On Entropy, Specific Heat, Susceptibility and Rushbrooke
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M. and cha n een ucpiiiyfor susceptibility redefine and heat fic gt ieetuieslt classes. universality different to ng ≈ H γ s pcfi et Padsusceptibil- and OP heat, specific es, β P stohg ooe I elzn hs draw- these Realizing RI. obey to high too is efidthat find we 0 + stertemlcutrat We counterpart. thermal their as ssgicnl ihadi h te phase other the in and high significantly is encutrsz stessetblt,the susceptibility, the is size cluster mean , d,Foia386 USA 32816, Florida ndo, γ ≥ p .Teraaousi per- in analogues Their 2. − H p and qaiyi percolation in equality c ihlna size linear with gladesh ξ P P p 1 t.Hwvr esild not do still we However, etc. c ohcno eextremely be cannot both ssgicnl iha ex- as high significantly is H twihteeapasa appears there which at q eso httere- the that show We . saePtsmdlonto model Potts -state H p c esrstedegree the measures sfudt erem- be to found is L ftelattice the of P ,we 0, = γ still ǫ i 2 of disorder and P measures the extent of order. Second, 12 12 L=150 t=8000 L=200 t=10000 we define specific heat and find positive critical expo- 10 L=250 10 t=12000 L=300 t=20000 nent α for both the lattices, which is in sharp contrast 8 8 to the existing value α = −2/3 for square lattice. We 6 6 also redefine susceptibility and show that it diverges at Entropy, H 4 Entropy, H 4 2 2 the critical point without having to exclude the spanning 0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 cluster. Besides, we also obtain its critical exponent γ 1-p 1-p and find that it is significantly smaller than the existing (a) (b) known value. The values of α and γ re-affirm our ear- lier findings that percolation on square and WPS lattices FIG. 1: Change in Shannon entropy H with 1 − p for (a) belong to two distinct universality classes although they square and (b) WPS lattices. The sharp rise in H occurs near q = 0.5 and q = 0.6543 for (a) and (b) respectively. are embedded in the same spatial dimension [12]. Finally, c c we find that the elusive RI holds in random percolation regardless of whether it is on square or on WPS lattices. convolution relation
M M − We find it worthwhile first to discuss the construction X(p)= pn(1 − p)M nX , (1) n n process of WPS lattice which we proposed in 2010 [13]. nX=1 It starts with a square of unit area which we call an ini- tiator. Then, in the first step, we divide the initiator ran- to obtain X as a function of p that helps obtaining a domly into four smaller blocks. In the second step and smooth curve for X(p). thereafter, only one of the blocks at each step is picked Phase transitions always entails a change in entropy from all the available blocks preferentially according to in the system regardless of whether the transition is first their respective areas and divide that randomly into four or second order in character. Thus, a model for CPT is smaller blocks. The details of the algorithm and image not complete without a proper definition of entropy. We of the lattice can be found in [13, 14]. Percolation on find that the Shannon entropy is the most appropriate such a lattice has already shown unique results [12]. For one for percolation as it is too probabilistic in nature. In instance, it is well-known that random percolation on general, it is defined as all lattice, regardless of the type of percolation and the m structural difference of the lattice, share the same set of H = −K µ log µ , (2) vis-a-vis i i critical exponents belong to the same universal- Xi ity class provided they share the same dimension. How- ever, we have recently shown that percolation on WPS where we set the constant K = 1 since it merely amounts lattice does not belong to the universality class of all the to a choice of a unit of measure of entropy [16]. In known two dimensional lattices [12]. information theory, it is a common practice to choose K = 1/ log 2. Although there is no explicit restriction per se on the choice of µi for Shannon entropy but when To study percolation, we use the Newman-Ziff (NZ) al- we use it to describe entropy for phase transition there gorithm which, apart from being the most efficient one, ought to have some implicit restrictions. We all know has also many other advantages [15]. For instance, it from thermodynamics and statistical mechanics that G helps calculating various observable quantities over the vs T should be a concave curve with negative slope to en- entire range of p in every realization instead of measur- sure that entropy S ≥ 0. On the other hand, the slope of ing them for a fixed probability p in each realization. the S vs T plot must have a sigmoid shape with positive According to the NZ algorithm, all the labeled bonds slope since C ≥ 0 and according to second law of thermo- i = 1, 2, 3, ..., M are first randomized and then arranged dynamics ∆S ≥ 0 [2]. The entropy for percolation too in the order in which they will be occupied. Using the must have the same generic features. periodic boundary condition we get M =2L2 for square The key question in percolation is not whether we can lattice and M ∼ 8t for WPS lattice where t is the time measure Shannon entropy or not. Rather, faced with a step. One advantage of using NZ algorithm is that we number of different normalized probabilities, the ques- can create percolation states consisting of n + 1 occupied tion is, can we use them? Earlier, some authors used ws, bonds simply by occupying one more bond to its imme- the probability that a site picked at random belongs to diate past state consisting of n occupied bonds. Initially, a cluster exactly of size s, to measure entropy using Eq. there are L2 and 3t + 1 clusters of size one in the square (2) and they both found a bell-shaped curve with peaks and WPS lattices respectively. Occupying the first bond at around tc [17, 18]. It implies that the system is in means forming a cluster of size two. Each time there- the most ordered state at p = 0 since entropy is mini- after, either the size of an existing cluster grows due to mally low there and at the same time it is in the most occupation of inter-cluster bond or remains the same due disordered state since the order parameter P is also zero to occupation of intra-cluster bond. We calculate an ob- there and hence we see a contradiction. The problem lies servable, say Xn, as a function of n and use it in the in using ws is that the sum in Eq. (2) is not over cluster 3 size s rather it is over cluster label i so that it measures 160 70 L=150 t=8000 140 L=200 t=10000 the amount of information conveyed by each cluster not L=250 60 t=12000 L=300 t=20000 120 50 by a class or group of cluster of size s. Thus, we have 100 40 80 to choose a probability that contains information about 30 60 20 individual clusters. To find the appropriate probability 40 Specific Heat, C Specific Heat, C 20 10 for µi of Eq. (2), we assume that for a given p there 0 0 0.4 0.42 0.44 0.46 0.48 0.5 0.52 0.54 0.56 0.58 0.6 0.2 0.25 0.3 0.35 0.4 0.45 0.5 are m distinct, disjoint, and indivisible labeled clusters Occupation Probability, p Occupation Probability, p i =1, 2, ..., m of size s1,s2, ...., sm respectively. We then (a) (b) propose the labeled picking probability (CPP) µi, that 4.4 4.5 a site picked at random belongs to cluster i. The most wps WPS square 4 SQUARE 4.3 generic choice for CPP would be µ ∝ s so that the 3.5 i i 4.2 3 ν ) m m / h 4.1 α 2.5 probability µ (p) = s / s where s = N is - i i j=1 j j=1 j 2
4 C L the normalization factor. ln(C 1.5 P P 3.9 1 Substituting µi(p)= si/N in Eq. (2) we first find mi- 3.8 0.5 0 3.7 -4 -2 0 2 4 5 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 ν crocanonical ensemble average of entropy Hn as a func- (p - p )L1/ ln(L) c tion of n from 10, 000 independent realizations. Then we (c) (d) use it in Eq. (1) to get the canonical ensemble average of H(p) as a function of p. In Figs. 1a and 1b we plot H(p) FIG. 2: Specific heat C(p) vs p for (a) square and (b) WPS versus q =1 − p and find that the curve has the desired lattices. (c) Slope of the plots of log(Ch) versus log(L) gives sigmoidal shape. We see that the entropy is maximum α/ν = 0.68(1) and α/ν = 0.5007(3) for square and WPS −α/ν 1/ν H = log(N) at q = 1 where µi = 1/N ∀ i as there are lattices respectively. (d) Plots of C(p)L vs (p−pc)L of N clusters of equal size (one). This is exactly like the the same data as in (a) and (b) shows excellent data-collapse. state of an isolated ideal gas where all microstates are equally likely and hence q = 1 corresponds to the most disordered state and it is consistent with the fact that with slopes α/ν = 0.68(1) for square lattice and α/ν = − P = 0 there. As we lower the q value from q = 1 we 0.5007(3) for WPS lattice. Note that CL α/ν and (p − 1/ν see H decreases slowly, however, as q approaches qc we pc)L are dimensionless quantities and hence if we now −α/ν 1/ν observe a sudden drops in entropy. This is because in the plot CL as a function of (p−pc)L then the distinct vicinity of pc coagulation of two moderately large clus- plots of C vs p should collapse onto a single universal ter happens so frequently that we already see a sign of curve φc. Indeed, Fig. (2d) shows that all the distinct the emergence of would spanning cluster that it causes a plots of Figs. (2a) and (2b) are collapse superbly into sharp rise of CPP which eventually becomes the spanning their own universal curve revealing that if we had data for −ν α/ν cluster. Already at qc the incipient spanning cluster be- infinite Using the relation L ∼ (p−pc) in C(p) ∼ L comes so prevailing that it embodies almost all the sites. we find that the specific heat diverges Lowering q further to below q , we see that the extent c ∼ − −α of uncertainty diminishes sharply. Exactly at q = 0, we C(p) (p pc) , (5) have µ1 = 1 and hence entropy H = 0. This situation is with α = 0.906(13) for square lattice and α = 0.816(4) like perfect crystal as the extent of uncertainty, which is for WPS lattice. Our value of α for square lattice is in synonymous to degree of disorder, completely ceases to sharp contrast to the existing value α = −2/3 [19]. Note exist. The term percolation thus refers to the transition that the negative value of α means specific heat does not across qc from ordered phase at low-q to disordered phase diverge at the critical point rather it behaves more like at high-q exactly like ferromagnetic transition. order parameter. To find the specific heat C(p) for percolation we just Recall that the order parameter P of percolation is de- need to use its thermodynamic definition C = T dS and dT fined as the the ratio of the size of the largest cluster replace S by H and T by 1 − p to obtain smax to the lattice size N. We then keep record of the dH successive jump size in P i.e., ∆P within the successive C(p)=(1 − p) . (3) interval ∆p = 1 where M is the total number of bonds. d(1 − p) M The idea of successive jump size in P was first studied by Manna in the context of explosive percolation [20]. We, In Figs. 2a and 2b we plot it for both the lattices as a ∆P function of p for different system sizes. Now following however, here consider the ratio ∆p and define it as the the finite-size scaling (FSS) hypothesis we can write susceptibility χ(p). In Figs. (3a) and (3b) we plot χ(p) as a function of p for both types of lattice and find that α/ν 1/ν C(p,L) ∼ L φC ((p − pc)L ), (4) χ(p) grows as we increase p and as we approach pc the growth is quite steep. However, beyond pc it decreases where φC is the scaling function for specific heat. To sharply without excluding the size of the spanning clus- find an estimate for the exponent α/ν, we measure the ter. This is in sharp contrast to the existing definition height of the peak Ch at pc as a function of L. Plotting of susceptibility as we know that the mean cluster size log(Ch) versus log(L) we get straight lines, see Fig. (2c), decreases beyond pc only if the spanning cluster size is 4
To check whether the critical exponents α, β and γ 16 25 L=150 t=12000 14 L=200 t=20000 L=250 t=30000 obey the Rushbrooke inequality we substitute their val- L=300 20 t=40000 12 ues in the Rushbrooke relation and the results are shown 10 15
χ 8 χ in table I. It clearly suggests that Rushbrooke inequality 6 10
4 5 holds in percolation. However, the point to emphasize is 2
0 0 that it obeys rather more as equality, within the limits 0.4 0.42 0.44 0.46 0.48 0.5 0.52 0.54 0.56 0.58 0.6 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Occupation Probability, p Occupation Probability, p of error, than as a inequality. Many experiments and ex- (a) (b) actly solvable models of thermal CPT too suggest that the Rushbrooke inequality actually holds as an equality 3.2 1.8 wps WPS 3.1 square 1.6 SQUARE [21]. Through this work we show that the entropy and 3 1.4 2.9 1.2 the order parameter complements each other exactly like ν / )
2.8 γ 1 - h χ 2.7 L 0.8 χ in the thermal counterpart. For instance, we find that
ln( 2.6 0.6 2.5 0.4 the order parameter P , which quantifies the degree of 2.4 0.2 2.3 0 ≥ -3 -2 -1 0 1 2 3 order, is equal to zero for q qc but entropy, which 2.2 1/ν 5 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 (p - pc) L ln(L) measures the degree of order, is significantly high reveal- (c) (d) ing the high-q is the disordered phase exactly like high- T phase in the ferromagnetic transition. On the other FIG. 3: Plots of χ vs p for (a) square and (b) WPS lat- hand, at q < qc we find that the entropy H is negligibly tices where data represents convolution of ensemble average of small where P increases with decreasing q to its maxi- 10, 000 independent realizations. (c) The slopes of the plots of mum value at q = 0 revealing that low-q is the ordered log(χh) vs log(L) gives γ/ν = 0.635(2) and γ/ν = 0.460(1) for phase which is once again like low-T phase in the fer- −γ/ν square and WPS lattices respectively. In (d) we plot χL romagnetic transition. It all implies that percolation is 1/ν vs (p − pc)L and find the same data as in (a) and (b) indeed an order-disorder transition. collapse into their respective universal curves.
To summarize, in this article we proposed the equiv- excluded. To find the exponent γ, we apply the FSS hy- alent counterpart of entropy and specific heat and re- pothesis. Following the same procedure done for C(p), we defined the susceptibility for percolation model. To mea- find γ/ν =0.635(2) for square lattice and γ/ν =0.460(1) sure entropy for percolation we proposed cluster picking − for WPS lattice (see Fig. (3c)). Plotting χL γ/ν versus probability (CPP) and shown that the Shannon entropy 1/ν (p − pc)L we find that all the distinct plots of Figs. for CPP has the same generic feature as the thermal en- (3a) and (3b) collapse into their own distinct universal tropy. Until now, we could only quantify the extent order curve as shown in Fig. (3d). It implies that of the ordered phase by measuring P and we could say nothing about the other phase since P = 0 there. Now, −γ χ(p) ∼ (p − pc) , (6) having known entropy, we can also quantify the other phase and regard it as the disordered phase as entropy is where γ =0.846(2) and γ =0.750(6) for square and WPS high and always keep increasing with 1−p. We have also lattices respectively. It suggests two important develop- shown that specific heat diverges with positive and sus- ments. Thus, we find that the susceptibility diverges at ceptibility both exhibit power-law near pc and divergence the critical point without having to exclude the spanning at pc without having to exclude the spanning cluster from cluster. Moreover, the value of the critical exponent γ is their calculations. Using the FSS hypothesis and the idea far too less than what we find from the mean cluster size. of data-collapse, we numerically obtained the critical ex- Lattice α β γ α + 2β + γ ponents α and γ for both the lattices. Their distinct Square 0.906 0.1388 0.846 2.029 values for the two lattices once again confirm that they WPS 0.816 0.222 0.750 2.01 belong to two different universality classes. Finally, we showed that the Rushbrooke inequality holds in percola- TABLE I: The critical exponents and Rushbrooke inequality tion on both the lattices albeit they belong to two distinct for random percolation on square and WPS lattices. universality classes.
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