Critical behavior of slider-block model. (Short title: Critical ) S G Abaimov

Transcription

1 Critical behavior of lider-bloc model (Short title: Critical ) S G Abaimov gabaimov@gmail.com. Abtract. Thi paper applie the theory of continuou phae tranition of tatitical mechanic to a lider-bloc model. The lider-bloc model i choen a a repreentative of ytem with avalanche. Similar behavior can be oberved in a foret-fire model and a and-pile model. Utilizing the welldeveloped theory of critical phenomena for percolating ytem a a foundation, a trong analogy for the lider-bloc model i developed. It i found that the lider-bloc model ha a critical point when the tiffne of the model i infinite. Critical exponent are found and it i hown that the behavior of the lider-bloc model and, particularly, the occurrence of ytem-wide event are trongly dominated by finite-ize effect. Alo the unnown before behavior of the frequency-ize ditribution i found for large tatitic of event.. Introduction Model with avalanche, recently introduced in the literature, exhibit complex behavior of event occurrence. A lider-bloc model [] (further on SBM) ha been invetigated by many tudie a a model repreenting the

2 recurrent earthquae occurrence [-5]. A foret-fire model repreent the occurrence of fire in foret [6, 7]. A and-pile model [8] i the main repreentative of the theory of elf-organized criticality. All thee model exhibit complex behavior. Energy (or another driving quantity) i pumped into a ytem. In repone the ytem organize it diipation through the complex behavior of avalanche. During the pat century major breathrough have been achieved in the theory of phae tranition in tatitical mechanic (for review ee, e.g. [9-]). The major concept of thi theory have been applied not only to the typical thermal ytem lie liquid-ga or magnetic ytem but alo to ytem without actual termalization lie percolation theory [3] or damage mechanic [4, 5]. In thi paper we apply the concept of continuou phae tranition to the lider-bloc model. For the foret-fire and and-pile model the application of tatitical mechanic i imilar and will be invetigated in future publication. Graberger P. [7] ha hown that critical exponent ignificantly depend on the model ize. In thi paper we invetigate thi effect for the lider-bloc model. We conider model with L = 5, 50, 00, 500, and 000 lider-bloc. Alo in preliminary tudie we dicovered that the number of event in tatitic alo ignificantly influence the critical behavior.

3 3 Particularly, we found that the dependence of a correlation length on a tuning field parameter exhibit ignificant non-mooth deviation for tatitic of 0,000 event in comparion with tatitic of,000,000 event which we ued a a reference. The frequency-ize behavior alo ignificantly depend on the ize of tatitic. We found unnown behavior when we ued large tatitic. We utilize the modification of the SBM which require integration of coupled ordinary differential equation. Therefore the ize of tatitic are limited by the time of numerical imulation. In pite of thi difficulty for large model ize of L = 500 and L = 000 bloc we have obtained large tatitic in the range from 70,000 up to,00,000 avalanche. Mot of ditribution have from 300,000 to 800,000 lip event. Thi let u obtain mooth caling dependence and accurate value of critical exponent. In Section we introduce the model. In Section 3 we invetigate it frequency-ize behavior. In Section 4 we conider an analogy with the theory of percolation and develop preliminary expectation what a critical point and a correlation length of the model are. In Section 5 we develop a rigorou expreion for the correlation length and conider it behavior. Alo we invetigate the finite-ize caling of the model and find that the dependence of correlation length for different model ize collape on a

4 4 ingle curve, repreenting a caling function. In Section 6 we invetigate the caling behavior of a uceptibility and alo find it caling function. In Section 7 we return to the frequency-ize ditribution and invetigate it caling behavior. Although all correlation length, uceptibility, and frequency-ize ditribution repreent correlation of fluctuation, the main repreentative i a correlation function. In Section 8 we invetigate it caling behavior.. The model In thi paper we utilize a modification of the lider-bloc model (SBM) with the inertia of bloc where the differential equation of motion are coupled []. Thi i the variation of the model which i the mot difficult to imulate numerically. However, it ha an advantage of the abence of multiple approximation that are ued in other modification. One of the mot important improvement i that the time evolution of an avalanche include coupled motion of all participating bloc in contrat to cellular-automata model where bloc move in equence (i.e., a bloc can move only when it neighbor top). A linear chain of L lider bloc of ma m i pulled over a urface at a contant velocity V L by a loader a illutrated in figure. Thi introduce a mechanim to pump energy into the ytem. Each bloc i connected to the

5 5 loader by a pring with tiffne L. Adjacent bloc are connected to each other by pring with tiffne C. Boundary condition are aumed to be periodic: the lat bloc i connected to the firt bloc. The bloc interact with the urface through tatic-dynamic friction. The tatic tability of each lider-bloc i given by L ( yi yi yi ) FSi, () y < i + C + where F Si i the maximum tatic friction force on bloc i holding it motionle and y i i the poition of bloc i relative to the loader. Thee threhold introduce the non-linearity of ytem behavior. During train accumulation due to the loader motion all bloc are motionle relative to the urface and have the ame increae of their coordinate relative to the loader plate dy i = V. () L dt When the cumulative force of the pring connecting to bloc i exceed the maximum tatic friction F Si, the bloc begin to lide. The dynamic lip of bloc i i controlled by it inertia d yi m + L yi + C + = dt ( yi yi yi ) FDi, (3) where F Di i the dynamic (liding) frictional force on bloc i. The loader velocity i aumed to be much maller than the lip velocity, o the

6 6 movement of the loader i neglected during a lip event. Thi i conitent with the concept that the lip duration of an earthquae i negligible in comparion with the interval of low tectonic tre accumulation between earthquae. The liding of one bloc can trigger intability of other bloc forming a multi-bloc event. When the velocity of a bloc decreae to zero it tic and witche from the dynamic to tatic friction. It i convenient to introduce the non-dimenional variable and parameter: L τ f = t for the fat time during avalanche evolution, m y Y i = for the coordinate of bloc. The ratio of tatic to dynamic friction F L i ref S F Si φ = i aumed to be the ame for all bloc =. 5 F Di φ but the value of friction F β i = vary from bloc to bloc with F ref S a a reference value of F Si ref S the tatic frictional force (F ref S i the minimum value of all F Si ). Particularly, the value of frictional parameter β i are aigned to bloc by the uniform random ditribution in the range < β i < 3.5. Thi quenched random diorder in the ytem i a noie required to generate event variability in C tiff ytem. Parameter α = i the tiffne of the ytem relative to the L

7 7 tiffne of ytem connection to the loader. Later we will ee that α play an important role of a tuning field parameter. For all model ize a value of α we will in general utilize,, 3, 4, 5, 6, 7, 8, 9, 0,,, 4, 6, 8, 0, 5, 30, 35, 40, 50, 60, 75, 00, 00, 500, 000, 000, 5000, 0000, and ome other, pecific for each particular model ize. Stre accumulation occur when all bloc are table; lip of bloc occur during the fat time τ f when the loader i aumed to be approximately motionle. In term of thee non-dimenional variable the tatic tability condition () become Y ( Yi Yi Yi ) βi, (4) α < i + + train accumulation () become dy dτ i S =, (5) and dynamical lip (3) become d Yi dτ f + α β φ i ( Y Y Y ) =. (6) + Yi i i i+ For numerical imulation a velocity-verlet numerical cheme i utilized which i a typical cheme for molecular-dynamic imulation [e.g., 6]. 3. Frequency-ize behavior

8 8 Figure (a-b) illutrate behavior of the SBM coniting of L = 500 and 000 bloc. The probability denity function of the frequency-ize ditribution i plotted on log-log axe for different value of the ytem tiffne α. A a ize of an event the number of different bloc participating in thi event i ued. During an avalanche a bloc can loe and gain it tability many time but i counted only once in the ize of thi event. Thi mae the ize of an event equal to it elongation in the model pace (equal to the number of conecutive bloc in a continuou chain which ha lot it tability). If the ize of an event equal the ize of the model we will refer to thee event a ytem-wide (SW) event. For large ize in figure (a-b) the liding average over 9 adjacent ize ha been ued to remove fluctuation. For mall value of α the SBM ha no SW event. The frequency-ize tatitic for mall event ha a tendency to be imilar to the Gutenberg- Richter power-law ditribution (traight line on the log-log axe) but for larger event it ha a roll-down. When α increae, the roll-down move to the right and finally goe beyond the ytem ize L. Alo the behavior of the ytem change: We ee the appearance of a pea of event whoe ize are about a half of the model ize. When α exceed ome critical value, the firt SW event tart to appear. The pea of the half-model-ize event become

9 9 narrower and diappear on ome tatitic ( α = 000 for L = 500 bloc and α = 5000 for L = 000 bloc). Intead, another pea appear which i adjacent to the SW limit. We believe that thi effect i oberved for the firt time in thi tudy due to the preence of large tatitic. Further increae of α i aumed to caue all complexitie of the curve to diappear and the frequency-ize dependence i aumed to become a perfect power-law plu a dicrete pea of SW event. However, we have not been able to oberve thi effect for the large ytem with L = 500 and L = 000 bloc becaue thi clean power-law dependence i uppoed to appear at very high value of α, where the differential equation become difficult to be olved numerically. Therefore we illutrate thi dependence for the model with L = 00 bloc in figure (c). The maximum lielihood fit give the value of the exponent of the power-law dependence τ =.08±0.09 which i very cloe to. Therefore we can ugget that in the limit of infinite tiffne the model exhibit the power-law dependence of non-sw event with the meanfield (rational) value τ = of the exponent. We illutrate thi model tendency in figure 3. The frequency-ize ditribution are normalized by the number of SW event. For all model ize we ued here the ame value of α = 000. When the ize of the model decreae we ee the tendency of the ditribution to attenuate the pea of half-model-ize

10 0 event and to become a power-law plu the dicrete pea of SW event. We ee that α = 000 i ufficient to reveal the power-law tendency for model ize L = 5, 50, and 00. However, for model ize L = 500 and 000 the ytem i not tiff enough to remove the influence of the pea of half-modelize event from the power-law dependence. Alo in figure 3 we ee that for the ame event ize the number of event with thi ize relative to the number of SW event increae with the increae of the model ize. However, thi increae i le than an order of amplitude and can be aociated with the deviation from the pure powerlaw dependence. Again, thee deviation are caued by the fact that the tiffne α of the ytem i not high enough. The dependence of α, at which the firt SW event appear, on the ize of the model i hown in figure 4. We ee that the appearance of the firt SW event depend on the ytem ize and i a reult of the finite-ize effect [3]. Therefore, it would be wrong to interpret the appearance of the firt SW event in a finite ytem a a critical point of the infinite model. What the meaning of thee value of α i and what the critical point of the model i, we will dicu in the next ection. 4. An analogy with the percolation theory

11 A a poible analogy we conider a percolating ytem. In the cae of ite percolation [3] a field parameter p i the probability for a lattice ite to be occupied. If N i the number of occupied ite on the lattice and N total i the total number of ite on the lattice then p = N / N total. For the rectangular (quare) d-dimenional lattice with the linear ize of L ite the total number of ite i N total = L d. For the given value of p we define a microtate a a particular microconfiguration of occupied ite realized on the lattice. For example, for N = there are N total microtate when there i only one occupied ite at any of N total poible location on the lattice. For N = N total there i only one microtate when all ite are occupied. Let u aume that p increae from 0 to. Then initially for p below the percolation threhold p C there i no percolating cluter on the infinite lattice. For the finite lattice with ize L for p < L / N total (for N < L) there i alo no percolating cluter. However, when p i greater than L / N total (when N L) the appearance of a percolating cluter among all microtate i poible. Particularly, percolating i any microtate which contain one row of the lattice completely occupied. For p ignificantly below the percolating threhold p C the number of thee percolating microtate i much maller than the total number of microtate for the given p. Therefore if an oberver

12 were looing at an enemble of all poible ytem realization for the given p (an enemble of all poible microtate) /he would count percolating microtate a highly improbable and their fraction among all microtate in the enemble a negligible. A correlation length ξ for thi value of p i much maller than the ytem ize. Even when, for the further increae of p, the fraction of percolating microtate become finite for a finite ytem, thi doe not guarantee that the infinite ytem percolate. In the finite ytem the fraction of percolating microtate become finite earlier than in the infinite ytem becaue of the finite-ize effect [3]. The finite ytem begin to percolate when the correlation length ξ reache the ize of the ytem L. But percolation of the infinite ytem require an infinite correlation length (which appear at higher value of p). Viually, the infinite ytem can be imagined a compoed by an enemble of finite ytem combined together (an enemble of all microtate of a finite ytem). If only the negligible fraction of thee microtate percolate the finite lattice, then the infinite ytem doe not have a percolating cluter. For the cae of the SBM the field parameter i the tiffne of the ytem α. For mall value of α there are no SW event in the ytem. If α increae and exceed ome threhold, the firt SW event appear in the

13 3 ytem. However, the fraction of thee event (e.g., 3 of 90,000 for the ytem ize L = 000 and α = 4) i very mall. The appearance of SW event i poible becaue the field parameter i above ome threhold. However, the correlation length ξ i till finite and, in fact, i very much maller than the ize of the ytem L. For further increae of the field parameter the correlation length reache the ize of the finite ytem, but i till much maller than the infinite correlation length, required for the infinite ytem to reach it critical point. Therefore, the firt appearance of SW event in the finite ytem mut not be confued with the cae of an infinite ytem at the critical point α C. Returning again to the percolating ytem, below the percolation threhold p < p C the behavior of the ytem i ignificantly different at different patial cale. For the cale maller than the correlation length ξ the ditribution of cluter i fractal and cale-invariant. The frequency-ize ditribution of cluter ize in thi cae i a power-law and again there i an analogy here with the frequency-ize ditribution of mall event in the SBM (traight line of the Gutenberg-Richter power-law ditribution for mall event on log-log axe). For the cale imilar or greater than the correlation length ξ the frequency-ize ditribution of cluter deviate from the powerlaw and ha an exponential roll-down. Again, there i an analogou roll-

14 4 down for the SBM for larger event. Therefore, preliminary, for the SBM the correlation length ξ can viually be found a being in the range where the frequency-ize ditribution change it behavior from the power-law to the roll-down. However, the frequency-ize behavior of the SBM i more complex than the behavior of the percolating ytem. Therefore later we will provide a more rigorou tatement. For a finite percolating ytem, when the correlation length become greater than the ytem ize ξ L, the ditribution of all non-percolating cluter i fractal and cale-invariant. The ame we can ee for the SBM for the range of high value of α when the roll-down ha moved completely beyond the ytem ize L and the frequency-ize ditribution for non-sw event become a power-law ditribution pdf ( S ) S (α = 000 in Fig. (c)). When the correlation length approache the ytem ize for a finite ytem, the fraction of percolating cluter become finite becaue of the finite-ize effect. Therefore for the SBM we can conclude that the appearance of the ignificant fraction of SW event i alo a reult of the finite-ize effect and i an indication that the correlation length ξ i reaching the ytem ize L. 5. Correlation length ξ

15 5 Firt we will conider the definition of a correlation length in the theory of percolation. For the infinite ytem the correlation length ξ may be defined a the averaged root mean quare ditance between two arbitrary occupied ite on the lattice under the condition that thee two ite mut belong to the ame cluter [3] r i, j < i, j > the ame cluter ξ, (7) < i, j > the ame cluter where indexe i and j enumerate occupied ite on the lattice, r i,j i the ditance between occupied ite i and j, and um < i, j > the ame cluter goe over all pair of occupied ite < i,j > under the condition that both ite in each pair mut belong to the ame cluter. Thi definition can be written a averaging over all cluter on the lattice r i, j < i, j > cluter ξ =, (8) < i, j > cluter where index enumerate all cluter on the lattice. The um over all cluter can be tranformed into the um over different cluter ize ( i the number of occupied ite in a cluter)

16 6 = = = = > < = = > < = > < = > < N j i j i N N j i j i N j i N j i j i N r r r ) ( ) ( cluter,, cluter,, cluter, cluter,, ξ, (9) where index enumerate N cluter of ize. The radiu of gyration of given cluter i ) ( cluter,, cluter, cluter,, = = > < > < > < r r R j i j i j i j i j i, (0) and the averaged root mean quare radiu of gyration for cluter of ize on the lattice i ) ( cluter,, = = = > < = N r N R R N j i j i N. () Therefore equation (9) can be written a = = n R n N R N ) ( ) ( ) ( ) ( ξ, () where n i the number of cluter of ize per lattice ite for the given p. So, the correlation length i the root mean quare of radii of all cluter averaged over all cluter in the lattice not directly but with the weight coefficient ( - ).

17 7 Becaue the SBM i a one-dimenional chain of bloc, each event i aumed to be continuou over the model pace (all bloc, which are untable during an avalanche, form a continuou chain). Therefore for the SBM the ize of an event (the number of bloc participating in an avalanche) i the elongation of thi event. Thi ignificantly implifie all further calculation. For any event of ize the firt ite mae ( ) pair with ( ) other ite. Then the econd ite mae ( ) pair with ( ) ite, and o on. Finally, the ite before the lat ite mae one pair with the lat ite. For the radiu of gyration of thi event it provide R = i ( i) ( ) ( + ) = =. (3) i ( i ) i= In the one-dimenional cae for the ame ize there i no variability of cluter. Therefore the averaged radiu of gyration R equal to the radiu of gyration of any cluter with ize : R = R. For the correlation length ξ in the imilar way we obtain L L pdf ( ) i ( i) pdf ( ) ( ) ( + ) = i= = ξ = =, (4) L L pdf ( ) i pdf ( ) ( ) = i= = where pdf() i the probability denity function to oberve an event with the elongation in the equence of avalanche.

18 8 Figure 5(a) preent the dependence of the correlation length ξ on the ytem tiffne α for different value of the model ize L. Behavior of the correlation length ugget that the critical point i located in the infinity of the field parameter α. Therefore further on we ue a field parameter t = / α intead of α and aume that the critical point i located at t = 0. Figure 5(b) preent on log-log axe the dependence of the correlation length ξ on the field parameter t for different model ize. The correlation length ξ increae monotonically with the decreae of t. Initially thi increae i influenced by non-linear effect becaue the ytem i far from the fixed point of a renormalization group. When the field parameter t reache the vicinity of the fixed point, the linearization of the renormalization group become poible. Starting from thi value of t the dependence of the correlation length on the field parameter become a power-law / t ν with the exponent ν =.85±0.03. Thi value wa obtained by the maximum lielihood fit of the power-law part of the curve for the SBM with 500 and 000 bloc. We ue for the fit only thee model ize becaue they provide the dependence which i the cleanet from the non-linear and croover effect. For the infinite ytem we would expect the power-law divergence of the correlation length at the critical point t = 0 with the ame value of the exponent ν.

19 9 However, our SBM are finite. Therefore, for further decreae of t the correlation length increae a a power-law and finally become of the order of the ytem ize L. Starting from thi value of the field parameter, the finite-ize effect, a a croover effect, influence the dependence of ξ on t. When the ytem approache the critical point, the correlation length reache the limit of the ytem ize and tay contant at thi limit [9]. More rigorouly, the averaged cluter elongation reache the ytem ize while the correlation length tay contant at a lower value due to the fact that correlation length (7) i alway lower than the averaged linear ize of cluter. From the theory of caling function [9,, 7] we expect that the functional dependence of the correlation length ξ on the field parameter t hould have the form ξ Ξ ν t ( Lt ν ), where cont, x >> Ξ( x ) = (5) x, x << i ome caling function. In the limit Lt ν >>, when ξ << L, thi function ha a contant limit, which doe not influence the power-law dependence / t ν. In the limit Lt ν <<, when the correlation length of the infinite ytem i ξ >> L, thi function generate a power-law dependence x t ν, which cancel the power-law dependence /t ν in front of the function Ξ(x) and provide the finite limit for the correlation length. To find the caling

20 0 function Ξ(x), we multiply the dependence ξ(t) by t ν and then plot the ν reulting dependence ξt a a function of the parameter ν x = Lt. The obtained caling function Ξ(x) i preented in figure 6. Alo we plot the dependence x to compare it with the caling function for low value of x. In figure 6 we ee that all curve perfectly collape on the caling dependence Ξ(x) except only for high value of t when the ytem i far from the critical point and the renormalization group cannot be linearized far from it fixed point. At thee high value of t the power-law divergence non-linear deviation, and the caling i not valid. 6. Suceptibility Κ / t ν of the correlation length ha Similarly to the correlation length, in thi ection we invetigate the behavior of the uceptibility a a meaure of fluctuation. In tatitical mechanic thi quantity i proportional to the variance of fluctuation; in the theory of percolation thi quantity i called a mean cluter ize [3]. Following the analogy with the percolation theory, we define uceptibility a Κ L = pdf ( ), (6) = a the averaged quared cluter ize. Here pdf() i the probability denity function to oberve an event with the elongation in the equence of avalanche.

21 Figure 7 preent the dependence of the uceptibility Κ on the field parameter t = / α on log-log axe for different model ize. The uceptibility Κ increae monotonically with the decreae of t. Initially thi increae i influenced by non-linear effect becaue the ytem i far from the fixed point of the renormalization group. When the field parameter t reache the vicinity of the fixed point, the linearization of the renormalization group become poible. Starting from thi value of t the dependence of the uceptibility on the field parameter become a powerlaw γ /t with the exponent γ =.94±0.03. Thi value wa obtained by the maximum lielihood fit of the power-law part of the curve for the SBM with 500 and 000 bloc. We ue for the fit only thee model ize becaue they provide the dependence which i the cleanet from the non-linear and croover effect. For the infinite ytem we would expect the power-law divergence of the uceptibility at the critical point t = 0 with the ame value of the exponent γ. However, our SBM are finite. Therefore, when the ytem approache the critical point and the correlation length reache the ize of the ytem, the uceptibility top to increae a a power-law and tay contant. Starting from thi value of the field parameter, the finite-ize

22 effect, a a croover effect, influence the dependence of Κ on t. In other word, the mean cluter ize reache the limit of the ytem ize. From the theory of caling function [9,, 7] we expect that the functional dependence of the uceptibility Κ on the field parameter t hould have the form Κ Ξ γ t ( Lt ν ), where cont, x >> Ξ( x) = (7) γ / ν x, x << i ome caling function. In the limit Lt ν >>, when ξ << L, thi function ha a contant limit, which doe not influence the power-law dependence γ / t. In the limit Lt ν <<, when the correlation length of the infinite ytem i ξ >> L, thi function generate a power-law dependence x t γ / ν γ, which cancel the power-law dependence γ /t in front of the function Ξ(x) and provide the finite limit for the uceptibility. To find the caling function Ξ(x) we multiply the dependence Κ(t) by t γ and then plot the reulting dependence γ Κt a a function of the parameter ν x = Lt. The obtained caling function Ξ(x) i preented in figure 8. Alo we plot the dependence γ / ν x to compare it with the caling function for low value of x. In figure 8 we ee that all curve perfectly collape on the caling dependence Ξ(x) except only for high value of t when the ytem i far from the critical point and the renormalization group cannot be linearized far from it fixed point. At thee

23 3 high value of t the power-law divergence γ /t of the uceptibility ha nonlinear deviation, and the caling i not valid. 7. Frequency-ize ditribution In ection 3 we dicued the frequency-ize behavior of the SBM. In thi ection we return to the frequency-ize ditribution to invetigate it caling. From the theory of caling function [9,, 7] we expect that the functional dependence of the frequency-ize ditribution on the field parameter t and on the ize of an event hould have the form ν ( t Lt ν ) FSD, τ Ξ (7) where τ i the caling exponent, dicued in Section 3. Further on we will ue τ =. To find the caling function Ξ(x,y) we hould multiply the FSD dependence by τ and then plot the reulting dependence τ FSD a a function of the parameter ν ν x = t and y = Lt. However, in contrat to other caling dependence dicued above, we encounter here a difficulty. If we were looing at a percolating ytem, the frequency-ize ditribution would be normalized by the ize of the lattice. In other word, the number of poibilitie to count a particular cluter configuration i limited by the lattice ize, which give a natural normalization for the ditribution. In the cae of the SBM we count cluter a they occur in time during the model evolution. The time of poible obervation i not limited, and in our model

24 4 we lot a natural normalization of the frequency-ize ditribution. Therefore, oberving the caling function Ξ(x,y), we can determine it only with the accuracy of a contant multiplier. In figure 9(a,b) we plot the obtained caling dependence on the log-log-log axe for all five model ize L = 5, 50, 00, 500, and 000, each above other. All obtained caling function Ξ(x,y) have imilar hape and imilar tendencie to become traight horizontal line when the tiffne of the model increae. 8. Correlation function Following the theory of percolation [3], we define the correlation function G(R) r a a probability that, if a given ite i occupied, the ite at ditance R r i alo occupied and belong to the ame cluter r G( R) = r < i, j = i+ R> cluter i cluter, (8) where the um goe over all cluter enumerated by the index, the um i cluter goe over all ite of cluter, and the um r < i, j = i+ R> cluter goe over all pair of ite <i,j> which belong to cluter and which are eparated by the ditance R r. Arranging cluter by their ize we obtain

25 5 r G( R) = N = r = < i, j = i+ R> cluter N, (9) where index enumerate N cluter of ize. For the one-dimenional SBM, when cluter are linear chain, we can ignificantly implify equation (9) a, R = 0 pdf ( ) 0, R > 0, < R + R, R > 0, R + G( R) =, (0) pdf ( ) The behavior of the correlation function G(R) a a function of ditance R for different tiffnee of the model of ize L = 000 i preented in Fig. (0a) on log-log axe. The obervation we can mae i that for the power-law part of the dependence, which on the log-log axe i uppoed to be a traight line, we oberve zero exponent, in other word, a horizontal line. Therefore, for the dependence η Ξ( R / ξ ) R, where (x) G ( R) / Ξ i ome near-exponential function, we expect the exponent η to be zero. Thi i confirmed by Fig. (0b), where we plot the correlation function on the emilog axe. The main attenuating dependence i near exponential without a power-law addition. However, it i not pure exponential and for high R ha attenuation fater than exponential.

26 6 From the theory of caling function [9,, 7] we expect that the functional dependence of the correlation function on the field parameter t and on the ditance R hould have the form ν ν ν ν ( Rt, Lt ) Ξ( Rt Lt ) G Ξ, when η = 0. () η R In Fig. (0c) we plot the correlation function a a function of the parameter ν ν x = Rt and y = Lt on the log-log-log axe for all five model ize L = 5, 50, 00, 500, and 000. All obtained caling function Ξ(x,y) collape on a ingle urface with minor non-linear deviation far from the critical point where the renormalization group cannot be linearized. 9. Concluion For different ize of the lider-bloc model we obtain the dependence of the correlation length on the tiffne of the ytem a a field parameter. The obtained caling ugget that the lider-bloc model ha a critical point when it tiffne i infinite. For the exponent of the correlation length and uceptibility we obtain value.85 and.94 repectively. Alo we invetigate the finite-ize caling function of the model and find that the dependence for different model ize collape onto a ingle curve. For the exponent of the frequency-ize ditribution and correlation function we find τ = and η = 0 repectively.

27 7 Reference [] Burridge R and Knopoff L, Model and theoretical eimicity, 967 Bull. Seim. Soc. Am [] Carlon J M and Langer J S, Mechanical model of an earthquae fault, 989 Phy. Rev. A [3] Abaimov S G, Turcotte D L, Shcherbaov R and Rundle J B, Recurrence and interoccurrence behavior of elf-organized complex phenomena, 007 Nonlinear Proc. Geophy [4] Abaimov S G, Turcotte D L, Shcherbaov R, Rundle J B, Yaovlev G, Goltz C and Newman W I, Earthquae: Recurrence and interoccurrence time, 008 Pure Appl. Geophy [5] Abaimov S G, Tiampo K F, Turcotte D L and Rundle J B, Recurrent frequency-ize ditribution of characteritic event, 009 Nonlinear Proc. Geophy [6] Droel B and Schwabl F, Self-organized critical foret-fire model, 99 Phy. Rev. Lett [7] Graberger P, Critical behaviour of the Droel-Schwabl foret fire model, 00 New J. Phy. 4 7 [8] Ba P, Tang C and Wieenfeld K, Self-organized criticality, 988 Phy. Rev. A

28 8 [9] Goldenfeld N, 99 Lecture on Phae Tranition and the Renormalization Group (Reading, MA: Addion Weley) [0] Cardy J, 996 Scaling and Renormalization in Statitical Phyic (Cambridge: Cambridge Univerity Pre) [] Pathria R K, 996 Statitical Mechanic (Oxford: Butterworth- Heinemann) [] Ma S K, 976 Modern Theory of Critical Phenomena (Reading, MA: Benjamin) [3] Stauffer D and Aharony A, 99 Introduction To Percolation Theory: Taylor&Franci) [4] Abaimov S G, Applicability and non-applicability of equilibrium tatitical mechanic to non-thermal damage phenomena, 008 J. Stat. Mech. P09005 [5] Abaimov S G, Applicability and non-applicability of equilibrium tatitical mechanic to non-thermal damage phenomena: II. Spinodal behavior, 008 J. Stat. Mech. P03039 [6] Thijen J M, 999 Computational Phyic (Cambridge: Cambridge Univerity Pre)

29 9 [7] Branov J G, 996 Introduction to Finite-Size Scaling, Leuven Note in Mathematical and Theoretical Phyic. Serie A: Mathematical Phyic vol 8 (Leuven: Leuven Univerity Pre)

30 Figure. A lider-bloc model. 30

31 3 (a) (b)

32 3 (c) Figure. Frequency-ize ditribution of the model with (a) L = 500 bloc, (b) L = 000 bloc, and (c) L = 00 bloc for different value of the model tiffne α. The value of α are hown in the legend and in the label for individual curve. Starting from (a) α = 6, (b) α = 35, and (c) α = 8, ytemwide (SW) event are hown a marer on the right ide of the plot.

33 Figure 3. The frequency-ize ditribution normalized by the number of SW event. For all model ize L the value of α i

34 Figure 4. The value of α, at which the firt SW event appear. The fit how that the dependence i cloe to the quare root of the ize of the model L. 34

35 35 (a) (b) Figure 5. Correlation length ξ a a function of the (a) field parameter α and (b) field parameter t = / α. Each marer repreent a eparate equence of avalanche obtained in numerical imulation. The dahed line i the

36 maximum lielihood fit for the power-law part of the curve for the SBM with 500 and 000 bloc. 36

37 Figure 6. Scaling function Ξ(x) of the correlation length ξ. For comparion, the dependence x i given a a dahed line. 37

38 Figure 7. Suceptibility Κ a a function of the field parameter t = / α. Each marer repreent a eparate equence of avalanche obtained in numerical imulation. The dahed line i the maximum lielihood fit for the powerlaw part of the curve for the SBM with 500 and 000 bloc. 38

39 Figure 8. Scaling function Ξ(x) of the uceptibility Κ. For comparion, the dependence x γ / ν i given a a dahed line. 39

40 40 (a) (b) Figure 9. Scaling function Ξ(x) of the frequency-ize ditribution. Each ditribution i labeled by it model ize L. Each olid line repreent a

41 particular ditribution over event ize for given value of the field parameter t and model ize L. Horizontal hift among olid line correpond to the change of the field parameter t; vertical hift correpond to the change of the model ize L. Dot-marer repreent SW event. 4

42 4 (a) (b)

43 43 (c) Figure 0. Correlation function. (a-b) For model ize L = 000 the dependence of the correlation function on ditance R i preented for different model tiffnee on (a) log-log and (b) emi-log axe. (c) The caling function of the correlation function. Each olid line repreent a particular correlation function over ditance R for given value of the field parameter t and model ize L. Horizontal hift among olid line correpond to the change of the field parameter t. All model ize are collaped on a ingle urface.