆ p → N → ∞ as∆ p = L . The idea of jump has been studied first byManna in the context of explosive percolation [27]. Theresulting susceptibility is shown in Fig. (7a) as a func-tion of p . We already know that γ/ν = 0 . χL − γ/ν vs( p − p c ) L /ν we find that all the distinct curves in Fig.(7a) collapse superbly. It confirms that the susceptibilityobeys the finite-size scaling χ ∼ L γ/ν φ χ (( p − p c ) L /ν ) . (11)It once again suggest that susceptibility diverges near thecritical point following a power-law χ ∼ ( p − p c ) − γ , (12)with γ = 0 . γ value. VII. CONCLUSIONS
In this article we first discussed entropy for bond per-colation and we measured it as a function of p . We havefound that it is consistent with the behaviour of the orderparameter and with the second law of thermodynamics.Essentially, entropy measures the degree of disorder whileorder parameter measures the extent of order. Thus,both cannot be minimum or maximum at the same statesince the system cannot be in the most disordered and inthe most ordered state at the same time. We have thenmeasured entropy and order parameter for site percola-tion using its existing definition. We have found that at p = 0 both order parameter and entropy equal to zero,which is contradictory. Besides, we have found that en-tropy first increases with 1 − p and then decreases again.This is a clear violation of the fact entropy can either in-crease or remain constant with temperature but cannotdecrease. We have therefore redefined the site percola-tion as follows. We occupy sites to connect bonds whichare assumed to exist already in the system and measurecluster sizes in terms of the number of contiguous bondsconnected by occupied sites.With the new definition we have found that the en-tropy behaves exactly in the same way as it does in thecase of its bond counterpart. Thus the conflict that thesystem is in ordered and disordered state at the sametime is resolved and it obeys the second law of thermo-dynamics too. One of the most important well-knownresults is that site and bond type percolation in a given d dimensional lattice belong to the same universality classregardless of the detailed nature of the structure of thelattice. We have shown that this is still true despite theold and new definitions are distinctly different. It provesthat changing local rules on a regular lattice does notchange the universality class of the system.Note that occupation of one isolated site forms a clus-ter of size four according to new definition while accord-ing to old definition it forms a cluster of size one only. Inthis sense the bond percolation is also different as we find that occupation of one isolated bond forms a cluster ofsize two. We hope the present work will have significantimpact in the future research of percolation theory. [1] D. Stauffer and A. Aharony, Introduction to PercolationTheory (Taylor & Francis, London, 1994).[2] A. A. Saberi, Phys. Rep.
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