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Effects of site dilution on the one-dimensional long-range bond-percolation problem
Physics Letters A 312 (2003) 331–335 www.elsevier.com/locate/pla
Effects of site dilution on the one-dimensional long-range bond-percolation problem U.L. Fulco a,b , L.R. da Silva b , F.D. Nobre b,∗ , H.H.A. Rego b , L.S. Lucena b a Departamento de Física, Universidade Federal do Piauí, 64049-550 Teresina PI, Brazil b Departamento de Física Teórica e Experimental, Universidade Federal do Rio Grande do Norte, 59072-970 Natal RN, Brazil
Received 12 December 2002; received in revised form 2 April 2003; accepted 8 April 2003 Communicated by C.R. Doering
Abstract The effects of the dilution of sites (with an occupancy probability ps for an active site) on the long-range bond-percolation problem, on a linear chain (d = 1), are analyzed by means of a Monte Carlo simulation. The occupancy probability for a bond between two active sites i and j , separated by a distance rij is given by pij = p/rijα (α 0), where p represents the usual occupancy probability between nearest-neighbor sites. The percolation order parameter, P∞ , is investigated numerically for different values of α and ps , in such a way that a crossover between a nonextensive regime and an extensive regime is observed. 2003 Elsevier Science B.V. All rights reserved. PACS: 05.20.-y; 05.70.Ce; 05.70.Jk; 64.60.Ak Keywords: Extensive and nonextensive behavior; d = 1 bond percolation; Long-range interactions
One of the main properties of the standard Boltzmann–Gibbs statistical-mechanics formalism [1–3] is the additivity of some thermodynamical quantities, like the internal and free energies. Essentially, this property asserts that for a system made of two parts X and Y , a given thermodynamic quantity B is considered additive if BX+Y = BX + BY . Although the two concepts should be considered in separate, the additivity property implies extensivity [4], which means that extensive quantities increase linearly with the size of the system (e.g., the number of particles, N ), in such a way that the internal energy per particle, or the
* Corresponding author.
E-mail address: [email protected] (F.D. Nobre).
free-energy per particle, approach well-defined values (independent of N ) in the thermodynamic limit. However, only a few textbooks [5] emphasize that the additivity property does not hold in general, being directly related to the characteristics of the system under consideration, e.g., the interactions among particles must be short-ranged. When long-range interactions are present, each particle interacts with all the others, in such a way that additivity does not hold, e.g., the total free energy is not equal to the sum of the free energies of the macroscopic parts of the system. In some cases, even though the system is nonadditive, the energy extensivity may be recovered by applying the Kac prescription [6], within which the potential is rescaled by an appropriate N -dependent factor.
0375-9601/03/$ – see front matter 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0375-9601(03)00642-X
332
U.L. Fulco et al. / Physics Letters A 312 (2003) 331–335
A lot of interest has been dedicated to the study of nonextensive systems lately. Nonextensive behavior has been found in many physical systems, and may be produced by a variety of physical ingredients, like long-range interactions and long-time memory, among others. Clear limitations have been identified on standard formalisms, e.g., the powerful Boltzmann–Gibbs statistical mechanics was shown to fail in providing a satisfactory description of a wide variety of theoretical models and experimental realizations [7–9], in such a way that a completely new physics is emerging for dealing with such systems, in the recent years. The most successful alternative for the standard Boltzmann–Gibbs formalism appears to be the nonextensive thermostatistics, proposed by Tsallis [10], characterized by an entropic index q (q ∈ ). The usual additivity property of thermodynamic quantities (e.g., entropy, internal energy) is broken for any q = 1. A Hamiltonian system that has been very much investigated recently is the inertial classical XY ferromagnet with long-range interactions [11–15]. It consists in a modification of the well-known standard model, where the spins are replaced by classical rotators. Such a system exhibits evident inconsistencies with the Boltzmann–Gibbs formalism: (i) the maximal Lyapunov exponent (that should be positive for a quick and uniform occupation of phase space) approaches zero in the thermodynamic limit [11,12]; (ii) a metastable state, whose duration diverges in the thermodynamic limit, is observed [13–15]. In such a state one finds a non-Gaussian particle distribution, that is reasonably well-fitted by a Tsallis distribution [14,15]; (iii) for a finite number of rotators, the system displays a crossover from the metastable state (where the nonextensive thermodynamics seems to apply) to the terminal equilibrium state (where extensivity is recovered) [14,15]. Another example of a crossover between nonextensive and extensive regimes has been found in the long-range one-dimensional bond-percolation problem [16]. In this problem, one considers a bond occupancy probability between any two sites i and j of the linear chain as pij =
p rijα
(0 p 1, α 0),
(1)
where rij represents the distance between sites (in crystalline units, rij = 0, 1, 2, . . .) and p denotes the first-neighbor bond occupancy probability. The exponent α controls the range of the bonds in the chain, with two well-known particular cases, namely, the first-neighbor (α → ∞) and infinite-range, i.e., mean-field-like (α = 0), bond-percolation limits. Such a crossover was found to occur at α ≡ α1 = 1, in such a way that for 0 α α1 (nonextensive regime) the percolation order parameter is always maximum. The extensive regime holds for all α > α1 , although only for α > α2 (α2 = 2), the percolation order parameter presents the usual behavior characteristic of the one-dimensional first-neighbor bond-percolation, vanishing ∀p < 1. In the interval α1 < α < α2 , the order parameter presents a nontrivial behavior, varying with p between its minimum and maximum possible values. In the present Letter we shall investigate the effects of a site-dilution (probabilities ps and 1 − ps for a site to be present or vacant, respectively), in the above-mentioned long-range one-dimensional bondpercolation problem. Physically, one expects the percolation order parameter, P∞ [17,18], to get weakened by the inclusion of diluted sites, for a fixed value of α; indeed, this is what we have found, as we report below. We have simulated open linear chains of several sizes varying from N = 100 up to N = 1000; in each case, we have computed P∞ by considering averages over 10 000 samples (i.e., different configurations of bonds and diluted sites). In Figs. 1–3 we present the percolation order parameter P∞ (p) for typical values of α, ps and N . In Fig. 1 we show P∞ (p) for fixed values of α (α = 0.8) and N (N = 300), and several values of ps (ps = 0.8, 0.6, 0.4, 0.2). One sees clearly that the order parameter is weakened by the dilution of sites, leading to an increase in the critical value of the first-neighbor bond occupancy probability, pc , below which P∞ becomes zero. As expected, fluctuations get enlarged, for higher fractions of diluted sites (cf. the case ps = 0.2 in Fig. 1), in such a way that a larger computational effort is required in order to get reliable results in such cases. In Fig. 2 the order parameter is exhibited for fixed values of ps (ps = 0.8) and N (N = 300) and several values of α, in the range α = 0.2 up to α = 2.0. One notices that, for small values of α one is found in the nonextensive
U.L. Fulco et al. / Physics Letters A 312 (2003) 331–335
Fig. 1. The percolation order parameter, P∞ , versus the first-neighbor bond-percolation occupancy probability, p, for fixed values of N and α, and several values of the site-occupancy probability ps .
Fig. 2. The percolation order parameter, P∞ , versus the first-neighbor bond-percolation occupancy probability, p, for fixed values of N and ps , and typical values of α.
regime, for which P∞ equals unit, ∀p > 0, leading to pc (α) = 0. However, for increasing values of α, one gets pc (α) > 0, in such a way that as one approaches the first-neighbor bond-percolation limit (typically, α ∼ 2), large fluctuations are observed, due to finite size effects; in such a case, one should get, in the
333
Fig. 3. The percolation order parameter, P∞ , versus the first-neighbor bond-percolation occupancy probability, p, for a fixed value of α, two different site-occupancy probabilities, and several system sizes.
thermodynamic limit, P∞ = 0 and pc (α) = 1. In Fig. 3 we show how the order parameter evolves with the size of the system; we consider two different site concentrations (ps = 0.6 and ps = 0.2) and a fixed value of α (α = 0.8). Once again, one observes that the finite-size effects become more relevant for lower site concentrations (cf. the different sizes considered for the case ps = 0.2 of Fig. 3), in such a way that more reliable estimates of P∞ and pc (α) are obtained at large system sizes (typically, N = 1000). By locating, for finite values of N , the abscissa points given by the linear continuations of the curves of P∞ (p), and then extrapolating for N → ∞, one may estimate the critical values of the first-neighbor bond occupancy probability, for fixed values of ps and different values of α, pc (α). In Fig. 4 we exhibit how pc (α) varies with α for a typical value of the site concentration: ps = 0.8. The error bars express the accuracy of the linear continuations, together with the thermodynamic-limit extrapolations, of the curves of the parameter P∞ . One notices that the two characteristic values of α, α1 (ps ), below which P∞ equals unit, ∀p > 0, and α2 (ps ), above which P∞ = 0, ∀p < 1, decrease as the site concentration decreases. In analogy to what happens
334
U.L. Fulco et al. / Physics Letters A 312 (2003) 331–335
ps = 0.8: α1 (ps ) = 0.7 ± 0.05, α2 (ps ) = 1.9 ± 0.06, ps = 0.6: α1 (ps ) = 0.6 ± 0.07, α2 (ps ) = 1.8 ± 0.07, ps = 0.4: α1 (ps ) = 0.5 ± 0.09, α2 (ps ) = 1.7 ± 0.10.
Fig. 4. The critical value of the first-neighbor bond occupancy probability versus α, for the case ps = 0.8.
in the case of no site dilution (ps = 1.0) [16], the characteristic value α1 (ps ) sets in a crossover between the nonextensive (0 α α1 (ps )) and extensive (α > α1 (ps )) regimes. On the other hand, α2 (ps ) defines the onset of the well-known first-neighbor onedimensional percolation behavior; as a consequence of this, the region right below α2 (ps ) is very delicate from the numerical point of view. In Fig. 4, the last point to the right corresponds to the estimate of pc (α) associated with the value of α that is as close as possible (in the sense that one may still get a reliable estimate, within the computational effort employed herein) to α2 (ps ). For slightly higher values of α, strong fluctuations, driven by finite-size effects, occur; in this region, in order to observe the behavior of the order parameter P∞ (i.e., whether it drops to zero ∀p < 1, or not) one needs to run much larger system sizes, as well as to increase reasonably the number of samples, as compared to those used in the present work. Besides the site-occupancy probability ps = 0.8, we have also investigated the cases ps = 0.6 and ps = 0.4, which lead to similar plots to the one exhibited in Fig. 4, except that the characteristic values α1 (ps ) and α2 (ps ) appear shifted to the left. Below, we present our estimates for the two characteristic values of α, for each one of site concentrations considered,
The above results, together with those for the case ps = 1.0, i.e., α1 (ps ) = 1.0 and α2 (ps ) = 2.0 [16], do surely indicate that the characteristic values of α decrease with the concentration of sites. To conclude, we have investigated the effects of the dilution of sites (with a site occupancy probability ps ) in the long-range one-dimensional bond-percolation problem, by means of Monte Carlo simulations. The range of the bonds is controlled by a parameter α, in such a way that two well-known limits may be recovered, namely, the mean-field (α = 0), and the first-neighbor bond percolation (α → ∞) ones. We have shown that two characteristic values of α occur, whose values depend on the site-occupancy probability, α1 (ps ) and α2 (ps ) (α2 (ps ) > α1 (ps ) 0). For 0 α α1 (ps ), the percolation order parameter P∞ is always maximum (equal to unit), indicating that one is in a nonextensive regime, whereas for α > α2 (ps ), the order parameter is always zero, in such a way that one is found in the well-known first-neighbor onedimensional bond-percolation regime. The crossover between the nonextensive and extensive regimes takes place at α = α1 (ps ) and for α1 (ps ) < α < α2 (ps ) the order parameter displays a familiar behavior, i.e., varying monotonically between its minimum and maximum possible values. We have shown that the two characteristic values of α, α1 (ps ) and α2 (ps ), decrease with the inclusion of diluted sites. Finally, it is important to mention that, apart from its relevance in the study of nonextensive systems, the site-diluted longrange bond-percolation models are closely related to a wide variety of complex networks, such as internet, www, and small-world economic networks [19].
Acknowledgements It is a pleasure to thank D. Stauffer for fruitful conversations. The partial financial supports from CNPq,
U.L. Fulco et al. / Physics Letters A 312 (2003) 331–335
CAPES, Pronex/MCT, and ANP/CTPETRO/CENPES/Petrobras (Brazilian agencies) are acknowledged.
References [1] R.K. Pathria, Statistical Mechanics, Butterworth, Washington, DC, 1996. [2] H.E. Stanley, Introduction to Phase Transitions and Critical Phenomena, Oxford Univ. Press, New York, 1971. [3] P.T. Landsberg, Thermodynamics and Statistical Mechanics, Dover, New York, 1991. [4] J. Barré, D. Mukamel, S. Ruffo, Phys. Rev. Lett. 87 (2001) 030601. [5] R. Balescu, Equilibrium and Nonequilibrium Statistical Mechanics, Wiley, New York, 1974. [6] M. Kac, G.E. Uhlenbeck, P.C. Hemmer, J. Math. Phys. 4 (1963) 216. [7] C. Tsallis, Brazilian J. Phys. 29 (1999) 1.
335
[8] C. Tsallis, in: S. Abe, Y. Okamoto (Eds.), Nonextensive Statistical Mechanics and its Applications, in: Lecture Notes in Physics, Springer-Verlag, Berlin, 2001. [9] C. Tsallis, Chaos Solitons Fractals 13 (2002) 371. [10] C. Tsallis, J. Stat. Phys. 52 (1988) 479. [11] C. Anteneodo, C. Tsallis, Phys. Rev. Lett. 80 (1998) 5313. [12] A. Campa, A. Giansanti, D. Moroni, C. Tsallis, Phys. Lett. A 286 (2001) 251. [13] A. Campa, A. Giansanti, D. Moroni, Physica 305 (2002) 137. [14] V. Latora, A. Rapisarda, C. Tsallis, Phys. Rev. E 64 (2001) 056134. [15] V. Latora, A. Rapisarda, C. Tsallis, Physica A 305 (2002) 129. [16] H.H.A. Rego, L.S. Lucena, L.R. da Silva, C. Tsallis, Physica A 266 (1999) 42. [17] D. Stauffer, A. Aharony, Introduction to Percolation Theory, 2nd Edition, Taylor and Francis, London, 1994. [18] G. Grimmet, Percolation, 2nd Edition, Springer-Verlag, Berlin, 1999. [19] G. Corso, L.S. Lucena, Z.D. Thomé, Physica A 324 (2003) 430.
Effects of site dilution on the one-dimensional long-range bond-percolation problem U.L. Fulco a,b , L.R. da Silva b , F.D. Nobre b,∗ , H.H.A. Rego b , L.S. Lucena b a Departamento de Física, Universidade Federal do Piauí, 64049-550 Teresina PI, Brazil b Departamento de Física Teórica e Experimental, Universidade Federal do Rio Grande do Norte, 59072-970 Natal RN, Brazil
Received 12 December 2002; received in revised form 2 April 2003; accepted 8 April 2003 Communicated by C.R. Doering
Abstract The effects of the dilution of sites (with an occupancy probability ps for an active site) on the long-range bond-percolation problem, on a linear chain (d = 1), are analyzed by means of a Monte Carlo simulation. The occupancy probability for a bond between two active sites i and j , separated by a distance rij is given by pij = p/rijα (α 0), where p represents the usual occupancy probability between nearest-neighbor sites. The percolation order parameter, P∞ , is investigated numerically for different values of α and ps , in such a way that a crossover between a nonextensive regime and an extensive regime is observed. 2003 Elsevier Science B.V. All rights reserved. PACS: 05.20.-y; 05.70.Ce; 05.70.Jk; 64.60.Ak Keywords: Extensive and nonextensive behavior; d = 1 bond percolation; Long-range interactions
One of the main properties of the standard Boltzmann–Gibbs statistical-mechanics formalism [1–3] is the additivity of some thermodynamical quantities, like the internal and free energies. Essentially, this property asserts that for a system made of two parts X and Y , a given thermodynamic quantity B is considered additive if BX+Y = BX + BY . Although the two concepts should be considered in separate, the additivity property implies extensivity [4], which means that extensive quantities increase linearly with the size of the system (e.g., the number of particles, N ), in such a way that the internal energy per particle, or the
* Corresponding author.
E-mail address: [email protected] (F.D. Nobre).
free-energy per particle, approach well-defined values (independent of N ) in the thermodynamic limit. However, only a few textbooks [5] emphasize that the additivity property does not hold in general, being directly related to the characteristics of the system under consideration, e.g., the interactions among particles must be short-ranged. When long-range interactions are present, each particle interacts with all the others, in such a way that additivity does not hold, e.g., the total free energy is not equal to the sum of the free energies of the macroscopic parts of the system. In some cases, even though the system is nonadditive, the energy extensivity may be recovered by applying the Kac prescription [6], within which the potential is rescaled by an appropriate N -dependent factor.
0375-9601/03/$ – see front matter 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0375-9601(03)00642-X
332
U.L. Fulco et al. / Physics Letters A 312 (2003) 331–335
A lot of interest has been dedicated to the study of nonextensive systems lately. Nonextensive behavior has been found in many physical systems, and may be produced by a variety of physical ingredients, like long-range interactions and long-time memory, among others. Clear limitations have been identified on standard formalisms, e.g., the powerful Boltzmann–Gibbs statistical mechanics was shown to fail in providing a satisfactory description of a wide variety of theoretical models and experimental realizations [7–9], in such a way that a completely new physics is emerging for dealing with such systems, in the recent years. The most successful alternative for the standard Boltzmann–Gibbs formalism appears to be the nonextensive thermostatistics, proposed by Tsallis [10], characterized by an entropic index q (q ∈ ). The usual additivity property of thermodynamic quantities (e.g., entropy, internal energy) is broken for any q = 1. A Hamiltonian system that has been very much investigated recently is the inertial classical XY ferromagnet with long-range interactions [11–15]. It consists in a modification of the well-known standard model, where the spins are replaced by classical rotators. Such a system exhibits evident inconsistencies with the Boltzmann–Gibbs formalism: (i) the maximal Lyapunov exponent (that should be positive for a quick and uniform occupation of phase space) approaches zero in the thermodynamic limit [11,12]; (ii) a metastable state, whose duration diverges in the thermodynamic limit, is observed [13–15]. In such a state one finds a non-Gaussian particle distribution, that is reasonably well-fitted by a Tsallis distribution [14,15]; (iii) for a finite number of rotators, the system displays a crossover from the metastable state (where the nonextensive thermodynamics seems to apply) to the terminal equilibrium state (where extensivity is recovered) [14,15]. Another example of a crossover between nonextensive and extensive regimes has been found in the long-range one-dimensional bond-percolation problem [16]. In this problem, one considers a bond occupancy probability between any two sites i and j of the linear chain as pij =
p rijα
(0 p 1, α 0),
(1)
where rij represents the distance between sites (in crystalline units, rij = 0, 1, 2, . . .) and p denotes the first-neighbor bond occupancy probability. The exponent α controls the range of the bonds in the chain, with two well-known particular cases, namely, the first-neighbor (α → ∞) and infinite-range, i.e., mean-field-like (α = 0), bond-percolation limits. Such a crossover was found to occur at α ≡ α1 = 1, in such a way that for 0 α α1 (nonextensive regime) the percolation order parameter is always maximum. The extensive regime holds for all α > α1 , although only for α > α2 (α2 = 2), the percolation order parameter presents the usual behavior characteristic of the one-dimensional first-neighbor bond-percolation, vanishing ∀p < 1. In the interval α1 < α < α2 , the order parameter presents a nontrivial behavior, varying with p between its minimum and maximum possible values. In the present Letter we shall investigate the effects of a site-dilution (probabilities ps and 1 − ps for a site to be present or vacant, respectively), in the above-mentioned long-range one-dimensional bondpercolation problem. Physically, one expects the percolation order parameter, P∞ [17,18], to get weakened by the inclusion of diluted sites, for a fixed value of α; indeed, this is what we have found, as we report below. We have simulated open linear chains of several sizes varying from N = 100 up to N = 1000; in each case, we have computed P∞ by considering averages over 10 000 samples (i.e., different configurations of bonds and diluted sites). In Figs. 1–3 we present the percolation order parameter P∞ (p) for typical values of α, ps and N . In Fig. 1 we show P∞ (p) for fixed values of α (α = 0.8) and N (N = 300), and several values of ps (ps = 0.8, 0.6, 0.4, 0.2). One sees clearly that the order parameter is weakened by the dilution of sites, leading to an increase in the critical value of the first-neighbor bond occupancy probability, pc , below which P∞ becomes zero. As expected, fluctuations get enlarged, for higher fractions of diluted sites (cf. the case ps = 0.2 in Fig. 1), in such a way that a larger computational effort is required in order to get reliable results in such cases. In Fig. 2 the order parameter is exhibited for fixed values of ps (ps = 0.8) and N (N = 300) and several values of α, in the range α = 0.2 up to α = 2.0. One notices that, for small values of α one is found in the nonextensive
U.L. Fulco et al. / Physics Letters A 312 (2003) 331–335
Fig. 1. The percolation order parameter, P∞ , versus the first-neighbor bond-percolation occupancy probability, p, for fixed values of N and α, and several values of the site-occupancy probability ps .
Fig. 2. The percolation order parameter, P∞ , versus the first-neighbor bond-percolation occupancy probability, p, for fixed values of N and ps , and typical values of α.
regime, for which P∞ equals unit, ∀p > 0, leading to pc (α) = 0. However, for increasing values of α, one gets pc (α) > 0, in such a way that as one approaches the first-neighbor bond-percolation limit (typically, α ∼ 2), large fluctuations are observed, due to finite size effects; in such a case, one should get, in the
333
Fig. 3. The percolation order parameter, P∞ , versus the first-neighbor bond-percolation occupancy probability, p, for a fixed value of α, two different site-occupancy probabilities, and several system sizes.
thermodynamic limit, P∞ = 0 and pc (α) = 1. In Fig. 3 we show how the order parameter evolves with the size of the system; we consider two different site concentrations (ps = 0.6 and ps = 0.2) and a fixed value of α (α = 0.8). Once again, one observes that the finite-size effects become more relevant for lower site concentrations (cf. the different sizes considered for the case ps = 0.2 of Fig. 3), in such a way that more reliable estimates of P∞ and pc (α) are obtained at large system sizes (typically, N = 1000). By locating, for finite values of N , the abscissa points given by the linear continuations of the curves of P∞ (p), and then extrapolating for N → ∞, one may estimate the critical values of the first-neighbor bond occupancy probability, for fixed values of ps and different values of α, pc (α). In Fig. 4 we exhibit how pc (α) varies with α for a typical value of the site concentration: ps = 0.8. The error bars express the accuracy of the linear continuations, together with the thermodynamic-limit extrapolations, of the curves of the parameter P∞ . One notices that the two characteristic values of α, α1 (ps ), below which P∞ equals unit, ∀p > 0, and α2 (ps ), above which P∞ = 0, ∀p < 1, decrease as the site concentration decreases. In analogy to what happens
334
U.L. Fulco et al. / Physics Letters A 312 (2003) 331–335
ps = 0.8: α1 (ps ) = 0.7 ± 0.05, α2 (ps ) = 1.9 ± 0.06, ps = 0.6: α1 (ps ) = 0.6 ± 0.07, α2 (ps ) = 1.8 ± 0.07, ps = 0.4: α1 (ps ) = 0.5 ± 0.09, α2 (ps ) = 1.7 ± 0.10.
Fig. 4. The critical value of the first-neighbor bond occupancy probability versus α, for the case ps = 0.8.
in the case of no site dilution (ps = 1.0) [16], the characteristic value α1 (ps ) sets in a crossover between the nonextensive (0 α α1 (ps )) and extensive (α > α1 (ps )) regimes. On the other hand, α2 (ps ) defines the onset of the well-known first-neighbor onedimensional percolation behavior; as a consequence of this, the region right below α2 (ps ) is very delicate from the numerical point of view. In Fig. 4, the last point to the right corresponds to the estimate of pc (α) associated with the value of α that is as close as possible (in the sense that one may still get a reliable estimate, within the computational effort employed herein) to α2 (ps ). For slightly higher values of α, strong fluctuations, driven by finite-size effects, occur; in this region, in order to observe the behavior of the order parameter P∞ (i.e., whether it drops to zero ∀p < 1, or not) one needs to run much larger system sizes, as well as to increase reasonably the number of samples, as compared to those used in the present work. Besides the site-occupancy probability ps = 0.8, we have also investigated the cases ps = 0.6 and ps = 0.4, which lead to similar plots to the one exhibited in Fig. 4, except that the characteristic values α1 (ps ) and α2 (ps ) appear shifted to the left. Below, we present our estimates for the two characteristic values of α, for each one of site concentrations considered,
The above results, together with those for the case ps = 1.0, i.e., α1 (ps ) = 1.0 and α2 (ps ) = 2.0 [16], do surely indicate that the characteristic values of α decrease with the concentration of sites. To conclude, we have investigated the effects of the dilution of sites (with a site occupancy probability ps ) in the long-range one-dimensional bond-percolation problem, by means of Monte Carlo simulations. The range of the bonds is controlled by a parameter α, in such a way that two well-known limits may be recovered, namely, the mean-field (α = 0), and the first-neighbor bond percolation (α → ∞) ones. We have shown that two characteristic values of α occur, whose values depend on the site-occupancy probability, α1 (ps ) and α2 (ps ) (α2 (ps ) > α1 (ps ) 0). For 0 α α1 (ps ), the percolation order parameter P∞ is always maximum (equal to unit), indicating that one is in a nonextensive regime, whereas for α > α2 (ps ), the order parameter is always zero, in such a way that one is found in the well-known first-neighbor onedimensional bond-percolation regime. The crossover between the nonextensive and extensive regimes takes place at α = α1 (ps ) and for α1 (ps ) < α < α2 (ps ) the order parameter displays a familiar behavior, i.e., varying monotonically between its minimum and maximum possible values. We have shown that the two characteristic values of α, α1 (ps ) and α2 (ps ), decrease with the inclusion of diluted sites. Finally, it is important to mention that, apart from its relevance in the study of nonextensive systems, the site-diluted longrange bond-percolation models are closely related to a wide variety of complex networks, such as internet, www, and small-world economic networks [19].
Acknowledgements It is a pleasure to thank D. Stauffer for fruitful conversations. The partial financial supports from CNPq,
U.L. Fulco et al. / Physics Letters A 312 (2003) 331–335
CAPES, Pronex/MCT, and ANP/CTPETRO/CENPES/Petrobras (Brazilian agencies) are acknowledged.
References [1] R.K. Pathria, Statistical Mechanics, Butterworth, Washington, DC, 1996. [2] H.E. Stanley, Introduction to Phase Transitions and Critical Phenomena, Oxford Univ. Press, New York, 1971. [3] P.T. Landsberg, Thermodynamics and Statistical Mechanics, Dover, New York, 1991. [4] J. Barré, D. Mukamel, S. Ruffo, Phys. Rev. Lett. 87 (2001) 030601. [5] R. Balescu, Equilibrium and Nonequilibrium Statistical Mechanics, Wiley, New York, 1974. [6] M. Kac, G.E. Uhlenbeck, P.C. Hemmer, J. Math. Phys. 4 (1963) 216. [7] C. Tsallis, Brazilian J. Phys. 29 (1999) 1.
335
[8] C. Tsallis, in: S. Abe, Y. Okamoto (Eds.), Nonextensive Statistical Mechanics and its Applications, in: Lecture Notes in Physics, Springer-Verlag, Berlin, 2001. [9] C. Tsallis, Chaos Solitons Fractals 13 (2002) 371. [10] C. Tsallis, J. Stat. Phys. 52 (1988) 479. [11] C. Anteneodo, C. Tsallis, Phys. Rev. Lett. 80 (1998) 5313. [12] A. Campa, A. Giansanti, D. Moroni, C. Tsallis, Phys. Lett. A 286 (2001) 251. [13] A. Campa, A. Giansanti, D. Moroni, Physica 305 (2002) 137. [14] V. Latora, A. Rapisarda, C. Tsallis, Phys. Rev. E 64 (2001) 056134. [15] V. Latora, A. Rapisarda, C. Tsallis, Physica A 305 (2002) 129. [16] H.H.A. Rego, L.S. Lucena, L.R. da Silva, C. Tsallis, Physica A 266 (1999) 42. [17] D. Stauffer, A. Aharony, Introduction to Percolation Theory, 2nd Edition, Taylor and Francis, London, 1994. [18] G. Grimmet, Percolation, 2nd Edition, Springer-Verlag, Berlin, 1999. [19] G. Corso, L.S. Lucena, Z.D. Thomé, Physica A 324 (2003) 430.