Elastic backbone defines a new transition in the percolation model
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1 APS Elastic backbone defines a new transition in the ercolation model Cesar I. N. Samaio Filho 1, José S. Andrade Jr. 1,2, Hans J. Herrmann 1,2, André A. Moreira 1 1 Deartamento de Física, Universidade Federal do Ceará, Fortaleza, Ceará, Brasil 2 Comutational Physics for Engineering Materials, IfB, ETH Zurich, Schafmattstrasse, 809 Zurich, Switzerland The elastic backbone is the set of all shortest aths. We found a new hase transition at eb above the classical ercolation threshold at which the elastic backbone becomes comact. At this transition in 2d its fractal dimension is 1.70±0.00, and one obtains a novel set of critical exonents β eb = 0.0 ± 0.02, = 1.97 ± 0.0, and ν eb = 2.0 ± 0.02 fulfilling consistent critical scaling laws. Interestingly, however, the hyerscaling relation is violated. Using Binder s cumulant, we determine, with high recision, the critical robabilities eb for the triangular and tilted square lattice for site and bond ercolation. This transition describes a sudden rigidification as a function of density when stretching a damaged tissue. PACS numbers:.0.ah,.0.al, 0.0.+q, 89.7.Da Desite being one of the simlest and most studied models, classical ercolation [1 ] still bears yet uncovered surrises. It is well known that at the ercolation threshold ( c ) the shortest ath between two oosite sides of the system is fractal, with a fractal dimension that is only known numerically to be d s = 1.107±0.000 [ 8] in two dimension and increases with dimension, until becoming two at and above the critical dimension d =. In fact the shortest ath is not unique: several shortest aths can exist simultaneously and the set of all shortest aths has been called elastic backbone in the ast [9], because it is the subset of the backbone that, when elongated, would give the first contribution to a restoring force. The elastic backbone indeed determines the first resistance that is felt, when stretching damaged [10 12] or biological tissues [1 18] and thus has exerimental relevance. It has been numerically established that at the ercolation threshold c its fractal dimension is indistinguishable from the one of the shortest ath [9]. Here we reort on the discovery that, above the classical ercolation threshold c, there exists another critical robability eb > c at which the elastic backbone becomes comact. While its dimension is known to be d s at c, our results in two dimensions show that it becomes unity between c and eb, it is 1.70 ± 0.00 at eb, and is equal to two above. At eb we reveal critical scaling laws and a set of exonents, however, violating hyerscaling. We simulated two-dimensional ercolation configurations at occuation robability for systems of size with eriodic boundary conditions in horizontal direction and oen boundaries at to and bottom. Using the burning algorithm, we identifieor each configuration the elastic backbone, i.e. the set of all shortest aths [9]. We considered site and bond ercolation on the triangular lattice and on two tyes of square lattices, namely, tilted and non-tilted. In Fig. 1 we see the elastic back- Corresondence to: cesar@fisica.ufc.br (a) (b) (c) FIG. 1: Shown in red are tyical elastic backbones obtained for site ercolation on a tilted square lattice of linear size and calculated with: (a) = ( c < < eb ), (b) = 0.70 ( = eb ), and (c) = ( > eb ). Occuied sites that are in the sanning cluster, but do not belong to the elastic backbone, are shown in blue. Other sites are reresented in white. bones for site ercolation on a tilted square lattice for three different robabilities below, at and above eb. In Fig. 2 we lot m = M eb /N, i.e. the robability that a site belongs to the elastic backbone against > c for site ercolation on the triangular lattice anind that there exists a clear hase transition at a value eb > c at which the elastic backbone becomes comact, with
2 () () β eb /ν eb M eb d = d = FIG. 2: Probability () that a site belongs to the elastic backbone as a function of the occuation robability for site ercolation on the triangular lattice for different system sizes. Inset: determination of the threshold using finite-size scaling. ln(1-1.u (q)) eb () FIG. : Binder s cumulant U () for different sizes as a function of the occuation robability for site ercolation on the triangular lattice. Inset: The deendence of eb () on 1 for site ercolation on the triangular lattice. eb () (circles) is obtainerom the crossing oint of U and U 2 for each air of successive values. Here we consider =, 128, 2, 12, 102, 208, 09, and Extraolating through the data oints (blue line) to the thermodynamic limit, we obtain eb ( ) = 0.70 ± (red dashed line). m acting as order arameter of this transition. Using finite size scaling, eb can be determined more recisely as shown in the inset, yielding also an estimate for the exonent β eb /ν eb = 0.2±0.02. The threshold eb can be determined even more recisely using Binder s cumulant FIG. : ogarithmic lot of the mass M eb of the elastic backbone as a function of the lattice size for site ercolation at eb on the tilted square lattice (orange circles) and on the triangular lattice (blue squares). The same is shown for the value of = 0., between c and eb, on the tilted square lattice for site ercolation (red stars) anor bond ercolation (green triangles), with ranging from to 18 sites. At = eb, the least-squares fit to the data of a ower law, M eb, gives the exonent 0 ± for the tilted square lattice (black line) and 0 ± for the triangular lattice (blue line). At = 0. on the tilted square lattice, the least-squares fit to the data of a ower law, M eb d, gives the exonent d = 00 ± for site ercolation (red line) and d = 00 ± for bond ercolation (green line). In all cases, the errors are smaller than the symbols. defined as m eb U () = 1 (1) m 2 eb 2 In Fig. we show the analysis of Binder s cumulant for site ercolation on the triangular lattice, where we obtain eb = 0.70 ± Moreover, by alying the same analysis, we finor site ercolation eb = 1 on the normal square lattice and eb = 0.70 ± on the tilted square lattice. For bond ercolation, we obtain eb = ± on the tilted square lattice and eb = 0.0 ± on the triangular lattice. At the threshold eb, we find that the elastic backbone is fractal with a fractal dimension 0 ± 0.00 for all studied models, while for c < < eb it is one-dimensional, as shown in Fig.. In this case, although the sanning cluster is comact, the existence of various holes revents the effective coalescence of the shortest aths resent in the system, therefore leading to an elastic backbone which is a fractal. The fractal dimension found at eb agrees within error bars well with the relation = d β eb /ν eb. Moreover, the resonse function of the order arameter, i.e. what would corresond in magnetic systems to the suscetibility,
3 χ ( c ) 7 /ν eb = /ν eb = () β eb /ν eb a) β eb = ε 1/ν eb FIG. : ogarithmic lot of the suscetibility χ of the elastic backbone as a function of the lattice size for site ercolation at eb, with ranging from to 18 sites. The leastsquares fit to the data of a ower law, χ( eb ) γ/ν, gives the exonent γ/ν = 0 ± 0.01 for the tilted square lattice (orange circles) and γ/ν = 0±0.02 for the triangular lattice (blue squares). χ = N( m 2 eb meb 2 ), diverges at the critical threshold eb with an exonent /ν eb = 0 ± 0.02 for all models considered, as shown in Fig. for site ercolation on the tilted square and triangular lattices. Finally, we also erform a full finite-size scaling analysis for () and for χ () of the form, () = β eb/ν eb m(ε 1/ν eb ), (2) χ () = /ν eb χ(ε 1/ν eb ), () where ε = ( eb ) is the distance from the critical threshold. The exonents β eb /ν eb, /ν eb, and ν eb are, resectively, associated with the decay of the order arameter, the divergence of the suscetibility, and the finite-size effects. As shown in Fig., for the secific case of site ercolation on the triangular lattice, we obtain excellent data collase for values β eb = 0.0 ± 0.0, = 1.97 ± 0.0 and ν eb = 2.0 ± 0.0. We note that hyerscaling relation 2β eb + = dν eb is violated [19, 20]. Similar data collase with the same exonents have been founor the other considered lattices. We also studied the elastic backbone transitions on a lattice model that mixes the features of normal square and triangular lattices, by adding to the normal square lattice with robability q some additional diagonals going from to-left to bottom-right. In this way, for q = 0, we obtain a normal square lattice and, for q = 1, a triangular lattice. For every value of q > 0, we found a value of c (q) < 1 at which the elastic backbone becomes χ () - /ν eb b) χ max ε 1/ν eb = 1.97 FIG. : (a) Results from the finite-size scaling analysis of the order arameter obtaineor site ercolation on the triangular lattice. The growing and decaying curves corresond to values above and below eb, resectively. The dashed line is the least-squares fit to data in the scaling region above eb for Eq. 2. (b) The same as in (a), but for the suscetibility χ. Blue symbols for < eb and red symbols for > eb. Here, the dashed line has sloe Inset of (b): og-log lot of the maximum of χ as a function of system size. comact. In Fig. 7a we show the finite-size scaling of the mass M eb of the elastic backbone for site ercolation at eb (q), for q = 0.0 and q = The results show that the fractal dimension is within error bars the same for both cases. Moreover, we calculated these fractal dimensions at c (q) for different values of q > 0 anound that they do not deend on q. This underlines, on one hand, the universality of d eb and questions, on the other hand, any relation to rigidity ercolation [21 2], which does exhibit a transition at an intermediate value of q. In Fig. 7b, we show the hase diagram between ercolation robability and the density of diagonals q, where three hases are identified. In the non-ercolating hase,
4 M () q = 0.0 q = Non-comact Phase Comact Phase Non-ercolating Phase (a) (b) q FIG. 7: (a) ogarithmic lot of the mass M eb of the elastic backbone as a function of the lattice size for site ercolation at eb (q) for two different values of q. The least-squares fit to the data of a ower law, M eb ( eb (q)), gives the exonent 0 ± 0.00 for q = 0.0 (orange circles) and 0 ± 0.00 for q = 0.80 (red squares). The results show that the fractal dimension is within error bars the same for both cases. (b) Phase diagram between ercolation robability and the density of diagonals q. Three hases can be identified. The non-ercolating hase, where the sanning cluster is absent, bounded on to by the curve c(q) (filled circles) which reresents the classical ercolation thresholds. The non-comact hase is bounded by the curve c(q) on the bottom and the curve eb (q) on the to (stars) that is the critical line along which the order arameter () vanishes and the elastic backbone is fractal. Finally, in the comact hase on the to the elastic backbone becomes comact. the sanning cluster is absent and consequently the elastic backbone does not ercolate. The non-ercolating hase is bounded by the curve c (q) which reresents the classical ercolation thresholds. The non-comact hase is characterized by the unitary dimension of the elastic backbone and it is bounded by the curves c (q) on the bottom and eb (q) on the to, which defines the critical line along which the order arameter () vanishes and the elastic backbone is fractal. In the comact hase, the dimension of the elastic backbone is equal to the dimension of the lattice considered. Concluding, we discovered that the mass of the elastic backbone serves as order arameter for a new transition within the connected hase of classical ercolation, exhibiting a new set of critical exonents β eb 1/2, 2, ν eb 2, and 7/ in two dimensions. Interestingly, however, hyerscaling is violated, being this to our knowledge the first examle for a violation of hyerscaling in a urely geometrical model. It would be also interesting to investigate higher dimensions and try to formulate a mean-field aroximation. Similar transitions for elastic backbones could be exected in models with tunable disorder [2 ]. Our findings have direct consequences to the stretching of random fibrous materials like biological tissues: when the first restoring force is felt, the resistance will grow very gently with dislacement below the threshold eb, while above eb the system will then to be instead very stiff. It is thus a transition between two very different stress-strain relations for a damaged tissue. Furthermore, there exists an interesting similarity between the elastic backbone ercolation, as introduced here, and the hase transition associated to rigidity ercolation, since the stressed backbone [ ] is a fractal whose dimension ( = 1.78 ± 0.02) is very close to the fractal dimension of the elastic backbone at eb. For non-tilted square lattices, in both cases, the transition is shifted to = 1 [7 9]. Nevertheless the two transitions describe different henomena, since they are in general located at different thresholds values and have a different set of critical exonents. Acknowledgments We thank the Brazilian agencies CNPq, CAPES, FUN- CAP, the National Institute of Science and Technology for Comlex Systems, and the Euroean Research Council (ERC) Advanced Grant 1998 FlowCCS for financial suort. [1] S. R. Broadbent and J. M. Hammersley, in Math. Proc. Cambridge (Cambridge University Press, 197), vol., [2] S. Kirkatrick, Rev. Mod. Phys., 7 (197). [] D. Stauffer and A. Aharony, Tailor & Francis, ondon (198). [] M. Sahimi, Alications of Percolation Theory (Taylor & Francis Grou, ondon, UK, 199).
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