# critical behavior for mixed site-bond directed percolation[精选推荐pdf]

arXiv:cond-mat/9505019v1 4 May 1995Critical behavior for mixed site-bond directed percolationA.Yu. Tretyakov School of Information Sciences, Tohoku University Aramaki, Aoba-ku, Sendai, 980-77, JAPANN. Inui Himeji Institute of Technology 2167, Shosha, Himeji, 671-22, JAPANFebruary 1, 2008AbstractWe study mixed site-bond directed percolation on 2D and 3D latticesby using time-dependent simulations. Our results are compared with rig-orous bounds recently obtained by Liggett and by Katori and Tsukahara.The critical fractions pcsiteand pcbondof sites and bonds are extremely wellapproximated by a relationship reported earlier for isotropic percolation,(logpcsite/log pc∗ site+ logpcbond/logpc∗ bond= 1), where pc∗ siteand pc∗ bondare thecritical fractions in pure site and bond directed percolation.classification numbers: 0250,0520,6460,64901The mixed site-bond percolation is a natural extension of the ordinary bondpercolation and site percolation models. The model is defined by opening sites of a lattice with probability psiteand bonds with probability pbond, clusters are defined as combinations of open sites connected by open bonds. The system is percolating in a region of psiteand pbondvalues in which an infinite cluster exists. The mixed site-bond percolation is a natural model for physical phenomena in randomly restricted media (accounted for by closed sites) with random iterations (accounted for by open and closed bonds). Introduced by Reynolds et. al. [1], it was used in treating polymer gelation, capillary phenomena and in fracture theory (see [2] and references in it) . Current work is concerned with the directed version of the mixed site-bondpercolation model (Fig. 1), first introduced by Kinzel [3, 4], in which bonds can be open only in a certain direction, physically it can be accounted for by the existenceof an external field, restricting interactions allowed in the system (as in the case oftransport of particles in porous media, induced by an external electrical field), or it could also be applied to time-dependent phenomena, when movement in a chosen direction corresponds to development in time (see [4, 5, 6]). Both pure bond and pure site directed percolation are known to belong to the same universality class known as DP (Directed Percolation) universality class. The DP universality class is known to be very robust; with a few exceptions all dynamical particle systems involving extinction-survival transition belong to DP universality class (see [7, 8] and references in it). The mixed directed side-bond percolation should be expected to belong to the DP universality, although no nu-merical check to confirm it has ever been conducted. Recently Katori and Tsukahara [9] proposed a lower boundary for the phase transition line formed by psiteand pbondvalues, at which the percolation transition is taking place in the case of mixed directed percolation in 1+1 dimensions. On the other hand, exact upper and lower bounds for the phase transition line in 1+1 dimensions have been recently proposed by Liggett [10]. Based on a heuristic argument, Yanuka and Englman [2] proposed an inter- polation formula for the phase transition line for ordinary (non-directed) mixed2site-bond percolation in any dimension, allowing to calculate the phase diagram for the mixed percolation from the critical fractions pc∗ siteand pc∗ bondfor pure site and pure bond percolation, corresponding to the end-points of the diagram. Because of the random spread on data, the Monte Carlo results for the intermediate region reported by the same authors (for a number of lattices in 2D and 3D) don’t allowto see any systematic difference from the phase transition line given by the formula, so that the precision of the critical line given by the formula is unclear. The principal aim of the current work is the evaluation of the phase transition line for the directed mixed site-bond percolation in 1+1 and in 2+1 dimensions using Monte Carlo simulations.For 1+1 dimensions we confirm the validity of upper and lower bounds suggested by Katori and Tsukahara [9] and by Liggett [10], and, for both 1+1 and 2+1 dimensions, show that an interpolation formula, similar to the one suggested by Yanuka and Englman [2] for the ordinary mixed site-bond percolation, is applicable with high precision, although we are able to detect small systematic deviations. The time-dependent simulation approach used in the current work allows, simultaneously with the transition line, the evaluation of dynamical critical exponents for the mixed directed percolation. Our results imply that, not surprisingly, the mixed case belongs to the same universality class to which both pure bond and pure site directed percolation belong. To decide the critical line, we use a time-dependent simulation [11]. Mixed directed percolation is regarded as a growth process, starting from a single site, the“origin”. For 1+1 dimensions, space and time are defined as coordinates of sites on a square lattice perpendicular and parallel to the (1,1) diagonal, and growth clusters are formed by sites that can be reached from the origin moving through open sites and bonds in the positive time direction (see Fig. 1). Similarly, for 2+1 case we consider growth on a simple cubic lattice with time axis directed along the (1,1,1) diagonal. This kind of growth can be realized by updating states of sites of a triangular lattice (see [12] for a discussion of the technique). We calculated the survival probability P(t) as the fraction of realizations containing sites connected to the origin at time t (realizations, surviving at time t); the average number of sites connected to the origin at time t, n(t), averaged over all realizations; and the3average square radius R2(t), averaged over surviving realizations. On the critical line, for big enough t, it is expected that these quantities are governed by power lawsP(t)∝t−δ n(t)∝tη R2(t)∝tz(1)The results of the time-dependent simulation are most conveniently represented as local slopes data (see [11] for a detailed description), for example, for the number of sites connected to the origin we plot η(t) ≡ ln[n(t)/n(t/m)]/ln[m] against 1/t, taking m=7. For psiteand pbondon the critical line the true η value is given by theintersection of the local slopes curve with ordinata, while in off-critical simulation the local slopes curves diverge for small 1/t. Survival probability P and the mean square radius R are treated in a similar way. Although the local slopes curves for R are quite insensitive to a change in psiteand pbondvalues and thus less useful in determining the critical line than in the cases of n and P, we used the mean square radius data to check the consistency of our simulation and to estimate the z values. An example of local slopes curves for time dependent simulation is given in Fig. 2. The data corresponds to pure site directed percolation, and allows us to support the critical point value provided by Onody and Neves [13] using series technique as opposed to the result obtained by D.ben-Avraham et.al. [14] using the transfer matrix technique. We only show the local slopes curves for the mean number of sites connected to the origin. Here, and in the rest of the current work, we take 1000 steps and we average over 105configurations. The result strongly implies that the the critical value obtained by Onody and Neves is correct, while the valueprovided by D.ben-Avraham et.al. [14] seems to be offthe critical point by more than the error margin, given in [14]. The overall results are given in Tab. 1 and in Tab. 2, all pcbondvalues given have an uncertainty of ±0.0002. The uncertainty has been determined by making sure that the local slopes curves for n and P, taken at pbondvalue, shifted from the one provided in Tab. 1 and Tab.2 by the value of the error margin, clearly diverge as 1/t approaches zero.4Repeating the argument proposed by Yanuka and Englman [2] in the case of mixed directed percolation, we havelogpcsite logpc∗ site+logpcbond logpc∗ bond= 1,(2)where logpc∗ siteand logpc∗ bondare the critical fractions of bonds and sites in pure bond and site directed percolation. We compare the critical line given by (2) with our Monte Carlo results in Fig. 3 and Fig.4.The formula works surprisingly well, although if the difference between the interpolation formula prediction and the simulation results is plotted separately, as is done in Fig. 5, one can see that it can not be explained by theflaws of the Monte Carlo simulation. Error bars, given in Fig. 5, are calculated as a combination of the uncertainty of the interpolation formula prediction due to the uncertainty of the threshold values for pure DP cases and of the error in pbondvalues, obtained in our simulations. The critical values for pure DP cases were taken as pc∗ site= 0.705489±0.000004 [13] and pc∗ bond= 0.644701±0.000001 [15] in 1+1 dimensions, and for 2+1 dimensions we used the values obtained by [12], pc∗ site= 0.43525 ± 0.00013 and pc∗ bond= 0.38216 ± 0.00006. Apart from the interpolation formula, in Fig. 3 we compare the Monte Carlo results with exact boundaries for 1+1 mixed directed percolation, proposed by Katori and Tsukahara and by Liggett. Clearly, our results are in agreement with the bounds, presumed to be exact. Concerning the universality of mixed site-bond directed percolation, both in 1+1 dimensions and in 2+1 dimensions, the dynamical exponents δ, η and t, ob- tained in our time-dependent simulations, were in agreement with the known DP values. For 1+1 dimensions, taking the DP exponents values given in [7], the error margins, estimated by analyzing the spread of the local slopes data, were given as: δ = 0.162 ± 0.005, η = 0.308 ± 0.005 and z = 1.263 ± 0.010. For 2+1 dimen- sions, using the DP exponents values from [12], the error margins were given as δ = 0.460 ± 0.020, η = 0.214 ± 0.025 and z = 1.134 ± 0.010. Thus, our results indicate that mixed directed percolation in 1+1 dimensions and in 2+1 dimensions belongs to the corresponding DP universality classes.5Concluding the results of our work, we would like to note that the interpolation formula suggested by Yanuka and Englman [2] for the ordinary mixed site-bond percolation works surprisingly accurately in the case of the directed mixed site-bond percolation as well. Nonetheless, the accuracy of our Monte-Carlo results allows usto show a systematic difference. Our results in 1+1 dimensions are in agreement with exact bounds suggested by Katori and Tsukahara [9] and by Liggett [10]. If the interpolation formula could be proven to be an upper boundary for the critical line as our results suggest, it would be a much closer bound than the Liggett’s bound, the best available so far.6References[1] Reynolds P J, Klein W and Stanley H E 1977 J. Phys. C: Solid State Phys. 10 L167-L172[2] Yanuka M and Englman R 1990 J. Phys. A: Math. Gen. 23 L339-L345[3] Kinzel W. 1985 Z. Phys. B58 229-244[4] Kinzel W and Yeomans J M 1981 J. Phys. A: Math. Gen. 14 L163-L168[5] Durrett R 1988 Lecture Notes on Particle Systems and Percolation (California: Wadsworth&Brooks)[6] Inui N, Katori M and Uzawa T Is to appear in J. Phys. A[7] Grassberger P and de la Torre A 1979 Ann. Phys. (NY) 122 373-396[8] Dickman R 1993 Int. J. Mod. Phys. C4 271-277[9] Katori M and Tsukahara H Submitted for publicaton to J. Phys. A[10] Liggett T M Submitted for publication[11] Jensen I 1993 Phys. Rev. Lett. 70 1465-1468[12] Grassberger P 1989 J. Phys. A: Math. Gen. 22 3673-3679[13] Onody R N and Neves U P C 1992 J. Phys A: Math. Gen. 25 6609-6615[14] ben-Avraham D, Bideaux R and Schulman L S 1991 Phys. Rev. A 43 7093- 7096[15] Baxter R J and Guttmann A J 1988 J. Phys. A 21, 3193-32047Figure captions1. Mixed directed site-bond percolation in 1+1 dimensions. Percolation is al- lowed in directions indicated by the arrows.One can clearly see a finite cluster, consisting of 9 sites, connected to the origin.2. 1+1 dimensions, local slopes for the mean number of sites connected to theorigin at different values of psite, pbond= 1 for all curves. Line shows the DP value of the exponent η, taken from [7]. 1- psite= 0.706522 (critical value according to [14]), 2- psite= 0.7057, 3- psite= 0.705489 (critical value according to [13]), 4- psite= 0.7053, 5- psite= 0.7051.3. Critical line in 1+1 dimensions. Monte Carlo results are shown as points. 1- The critical line given by Eq. 2. 2- The lower bound by Katori and Tsukahara [9], given as a solution of α3β4− αβ2+ 2αβ = 1 with α = pcsiteand β = pcbond. 3- The upper and lower bounds by Liggett [10], given by α =2+β 4βand α =2 β(4−β), respectively.4. Critical line in 2+1 dimensions, for a simple cubic lattice.5. The difference between the predictions of Eq. 2 and the Monte Carlo results. Crosses indicate the uncertainty. a)1+1 dimensions. b)2+1 dimensions, simple cubic lattice.8Table captions1. Critical line in 1+1 dimensions, Monte Carlo results.∗Values for ordinary directed percolation are taken from [15] and [13].2. Critical line in 2+1 dimensions, for a simple cubic lattice.Monte Carlo results.∗Values for ordinary directed percolation are taken from [12].9Table 1pcbondpcsite 0.644701∗1 0.70.93585 0.750.88565 0.80.84135 0.850.80190 0.90.76645 0.950.73450 10.705489∗Table 2pcbondpcsite 0.38216∗1 0.450.86410 0.50.78725 0.60.67115 0.70.58770 0.80.52460 0.90.47510 10.43525∗10