- Email: [email protected]
Crossover from extensive to nonextensive behavior driven by long-range d=1 bond percolation
Physica A 266 (1999) 42– 48
Crossover from extensive to nonextensive behavior driven by long-range d = 1 bond percolation Henio H.A. Regoa; ∗ , Liacir S. Lucenaa; 1 , Luciano R. da Silvaa;b; 2 , Constantino Tsallisa;b; 3
a Departamento b Centro
de FÃsica, Universidade Federal do Rio Grande do Norte, 59072-970 Natal-RN, Brazil Brasileiro de Pesquisas FÃsicas, Xavier Sigaud 150, 22290-180 Rio de Janeiro-RJ, Brazil
Abstract We present a Monte Carlo study of a linear chain (d = 1) with long-range bonds whose occupancy probabilities are given by pij = p=rij (06p61; ¿0) where rij = 1; 2; : : : is the distance between sites. The → ∞ ( = 0) corresponds to the rst-neighbor (“mean eld”) particular case. We exhibit that the order parameter P∞ equals unity ∀p ¿ 0 for 06 61, presents a familiar behavior (i.e., 0 for p6pc ( ) and nite otherwise) for 1 ¡ ¡ 2, and vanishes ∀p ¡ 1 for ¿ 2. Our results con rm recent conjecture, namely that the nonextensive region (06 61) can be meaningfully unfolded, as well as uniÿed with the extensive region ( ¿ 1), by exhibiting ∗ P∞ as a function of p∗ where (1 − p∗ ) = (1 − p)N (N ∗ ≡ (N 1− =d − 1)=(1 − =d); N being the number of sites of the chain). A corollary of this conjecture, now numerically veri ed, is that c 1999 Elsevier Science B.V. All rights reserved. pc ˙ ( − 1) in the → 1 + 0 limit. 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
The e ect of long-range interactions in Hamiltonian systems is nowadays actively studied (Ref. [1] and references therein). Indeed, they present a variety of interesting dynamical and thermodynamical phenomena, such as anomalous Lyapunov exponents, breakdown of standard thermodynamical extensivity and intensivity, anomalous di usion, and others. This intense activity is very well justi ed. Indeed, strong evidence is ∗
Corresponding author; e-mail: [email protected]. E-mail: [email protected]. 2 E-mail: [email protected]. 3 E-mail: [email protected]. 1
c 1999 Elsevier Science B.V. All rights reserved. 0378-4371/99/$ – see front matter PII: S 0 3 7 8 - 4 3 7 1 ( 9 8 ) 0 0 5 7 2 - X
H.H.A. Rego et al. / Physica A 266 (1999) 42– 48
43
accumulating in the sense that nontrivial violation of ergodicity occurs in such systems. This would possibly demand deep reformulation of our concepts concerning powerful formalisms such as statistical mechanics, and of course thermodynamics itself. In fact, a proposal along these lines has been advanced a decade ago [2,20,21], which uses a nonextensive entropic form (which recovers the usual Boltzmann–Gibbs entropy as a particular case, namely whenever extensivity is to be expected). The present work is related to this kind of approach, in a way that will become more clear later on. Since anomalies are present in long-range Hamiltonian systems, and since the Kasteleyn and Fortuin theorem [3] (among other results) enlightens us about the deep connections that can exist between thermal and geometrical systems, it is natural to expect similar thermodynamical anomalies also in long-range geometrical problems such as percolation. The purpose of the present e ort is to numerically study (Monte Carlo simulations) bond percolation in a N -sized open linear chain, all the sites of which are connected to all other sites through bonds which are present with probabilities pij between the sites i and j, separated by a distance rij (in crystalline units, i.e., rij = 0; 1; 2; : : :). The full problem is of course untractable since it would consist in establishing quantities such as the order parameter P∞ (probability of a randomly chosen site to belong, in the N → ∞ limit, to the in nite cluster that might be present) as a function of the whole set {pij }. We shall focus here a relevant particular case, namely that in which the bond occupancy probabilities are given by pij = p=rij where the occupancy probability p between rst neighbors satis es 06p61, and ¿0. We can see that the → ∞ limit corresponds to the standard, rst-neighbor, linear chain, for which it is well known that no phase transition exists at any p ¡ 1 (in fact, pc = 1). On the nontrivial cases of course are those with 06 6∞, to which the present study is dedicated, very especially the region in the neighborhood of = 1, value at which an extensive–nonextensive crossover occurs. Let us explain more. The thermodynamics (N → ∞) of the system is basically controlled by the following integral: Z N 1=d N 1− =d − 1 ∗ : (1) dr r d−1 r − = N ≡d 1 − =d 1 We can check that, in the N → ∞ limit, we have N∗ ∼
1 =d−1
N ∗ ∼ ln N ∗
N ∼
N 1− =d 1− =d
if
=d ¿ 1 ;
if
=d = 1 ;
if
06 =d ¡ 1 :
(2)
As will become transparent later on, what these regimes imply is that the system is extensive for =d ¿ 1 (hence standard thermodynamics apply), whereas it is nonextensive for 06 =d61, and special scalings become necessary [4,5,22–25] in order to have both a mathematically well posed problem, and a physical unfolding (or quali cation) of the nonextensive region. Furthermore, let us anticipate that, for our present d=1 case, two physically di erent subregions exist within the extensive region, namely
44
H.H.A. Rego et al. / Physica A 266 (1999) 42– 48
Fig. 1. p-dependence of P∞ for typical values of ( ; N ).
Fig. 2. -dependence of the N → ∞ extrapolated critical probability pc .
1 ¡ 62, for which 0 ¡ pc ¡ 1, and ¿ 2, for which pc = 1. A peculiarity occurs at = 2. Indeed, at this value, pc ( ) is discontinuous: the exact value pc (2) is still 2 unknown, but it has been conjectured [6,7] to be given by pc (2)=1−e−12= ' 0:703 : : : We present, in Fig. 1, P∞ (p) for typical values of ( ; N ). We notice that, while N increases, (i) P∞ (p) collapses towards the “left” in such a way that pc = 0 emerges, for 06 61; (ii) P∞ (p) converges towards an intermediate location, for 1 ¡ 62; (iii) P∞ (p) collapses towards the “right” in such a way that pc = 1 emerges, for ¿ 2. By locating, at nite values of N , the abcissa points determined through the linear continuations of the curves appearing in Fig. 1, and then extrapolating for N → ∞, we obtained the results presented in Fig. 2 (whose error bars have been obtained by evaluating the accuracy of this and similar extrapolation procedures). The region 1:5 ¡ ¡ 2 is numerically very delicate because of the vicinity of the = 2 anomaly.
H.H.A. Rego et al. / Physica A 266 (1999) 42– 48
45
In the region just above = 1, we have obtained pc ∼ A( − 1)
( → 1 + 0; A ' 0:9) :
(3)
This result con rms a conjecture made some time ago by one of us [4]. Indeed, it was argued that the mathematically well-de ned variables for expressing equations of states are not, generically speaking (i.e., ∀ ), variables such as the temperature T but rather the N ∗ -rescaled variables, like the rescaled temperature T ∗ ≡ T=N ∗ . If expressed in this variable, the successive approximations for the critical points are generically expected to remain ÿnite while N diverges. In other words, (Tc ( ))=N ∗ is expected to be nite, particularly in the neighborhood of the extensive–nonextensive crossover ( = d). For =d → 1 + 0, N ∗ = 1=( =d − 1), hence Tc ( ) ˙ 1=( =d − 1). If we use now the Kasteleyn and Fortuin isomorphism 1 p = 1 − exp(− ) ; T
(4)
where T is expressed in units of the magnetic coupling constant (J=kB ), we straightforwardly obtain −1 →1+0 : (5) pc ( ) ˙ d d It is precisely this conjecture that has been numerically con rmed in our present d = 1 model (see Eq. (3)). Let us now focus the nonextensive region. More precisely, let us unfold the pc = 0 result we have indistinctively obtained for all 61. The use of the variable T ∗ within Eq. (4) provides the following de nition for a rescaled occupancy probability: 1 − p∗ = (1 − p)N
∗
(6)
which, interestingly enough, is the composition law of N ∗ parallel bonds (see, for instance, [8]). In Fig. 3 we have presented, for typical values of in the nonextensive region, P∞ (p∗ ): we see now that, excepting in the vicinity of the still emerging critical point pc∗ , the curves associated with di erent values of N have collapsed. Finally, in Fig. 4 we present our extrapolated results for pc∗ . Summarizing, we have shown, for a geometrical model, that long-range interactions can generate, very analogously to what happens in Hamiltonian systems, extensive– nonextensive crossovers. In particular, the interesting data-collapsing and unfolding roles of the variable N ∗ unify the present bond percolation problem with long-range Lennard–Jones-like uids and Ising as well as Potts magnets [5,22–25]. Before concluding, let us comment on how the present study is possibly placed in the context of the previously mentioned nonextensive thermostatistics [2,21,22]. This statisP tics is based on a generalized entropic form, namely Sq ≡ (1 − i piq )=q − 1 (q ∈ R) P (hence, S1 = − i pi ln pi , the standard entropy). The nonextensivity of this form can be seen from the fact that, if A and B are two independent systems (in the sense that the probabilities associated with A+B factorize into those of A and B), then Sq (A+B)= Sq (A)+Sq (B)+(1−q) Sq (A) Sq (B). This formalism has received applications in a variety
46
H.H.A. Rego et al. / Physica A 266 (1999) 42– 48
Fig. 3. p∗ -dependence of P∞ for typical values of ( ; N ) (the numerical data are the same used in Fig. 1).
Fig. 4. -dependence of the N → ∞ extrapolated N ∗ -rescaled critical probability pc∗ (the numerical data are the same used in Fig. 3).
of situations such as self-gravitating systems [9,10,26,27], two-dimensional-like turbulence in pure-electron plasma [10,11], nonlinear maps [12,28,29], Levy-like [13,30 –33] and correlated-like [14,34 –36] anomalous di usions, solar neutrino problem [15,37], peculiar velocity distribution of galaxy clusters [16], cosmology [17], linear response theory [18], optimization techniques [19,38– 43], among others. The scenario which emerges is that, for nonextensive systems, the N → ∞ and the t → ∞ limits (t being the time) are not interchangeable. The present calculation is a q = 1 one because our Monte Carlo simulations rst apply the t → ∞ limit and only then apply the N → ∞ limit, whereas the q 6= 1 case is expected to appear when the limits are taken the other way around.
H.H.A. Rego et al. / Physica A 266 (1999) 42– 48
47
We gratefully acknowledge A.C.N. de Magalh˜aes and J.L. Lebowitz for useful remarks. The present work has received partial nancial support by CNPq and PRONEX (Brazilian agencies). References [1] C. Anteneodo, C. Tsallis, Phys. Rev. Lett. 80 (1998) 5313. [2] C. Tsallis, J. Stat. Phys. 52 (1988) 479; A regularly updated Bibliography on the subject is accessible at http://tsallis.cat.cbpf.br/biblio.htm. [3] P. Kasteleyn, C. Fortuin, J. Phys. Soc. Jap. (suppl.) 26 (1969) 11. [4] C. Tsallis, Fractals 3 (1995) 541. [5] P. Jund, S.G. Kim, C. Tsallis, Phys. Rev. B 52 (1995) 50. [6] M. Aizenman, J.T. Chayes, L. Chayes, C.M. Newman, J. Stat. Phys. 50 (1988) 1. [7] S.A. Cannas, A.C.N. de Magalh˜aes, J. Phys. A 30 (1997) 3345. [8] C. Tsallis, A.C.N. de Magalh˜aes, Phys. Rep. 268 (1996) 305. [9] A. R. Plastino, A. Plastino, Phys. Lett. A 174 (1993) 384. [10] B.M. Boghosian, Phys. Rev. E 53 (1996) 4754. [11] C. Anteneodo, C. Tsallis, J. Mol. Liq. 71 (1997) 255. [12] C. Tsallis, A.R. Plastino, W.-M. Zheng, Chaos, Solitons and Fractals 8 (1997) 885. [13] P.A. Alemany, D.H. Zanette, Phys. Rev. E 49 (1994) R956. [14] A.R. Plastino, A. Plastino, Physica A 222 (1995) 347. [15] G. Kaniadakis, A. Lavagno, P. Quarati, Phys. Lett. B 369 (1996) 308. [16] A. Lavagno, G. Kaniadakis, M. Rego-Monteiro, P. Quarati, C. Tsallis, Astrophys. Lett. and Comm. 35 (1998) 449. [17] V.H. Hamity, D.E. Barraco, Phys. Rev. Lett. 76 (1996) 4664. [18] A.K. Rajagopal, Phys. Rev. Lett. 76 (1996) 3469. [19] D.A. Stariolo, C. Tsallis, in: D. Stau er (Ed.), Ann. Rev. Comp. Phys., vol. II, World Scienti c, Singapore, 1995, p. 343. [20] E.M.F. Curado, C. Tsallis, J. Phys. A 24 (1991) L69 [Corrigenda: 24 (1991) 3187 and 25 (1992) 1019.] [21] C. Tsallis, Phys. Lett. A 206 (1995) 389. [22] J.R. Grigera, Phys. Lett. A 217 (1996) 47. [23] S.A. Cannas, F.A. Tamarit, Phys. Rev. B 54 (1996) R12661. L.C. Sampaio, M.P. de Albuquerque, F.S. de Menezes, Phys. Rev. B 55 (1997) 5611. [24] S.A. Cannas, F.A. Tamarit, A.C.N. de Magalh˜aes, XX Brazilian Meeting of Condensed Matter Physics (10-14 June 1997, Caxambu, Brazil) communicated. [25] S. Curilef, C. Tsallis, Critical behavior in Lennard-Jones-like uids: Long-range interactions and nonextensivity, 1998, preprint. [26] J.J. Aly, in: F. Combes, E. Athanassoula (Eds.), N-body problems and gravitational dynamics, Proceedings of the Meeting held at Aussois-France (21–25 March 1993), Publications de l’Observatoire de Paris, Paris, 1993, p. 19. [27] A.R. Plastino, A. Plastino, Phys. Lett. A 193 (1994) 251. [28] U.M.S. Costa, M.L. Lyra, A.R. Plastino, C. Tsallis, Phys. Rev. E 26 (1997) 245. [29] M.L. Lyra, C. Tsallis, Phys. Rev. Lett. 80 (1998) 53. [30] D.H. Zanette, P.A. Alemany, Phys. Rev. Lett. 75 (1995) 366. [31] C. Tsallis, S.V.F. Levy, A.M.C. de Souza, R. Maynard, Phys. Rev. Lett. 75 (1995) 3589. [Erratum: Phys. Rev. Lett. 27 (1996) 5442.] [32] M.O. Caceres, C.E. Budde, Phys. Rev. Lett. 77 (1996) 2589. [33] D.H. Zanette, P.A. Alemany, Phys. Rev. Lett. 77 (1996) 2590. [34] C. Tsallis, D.J. Bukman, Phys. Rev. E 54 (1996) R2197. [35] A. Compte, D. Jou, J. Phys. A 29 (1996) 4321. [36] D.A. Stariolo, Phys. Rev. E 55 (1997) 4806. [37] P. Quarati, A. Carbone, G. Gervino, G. Kaniadakis, A. Lavagno, E. Miraldi, Nucl. Phys. A 621 (1997) 345c.
48 [38] [39] [40] [41] [42] [43]
H.H.A. Rego et al. / Physica A 266 (1999) 42– 48 T.J.P. Penna, Phys. Rev. E 51 (1995) R1. T.J.P. Penna, Comput. Phys. 9 (1995) 341. K.C. Mundim, C. Tsallis, Int. J. Quantum Chem. 58 (1996) 373. J. Schulte, Phys. Rev. E 53 (1996) 1348. I. Andricioaei, J.E. Straub, Phys. Rev. E 53 (1996) R3055. P. Serra, A.F. Stanton, S. Kais, Phys. Rev. E 55 (1997) 1162.
Crossover from extensive to nonextensive behavior driven by long-range d = 1 bond percolation Henio H.A. Regoa; ∗ , Liacir S. Lucenaa; 1 , Luciano R. da Silvaa;b; 2 , Constantino Tsallisa;b; 3
a Departamento b Centro
de FÃsica, Universidade Federal do Rio Grande do Norte, 59072-970 Natal-RN, Brazil Brasileiro de Pesquisas FÃsicas, Xavier Sigaud 150, 22290-180 Rio de Janeiro-RJ, Brazil
Abstract We present a Monte Carlo study of a linear chain (d = 1) with long-range bonds whose occupancy probabilities are given by pij = p=rij (06p61; ¿0) where rij = 1; 2; : : : is the distance between sites. The → ∞ ( = 0) corresponds to the rst-neighbor (“mean eld”) particular case. We exhibit that the order parameter P∞ equals unity ∀p ¿ 0 for 06 61, presents a familiar behavior (i.e., 0 for p6pc ( ) and nite otherwise) for 1 ¡ ¡ 2, and vanishes ∀p ¡ 1 for ¿ 2. Our results con rm recent conjecture, namely that the nonextensive region (06 61) can be meaningfully unfolded, as well as uniÿed with the extensive region ( ¿ 1), by exhibiting ∗ P∞ as a function of p∗ where (1 − p∗ ) = (1 − p)N (N ∗ ≡ (N 1− =d − 1)=(1 − =d); N being the number of sites of the chain). A corollary of this conjecture, now numerically veri ed, is that c 1999 Elsevier Science B.V. All rights reserved. pc ˙ ( − 1) in the → 1 + 0 limit. 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
The e ect of long-range interactions in Hamiltonian systems is nowadays actively studied (Ref. [1] and references therein). Indeed, they present a variety of interesting dynamical and thermodynamical phenomena, such as anomalous Lyapunov exponents, breakdown of standard thermodynamical extensivity and intensivity, anomalous di usion, and others. This intense activity is very well justi ed. Indeed, strong evidence is ∗
Corresponding author; e-mail: [email protected]. E-mail: [email protected]. 2 E-mail: [email protected]. 3 E-mail: [email protected]. 1
c 1999 Elsevier Science B.V. All rights reserved. 0378-4371/99/$ – see front matter PII: S 0 3 7 8 - 4 3 7 1 ( 9 8 ) 0 0 5 7 2 - X
H.H.A. Rego et al. / Physica A 266 (1999) 42– 48
43
accumulating in the sense that nontrivial violation of ergodicity occurs in such systems. This would possibly demand deep reformulation of our concepts concerning powerful formalisms such as statistical mechanics, and of course thermodynamics itself. In fact, a proposal along these lines has been advanced a decade ago [2,20,21], which uses a nonextensive entropic form (which recovers the usual Boltzmann–Gibbs entropy as a particular case, namely whenever extensivity is to be expected). The present work is related to this kind of approach, in a way that will become more clear later on. Since anomalies are present in long-range Hamiltonian systems, and since the Kasteleyn and Fortuin theorem [3] (among other results) enlightens us about the deep connections that can exist between thermal and geometrical systems, it is natural to expect similar thermodynamical anomalies also in long-range geometrical problems such as percolation. The purpose of the present e ort is to numerically study (Monte Carlo simulations) bond percolation in a N -sized open linear chain, all the sites of which are connected to all other sites through bonds which are present with probabilities pij between the sites i and j, separated by a distance rij (in crystalline units, i.e., rij = 0; 1; 2; : : :). The full problem is of course untractable since it would consist in establishing quantities such as the order parameter P∞ (probability of a randomly chosen site to belong, in the N → ∞ limit, to the in nite cluster that might be present) as a function of the whole set {pij }. We shall focus here a relevant particular case, namely that in which the bond occupancy probabilities are given by pij = p=rij where the occupancy probability p between rst neighbors satis es 06p61, and ¿0. We can see that the → ∞ limit corresponds to the standard, rst-neighbor, linear chain, for which it is well known that no phase transition exists at any p ¡ 1 (in fact, pc = 1). On the nontrivial cases of course are those with 06 6∞, to which the present study is dedicated, very especially the region in the neighborhood of = 1, value at which an extensive–nonextensive crossover occurs. Let us explain more. The thermodynamics (N → ∞) of the system is basically controlled by the following integral: Z N 1=d N 1− =d − 1 ∗ : (1) dr r d−1 r − = N ≡d 1 − =d 1 We can check that, in the N → ∞ limit, we have N∗ ∼
1 =d−1
N ∗ ∼ ln N ∗
N ∼
N 1− =d 1− =d
if
=d ¿ 1 ;
if
=d = 1 ;
if
06 =d ¡ 1 :
(2)
As will become transparent later on, what these regimes imply is that the system is extensive for =d ¿ 1 (hence standard thermodynamics apply), whereas it is nonextensive for 06 =d61, and special scalings become necessary [4,5,22–25] in order to have both a mathematically well posed problem, and a physical unfolding (or quali cation) of the nonextensive region. Furthermore, let us anticipate that, for our present d=1 case, two physically di erent subregions exist within the extensive region, namely
44
H.H.A. Rego et al. / Physica A 266 (1999) 42– 48
Fig. 1. p-dependence of P∞ for typical values of ( ; N ).
Fig. 2. -dependence of the N → ∞ extrapolated critical probability pc .
1 ¡ 62, for which 0 ¡ pc ¡ 1, and ¿ 2, for which pc = 1. A peculiarity occurs at = 2. Indeed, at this value, pc ( ) is discontinuous: the exact value pc (2) is still 2 unknown, but it has been conjectured [6,7] to be given by pc (2)=1−e−12= ' 0:703 : : : We present, in Fig. 1, P∞ (p) for typical values of ( ; N ). We notice that, while N increases, (i) P∞ (p) collapses towards the “left” in such a way that pc = 0 emerges, for 06 61; (ii) P∞ (p) converges towards an intermediate location, for 1 ¡ 62; (iii) P∞ (p) collapses towards the “right” in such a way that pc = 1 emerges, for ¿ 2. By locating, at nite values of N , the abcissa points determined through the linear continuations of the curves appearing in Fig. 1, and then extrapolating for N → ∞, we obtained the results presented in Fig. 2 (whose error bars have been obtained by evaluating the accuracy of this and similar extrapolation procedures). The region 1:5 ¡ ¡ 2 is numerically very delicate because of the vicinity of the = 2 anomaly.
H.H.A. Rego et al. / Physica A 266 (1999) 42– 48
45
In the region just above = 1, we have obtained pc ∼ A( − 1)
( → 1 + 0; A ' 0:9) :
(3)
This result con rms a conjecture made some time ago by one of us [4]. Indeed, it was argued that the mathematically well-de ned variables for expressing equations of states are not, generically speaking (i.e., ∀ ), variables such as the temperature T but rather the N ∗ -rescaled variables, like the rescaled temperature T ∗ ≡ T=N ∗ . If expressed in this variable, the successive approximations for the critical points are generically expected to remain ÿnite while N diverges. In other words, (Tc ( ))=N ∗ is expected to be nite, particularly in the neighborhood of the extensive–nonextensive crossover ( = d). For =d → 1 + 0, N ∗ = 1=( =d − 1), hence Tc ( ) ˙ 1=( =d − 1). If we use now the Kasteleyn and Fortuin isomorphism 1 p = 1 − exp(− ) ; T
(4)
where T is expressed in units of the magnetic coupling constant (J=kB ), we straightforwardly obtain −1 →1+0 : (5) pc ( ) ˙ d d It is precisely this conjecture that has been numerically con rmed in our present d = 1 model (see Eq. (3)). Let us now focus the nonextensive region. More precisely, let us unfold the pc = 0 result we have indistinctively obtained for all 61. The use of the variable T ∗ within Eq. (4) provides the following de nition for a rescaled occupancy probability: 1 − p∗ = (1 − p)N
∗
(6)
which, interestingly enough, is the composition law of N ∗ parallel bonds (see, for instance, [8]). In Fig. 3 we have presented, for typical values of in the nonextensive region, P∞ (p∗ ): we see now that, excepting in the vicinity of the still emerging critical point pc∗ , the curves associated with di erent values of N have collapsed. Finally, in Fig. 4 we present our extrapolated results for pc∗ . Summarizing, we have shown, for a geometrical model, that long-range interactions can generate, very analogously to what happens in Hamiltonian systems, extensive– nonextensive crossovers. In particular, the interesting data-collapsing and unfolding roles of the variable N ∗ unify the present bond percolation problem with long-range Lennard–Jones-like uids and Ising as well as Potts magnets [5,22–25]. Before concluding, let us comment on how the present study is possibly placed in the context of the previously mentioned nonextensive thermostatistics [2,21,22]. This statisP tics is based on a generalized entropic form, namely Sq ≡ (1 − i piq )=q − 1 (q ∈ R) P (hence, S1 = − i pi ln pi , the standard entropy). The nonextensivity of this form can be seen from the fact that, if A and B are two independent systems (in the sense that the probabilities associated with A+B factorize into those of A and B), then Sq (A+B)= Sq (A)+Sq (B)+(1−q) Sq (A) Sq (B). This formalism has received applications in a variety
46
H.H.A. Rego et al. / Physica A 266 (1999) 42– 48
Fig. 3. p∗ -dependence of P∞ for typical values of ( ; N ) (the numerical data are the same used in Fig. 1).
Fig. 4. -dependence of the N → ∞ extrapolated N ∗ -rescaled critical probability pc∗ (the numerical data are the same used in Fig. 3).
of situations such as self-gravitating systems [9,10,26,27], two-dimensional-like turbulence in pure-electron plasma [10,11], nonlinear maps [12,28,29], Levy-like [13,30 –33] and correlated-like [14,34 –36] anomalous di usions, solar neutrino problem [15,37], peculiar velocity distribution of galaxy clusters [16], cosmology [17], linear response theory [18], optimization techniques [19,38– 43], among others. The scenario which emerges is that, for nonextensive systems, the N → ∞ and the t → ∞ limits (t being the time) are not interchangeable. The present calculation is a q = 1 one because our Monte Carlo simulations rst apply the t → ∞ limit and only then apply the N → ∞ limit, whereas the q 6= 1 case is expected to appear when the limits are taken the other way around.
H.H.A. Rego et al. / Physica A 266 (1999) 42– 48
47
We gratefully acknowledge A.C.N. de Magalh˜aes and J.L. Lebowitz for useful remarks. The present work has received partial nancial support by CNPq and PRONEX (Brazilian agencies). References [1] C. Anteneodo, C. Tsallis, Phys. Rev. Lett. 80 (1998) 5313. [2] C. Tsallis, J. Stat. Phys. 52 (1988) 479; A regularly updated Bibliography on the subject is accessible at http://tsallis.cat.cbpf.br/biblio.htm. [3] P. Kasteleyn, C. Fortuin, J. Phys. Soc. Jap. (suppl.) 26 (1969) 11. [4] C. Tsallis, Fractals 3 (1995) 541. [5] P. Jund, S.G. Kim, C. Tsallis, Phys. Rev. B 52 (1995) 50. [6] M. Aizenman, J.T. Chayes, L. Chayes, C.M. Newman, J. Stat. Phys. 50 (1988) 1. [7] S.A. Cannas, A.C.N. de Magalh˜aes, J. Phys. A 30 (1997) 3345. [8] C. Tsallis, A.C.N. de Magalh˜aes, Phys. Rep. 268 (1996) 305. [9] A. R. Plastino, A. Plastino, Phys. Lett. A 174 (1993) 384. [10] B.M. Boghosian, Phys. Rev. E 53 (1996) 4754. [11] C. Anteneodo, C. Tsallis, J. Mol. Liq. 71 (1997) 255. [12] C. Tsallis, A.R. Plastino, W.-M. Zheng, Chaos, Solitons and Fractals 8 (1997) 885. [13] P.A. Alemany, D.H. Zanette, Phys. Rev. E 49 (1994) R956. [14] A.R. Plastino, A. Plastino, Physica A 222 (1995) 347. [15] G. Kaniadakis, A. Lavagno, P. Quarati, Phys. Lett. B 369 (1996) 308. [16] A. Lavagno, G. Kaniadakis, M. Rego-Monteiro, P. Quarati, C. Tsallis, Astrophys. Lett. and Comm. 35 (1998) 449. [17] V.H. Hamity, D.E. Barraco, Phys. Rev. Lett. 76 (1996) 4664. [18] A.K. Rajagopal, Phys. Rev. Lett. 76 (1996) 3469. [19] D.A. Stariolo, C. Tsallis, in: D. Stau er (Ed.), Ann. Rev. Comp. Phys., vol. II, World Scienti c, Singapore, 1995, p. 343. [20] E.M.F. Curado, C. Tsallis, J. Phys. A 24 (1991) L69 [Corrigenda: 24 (1991) 3187 and 25 (1992) 1019.] [21] C. Tsallis, Phys. Lett. A 206 (1995) 389. [22] J.R. Grigera, Phys. Lett. A 217 (1996) 47. [23] S.A. Cannas, F.A. Tamarit, Phys. Rev. B 54 (1996) R12661. L.C. Sampaio, M.P. de Albuquerque, F.S. de Menezes, Phys. Rev. B 55 (1997) 5611. [24] S.A. Cannas, F.A. Tamarit, A.C.N. de Magalh˜aes, XX Brazilian Meeting of Condensed Matter Physics (10-14 June 1997, Caxambu, Brazil) communicated. [25] S. Curilef, C. Tsallis, Critical behavior in Lennard-Jones-like uids: Long-range interactions and nonextensivity, 1998, preprint. [26] J.J. Aly, in: F. Combes, E. Athanassoula (Eds.), N-body problems and gravitational dynamics, Proceedings of the Meeting held at Aussois-France (21–25 March 1993), Publications de l’Observatoire de Paris, Paris, 1993, p. 19. [27] A.R. Plastino, A. Plastino, Phys. Lett. A 193 (1994) 251. [28] U.M.S. Costa, M.L. Lyra, A.R. Plastino, C. Tsallis, Phys. Rev. E 26 (1997) 245. [29] M.L. Lyra, C. Tsallis, Phys. Rev. Lett. 80 (1998) 53. [30] D.H. Zanette, P.A. Alemany, Phys. Rev. Lett. 75 (1995) 366. [31] C. Tsallis, S.V.F. Levy, A.M.C. de Souza, R. Maynard, Phys. Rev. Lett. 75 (1995) 3589. [Erratum: Phys. Rev. Lett. 27 (1996) 5442.] [32] M.O. Caceres, C.E. Budde, Phys. Rev. Lett. 77 (1996) 2589. [33] D.H. Zanette, P.A. Alemany, Phys. Rev. Lett. 77 (1996) 2590. [34] C. Tsallis, D.J. Bukman, Phys. Rev. E 54 (1996) R2197. [35] A. Compte, D. Jou, J. Phys. A 29 (1996) 4321. [36] D.A. Stariolo, Phys. Rev. E 55 (1997) 4806. [37] P. Quarati, A. Carbone, G. Gervino, G. Kaniadakis, A. Lavagno, E. Miraldi, Nucl. Phys. A 621 (1997) 345c.
48 [38] [39] [40] [41] [42] [43]
H.H.A. Rego et al. / Physica A 266 (1999) 42– 48 T.J.P. Penna, Phys. Rev. E 51 (1995) R1. T.J.P. Penna, Comput. Phys. 9 (1995) 341. K.C. Mundim, C. Tsallis, Int. J. Quantum Chem. 58 (1996) 373. J. Schulte, Phys. Rev. E 53 (1996) 1348. I. Andricioaei, J.E. Straub, Phys. Rev. E 53 (1996) R3055. P. Serra, A.F. Stanton, S. Kais, Phys. Rev. E 55 (1997) 1162.