Non-extensive block entropy statistics of Cantor fractal sets

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Physica A 381 (2007) 148–154 www.elsevier.com/locate/physa

Non-extensive block entropy statistics of Cantor fractal sets A. Provata Institute of Physical Chemistry, National Center for Scientific Research ‘‘Demokritos’’, 15310 Athens, Greece Received 30 January 2007; received in revised form 26 March 2007 Available online 13 April 2007

Abstract By using non-extensive block entropy statistics, we demonstrate analytically that the static structures of deterministic Cantor sets with fractal dimension d f are characterised by a non-extensive q-exponent q ¼ 1=ðd f  dÞ, for d f od (where d is the embedding dimensions of the fractal set). To calculate the S q entropy we use the block entropy method based on nonoverlapping windows and standard exact enumeration on Cantor sets. This result indicates that fractal structures are formed via dynamical processes which operate ‘‘at the edge of chaos’’. r 2007 Elsevier B.V. All rights reserved. Keywords: Fractals; Non-extensive statistics; Block entropy

1. Introduction In recent studies particular attention has been paid to phenomena ‘‘at the edge of chaos’’, termed also as ‘‘weak chaos’’, where the trajectories in the phase space do not diverge exponentially, but follow a more weak, power law divergence. This regime is rather frequently manifested in nature, especially in out-of-equilibrium phenomena, and in equilibrium near criticality. Distinct features of the edge of chaos regime are the emergence of power law distributions and fractals. Studies on diverse fields in statistical mechanics, such as critical phenomena [1], anomalous diffusion [2], reactive dynamics [3], dynamics of growth [4] population and urban dynamics [5–8] and DNA sequences [9–12] have demonstrated power law and fractal behavior which may be attributed to the edge of chaos dynamics. To study such complex phenomena, the concept of non-extensive entropy was introduced by Tsallis in 1988 [13] and was later adopted and refined in diverse fields demonstrating power law dynamics and fractality [14]. The non-extensive entropy is defined as P q 1 W i¼1 pi Sq ¼ for qa1 q1 W X ¼  pi ln pi for q ¼ 1, i¼1

E-mail address: [email protected]. 0378-4371/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2007.03.055

ð1Þ

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where pi denotes the probabilities of the ith microscopic states, the average runs over the total number of states W and q is the exponent which characterises the particular statistics. Note that for q ¼ 1 the classical Boltzmann–Gibbs (BG) statistics is recovered and thus departure of the exponent q from the value 1 signals departure from BG statistics. The interpretation of pi in Eq. (1) is not always easy and it depends strongly on the specific definition of a microscopic state and hence on the level of description of the system. For Cantor sets, which we will study in the sequel, it is natural to use the block entropy concept, since a Cantor set at the jth generation is constructed as an iteration of a given number of known blocks. The classic Shannon block entropy based on blocks of size s is given in terms of pi ðsÞ by Hs ¼ 

W X

pi ðsÞ ln pi ðsÞ

(2)

i¼1

and has been used extensively in the literature for the study of binary sequences, symbolic sequences and applications [15–18]. By extending this to the non-extensive entropy we obtain the non-extensive block entropy P q 1 W i¼1 pi ðsÞ for qa1 Sq ðsÞ ¼ q1 W X ¼  pi ðsÞ ln pi ðsÞ for q ¼ 1 ð3Þ i¼1

which is now a function of the block size s and the average runs over all windows. The non-extensive block entropy has been used in the study of symbolic sequences and applications [19] and for the temporal evolution of entropy in time-dependent systems [20–23]. Using this non-extensive block entropy we will demonstrate analytically that all Cantor sets, in the limit of infinite system size are described by a single q exponent related directly to their fractal dimension as q ¼ 1=ðd f  dÞ, where d is the embedding dimensions. This demonstrates directly that there is a one-to-one correspondence between a fractal of dimension d f and the corresponding qðd f Þ-entropy. We may further say that the construction of a fractal may be attributed to a dynamical process whose entropy is governed by the qðd f Þ-exponent. Fractals then may emerge in nature as expressions of edge of chaos dynamics characterised by the appropriate non-extensive entropy production. In the next section we introduce the construction of fractal sets and the calculation of the pi ðsÞ. As a working example we consider always the triadic Cantor set, and we give also the analytic expressions for any Cantor set of dimension d f o1. Using analytical arguments and numerical integrations, we extract the relation between the exponent q and the fractal dimension d f . In Section 3 we study the case of d f 41. In the final section we draw our main conclusions and we discuss open problems. 2. Block entropy for Cantor sets with d f o1 Consider first a one-dimensional symbolic sequence of f1; 0g, as the Nth iteration of a Cantor set. As an example consider the triadic Cantor set (TCS) given, for the first three stages of iteration, as 0th iteration (1), 1st iteration (101), 2nd iteration (101000101), 3rd iteration (101000101000000000101000101) etc. In the general case, the ðm; rÞ-Cantor set (mpr), at the Nth level of iteration has size L ¼ rN , contains M ¼ mN times the symbol ‘‘1’’, and L  M times the symbol ‘‘0’’. The positions of ‘‘1’’ and ‘‘0’’ are deterministically defined on the symbolic sequence for all values of N. In the limit of N ! 1 the ðm; rÞ-Cantor sets correspond to fractals with fractal dimension d f ¼ ln m= ln r. In the case of the TCS, d f ¼ ln 2= ln 3. To these sets, seen as one-dimensional structures, an entropy may be attributed, which is a measure of their order, or a measure of their distance from a completely ordered, or a completely disordered structure.

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As entropy appropriate for describing the Cantor sets we propose the block entropy, Eq. (3). We define sliding, non-overlapping windows (blocks) of size s ¼ rs oL ¼ rN and we calculate the frequency of occurrence of all different window configurations. Note that at the first level of iteration N ¼ 1, there is only one possible configuration, which also describes the iteration law and which we call 1-segment. For NX2 the Cantor set consists of a deterministic collection of two types of segments, the 1-segments and the 0-segments. It is only possible to calculate the block entropy for block sizes s ¼ rs , where 1ps5N. In particular, at the Nth level of iteration it is obvious to show that we only have two types of segments of size s ¼ rs which occur with the following probability: p1 ðsÞ ¼ ðm=rÞNs for 1segments, p0 ðsÞ ¼ 1  p1 ðsÞ for 0segments.

ð4Þ

For example, for the TCS ½r ¼ 3; m ¼ 2, at the N ¼ 3 level of iteration, the system size is L ¼ rN ¼ 27 and if we use blocks of sizes s ¼ 3, ðs ¼ 1Þ, we have two types of segments: the (101) ð¼ 1segmentsÞ which appear with probability p1 ¼ ðm=rÞNs ¼ ð2=3Þ2 and the (000) ð¼ 0segmentsÞ which appear with probability p0 ¼ 1  p1 . If we use blocks of sizes s ¼ 9, ðs ¼ 2Þ, we similarly have two types of segments: the (101000101) ð¼ 1segmentsÞ which appear with probability p1 ¼ ðm=rÞNs ¼ ð2=3Þ1 and the (000000000) ð¼ 0segmentsÞ which appear with probability p0 ¼ 1  p1 . Combining Eqs. (3) and (4) we obtain the exact formula for the non-extensive block entropy of any ðm; rÞCantor set of size L based on blocks of sizes s (see Eq. (5)). Sq ðs; LÞ ¼

1  ðm=rÞqðln Lln sÞ= ln r  ð1  ðm=rÞðln Lln sÞ= ln r Þq q1

for qa1,

      m ðln Lln sÞ= ln r m ðln Lln sÞ= ln r S1 ðs : LÞ ¼  ln r r  mðln Lln sÞ= ln r   mðln Lln sÞ= ln r   1 ln 1  r r

for q ¼ 1.

ð5Þ

Eq. (5) holds provided that s ¼ rs and soN and that all fs; r; Ng are positive integers. These formulas are complicated and it is difficult to further simplify them. To obtain the proper q-value we calculate S q for various values of q and L. The appropriate q is the one for which the corresponding entropy increases linearly with the system size L. 2e+08

q=-1.6 q=-1.5

Sq

1.5e+08

1e+08

q=-1.4 5e+07

q=-1.3 q=-1.2 0 0

5e+08

1e+09

1.5e+09

2e+09

2.5e+09

L Fig. 1. The values of the entropy Sq as a function of L for the deterministic (2,12)-Cantor set. Sq demonstrates linear increase with L for q ¼ 1:4.

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In Fig. 1 we depict the characteristic L dependence of the non-extensive block entropy S q for the (2,12)Cantor set, for s ¼ 12 and for various values of q. For qo  1:4 the corresponding Sq increases exponentially, while for q4  1:4 the S q increase is slower than linear. Thus the appropriate q which attributes to S q linear increase with the system size is q ¼ 1:4  0:05. The error in the value of q is from fitting estimation. The estimated value of q does not change with the window size s, provided that soL. In Fig. 2 we present collective results from fitting data for several values of d f ¼ ln m= ln r. In the same figure we also plot the function qth ¼ 1=ðd f  1Þ. The fit is remarkably good and demonstrates that the q-exponent which describes the spatial distribution of monofractals depends solely on the fractal dimension d f as q¼

1 df  1

for d f a1.

(6)

Eq. (6) is remarkably simple and seems to hold for all mono-fractals and connects the spatial inhomogeneity with the distance of q from the value 1, which correspond to homogeneous distributions. Moreover, q depends only on d f independently of the specific values of m and r. We present here a theoretical argument which further supports the relation, Eq. (6), between the q-exponent and the fractal dimensions. From Eq. (5) the second term related to the probability for the 1-segments may be expressed as mqðln Lln sÞ= ln r pq1 ¼ . (7) r Using the identity xln y ¼ yln x , Eq. (7) reads m

m

pq1 ¼ Lq lnð r Þ= ln r  sq lnð r Þ= ln r ¼ Lðd f 1Þq  sðd f 1Þq .

(8)

Using Eq. (8), Eq. (5) reads Sq ðs; LÞ ¼

1  ðs=LÞqð1d f Þ  ð1  ðs=LÞð1d f Þ Þq q1

for qa1.

(9)

For q ¼ 1, Eq. (5) reads         s ð1d f Þ    s ð1d f Þ  s ð1d f Þ s ð1d f Þ S1 ðs : LÞ ¼  ln  1 ln 1  L L L L

for q ¼ 1.

(10)

0 -1 -2 -3

q-value

-4 -5 -6

qcalc

-7

qth -8 -9 -10 0

0.2

0.4

0.6

0.8

1

df Fig. 2. The values of the exponent q as a function of the fractal dimension d f for deterministic Cantor sets. The points (with error bars) correspond to linear Sq in Eq. (5), while the continuous curve represents the theoretical results, Eq. (6).

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152 10 10

q=-1.6, y=1.13

Sq

q=-1.5,y=1.05 10 8

q=-1.4,y=0.98 q=-1.3,y=0.91 q=-1.2,y=0.84

10

q=-1.1,y=0.77

6

q=-1.0,y=0.70

10 4

102

-8

10

-6

10

-4

10

-2

100

10

s-1 Fig. 3. The values of the entropy Sq as a function of s1 for the deterministic (2,12)-Cantor set. Sq demonstrates linear increase with s1 for q ¼ 1:4. y denotes the corresponding correlation coefficient.

For finite window size s and large values of the system size L, with Lbs , Eq. (9) can be expanded in terms of s=L and reduces to Sq ðs; LÞ ¼

qðs=LÞ1d f  ðs=LÞqð1d f Þ q1

for qa1.

(11)

If we assume linear increase of the non-extensive entropy with the system size then from the second term we obtain q ¼ 1=ðd f  1Þ as was also shown earlier, while the first term vanishes for large system sizes, Lbs. This formula also indicates that the entropy S q increases inversely with s for constant L and for the same values of q ¼ 1=ðd f  1Þ. This may be demonstrated in Fig. 3 where we present the entropy S q as a function of s1 , for the (2,12)-Cantor set and for lattice size L ¼ 1210 . In a double logarithmic scale, Sq is plotted for various values of q. The dotted line in Fig. 3 denotes a linear increase in s1 (or inverse linear increase in s). The value of the correlation coefficients y for linear fits to the curves are also calculated and they are reported in the same figure. The correlation coefficient y is a measure of the quality of the fit and perfect agreement between the data and the fit is obtained when the correlation coefficient is 1. Comparing the different S q curves it is obvious that linear increase (correlation coefficient closest to 1) is achieved near q ¼ 1:4. The theoretical value of q obtained from Eq. (6), is q ¼ 1:38685. 3. Block entropy of higher dimensional Cantor sets The previous discussion holds also for higher dimensional Cantor fractals. We first discuss the case of an ðm; rÞ-Cantor set embedded in d ¼ 2 dimensions and later we generalise for arbitrary d. To construct the ðm; rÞCantor set in 2 dimensions we iterate 2-d windows of linear size r which contain m times the symbol ‘‘1’’ and r2  m times the symbol ‘‘0’’. At the Nth level of iteration the set has size L  L ¼ rN  rN , contains M ¼ mN times the symbol ‘‘1’’, and L2  M times the symbol ‘‘0’’. In the limit of N ! 1 the ðm; rÞ-Cantor sets have fractal dimension d f ¼ ln m= ln r; with d f p2. To study the Sq entropy we use here again the non-overlapping windows method using now blocks of size s  s ¼ r2s oL2 ¼ r2N . For NX2 the Cantor set consists of a deterministic collection of two types of blocks: (a) the 0-blocks (blocks which contain only the symbol ‘‘0’’and (b) 1-blocks containing the symbols ‘‘1’’ and ‘‘0’’. The same procedure holds for arbitrary embedding dimensions d: the system has size Ld ¼ rNd , while the non-overlapping windows are d-dimensional blocks of size sd ¼ rsd . In particular, at the Nth level of iteration it is obvious to show that we only have two types of blocks of linear size s ¼ rs which occur with

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the following probability: p1 ðsÞ ¼ ðm=rd ÞNs for 1blocks, p0 ðsÞ ¼ 1  p1 ðsÞ for 0blocks.

ð12Þ

Using these probabilities we can write the non-extensive entropy Sq , in analogy with Eqs. (5), for Cantor sets embedded in d-dimensions 1  ðm=rd Þqðln Lln sÞ= ln r  ð1  ðm=rd Þðln Lln sÞ= ln r Þq for qa1 q1       m ðln Lln sÞ= ln r m ðln Lln sÞ= ln r ¼  ln rd rd   m ðln Lln sÞ= ln r    m ðln Lln sÞ= ln r   1 d ln 1  d for q ¼ 1. r r

Sq ðs; LÞ ¼

ð13Þ

Similar to the case d ¼ 1, Eqs. (13) may be simplified as Sq ðs; LÞ ¼

1  ðs=LÞqðdd f Þ  ð1  ðs=LÞðdd f Þ Þq q1

for qa1

and for q ¼ 1, Eq. (13) reads         s ðdd f Þ    s ðdd f Þ  s ðdd f Þ s ðdd f Þ S1 ðs : LÞ ¼  ln  1 ln 1  for q ¼ 1. L L L L

(14)

(15)

Following the same procedure and arguments as in the previous section we conclude that for large system sizes the S q entropy of Cantor sets embedded in d-dimensions reduces to Sq ðs; LÞ ¼

qðs=LÞdd f  ðs=LÞqðdd f Þ q1

for qa1.

(16)

If again we assume linear increase with the system size L we conclude for the general case of a fractal Cantor set embedded in d-dimensional space that q¼

1 df  d

for d f ad.

(17)

Eq. (17) connects directly fractality with non-extensivity. As in the case d ¼ d f ¼ 1 (in Section 2), the formula (17) diverges when the fractal tends to cover homogeneously the d-dimensional space. It should be added here that the BG regime cannot be achieved by the present model system (Cantor sets). The BG regime should correspond to a short range correlated sequence. The case d f ¼ d, which corresponds to a fully covered system, results to entropy strictly null, see Eq. (14). The same is true in the case of Cantor sets in one-dimensions, in Section 2, for d f ¼ 1. 4. Conclusions Using the block entropy method we study the non-extensive properties of deterministic Cantor fractal sets. We show that the entropy associated with the structure of fractals having dimension d f and being embedded in a d-dimensional space is characterised by a single, non-extensive exponent q ¼ 1=ðd f  dÞ for d f od. We may then expect that fractals in nature result due to processes which have non-extensive entropic character and ‘‘edge of chaos’’ dynamics. Note that this approach is different from the approach in Ref. [24] where the case of multifractals is studied. In Ref. [24] a multifractal is considered as a collection of sites which contain non-equal masses. The mass takes specific values defined through an iterative procedure. Each specific mass value is considered as a state and the probabilities are defined as the number of times a specific mass is found in the system. In other words only windows of size s ¼ 1 are used, but the values assigned on each window vary. [In our approach the system is covered by windows (blocks) of size sd and a state is defined as a particular configuration of a block.] Likewise,

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a different approach is proposed in Ref. [25], where an energy E i is associated with a certain configuration of a fractal binary sequence. Then the equilibrium probabilities are defined by the associated Gibbs distributions and using the non-extensive approach, Eq. (1), a q-exponent may be calculated for the fractal sequence. If the frequency is associated with the probability and its rank with the energy, the corresponding Zipf exponent can also be calculated. For the sake of completeness it is necessary to add here that exact relations between the entropic parameter q and characteristics of the multifractal spectrum of the attractors in some conservative and dissipative maps/ models have been recently reported in the literature [26–28]. The formalism presented in the current study concerns deterministic Cantor sets. Similar calculations are possible for random Cantor sets, where the positions of the symbols ‘‘1’’ and ‘‘0’’ are randomly chosen, but their populations (numbers) are given by the usual iteration process. Acknowledgments The author would like to thank Prof. C. Tsallis for helpful discussions and for a critical reading of the manuscript and an anonymous referee for constructive comments and suggestions. References [1] H.E. Stanley, Introduction to Phase Transitions and Critical Phenomena, Oxford University Press, New York, 1971. [2] J.P. Bouchaud, A. Georges, Phys. Rep. 195 (1991) 127; J.-P. Bouchaud, A. Georges, J. Koplik, A. Provata, S. Redner, Phys. Rev. Lett. 64 (1990) 2503. [3] G.A. Tsekouras, A. Provata, Phys. Rev. E 65 (2002) art. no. 016204. [4] T. Vicsek, Fractal Growth Phenomena, World Scientific, Singapore, 1989; A.-L. Barabasi, H.E. Stanley, Fractal Concepts in Surface Growth, Cambridge University Press, Cambridge, 1995. [5] C. Andersson, S. Rasmussen, R. White, Envir. Plann. B 29 (2002) 841. [6] Y. Xiao, D.Z. Cheng, S.Y. Tang, Chaos, Solitons Fractals 14 (2002) 1403. [7] Th. Keitt, A.R. Johnson, J. Theor. Biol. 172 (1995) 127. [8] E. Bonabeau, L. Dagorn, P. Freon, Proc. Natl. Acad. Sci. USA 96 (1999) 4472. [9] C.-K. Peng, S.V. Buldyrev, A.L. Goldberger, S. Havlin, F. Sciortino, M. Simons, H.E. Stanley, Nature 356 (1992) 168. [10] W. Li, K. Kaneko, Europhys. Lett. 17 (1992) 655. [11] R.N. Mantegna, S.V. Buldyrev, A.L. Goldberger, S. Havlin, C.-K. Peng, M. Simons, H.E. Stanley, Phys. Rev. E 52 (1995) 2939. [12] A. Provata, Y. Almirantis, Physica A 247 (1997) 482. [13] C. Tsallis, J. Stat. Phys. 52 (1988) 479; E.M.F. Curado, C. Tsallis, J. Phys. A 24 (1991) L69. [14] D. Prato, C. Tsallis, Phys. Rev. E 60 (1999) 2398; C. Tsallis, S.V.F. Levy, A.M.C. Souza, R. Maynard, Phys. Rev. Lett. 75 (1995) 3589; C. Tsallis, D.J. Bukman, Phys. Rev. E 54 (1996) R2197; P.A. Alemany, D.H. Zanette, Phys. Rev. E 49 (1994) R956; M.L. Lyra, C. Tsallis, Phys. Rev. Lett. 80 (1998) 53; G. Drazer, H.S. Wio, C. Tsallis, Phys. Rev. E 61 (2000) 1417; C. Tsallis, G. Bemski, R.S. Mendes, Phys. Lett. A 257 (1999) 93; F.A. Tamarit, S.A. Cannas, C. Tsallis, Eur. Phys. J. B 1 (1998) 545. [15] D. Holste, I. Grosse, H. Herzel, Phys. Rev. E 64 (2001) art. no. 041917. [16] P. Lio, A. Politi, M. Buiatti, S. Ruffo, J. Theor. Biol. 180 (1996) 151. [17] K. Rateittschak, W. Ebeling, J. Freund, Europhys. Lett. 35 (1996) 401. [18] W. Ebeling, G. Nicolis, Europhys. Lett. 14 (1991) 191. [19] M. Buiatti, P. Grigolini, L. Palatella, Physica A 268 (1999) 214. [20] A. Capurro, L. Diambra, D. Lorenzo, O. Macadar, M.T. Martin, C. Mostaccio, A. Plastino, J. Perez, E. Rofman, M.E. Torres, J. Velluti, Physica A 265 (1999) 235. [21] S. Tong, A. Bezerianos, J. Paul, Y. Zhu, N. Thakor, Physica A 305 (2002) 619. [22] G.A. Tsekouras, A. Provata, C. Tsallis, Phys. Rev. E 69 (2004) art. no. 016120. [23] C. Anteneodo, Eur. Phys. J. B 42 (2004) 271. [24] C. Tsallis, Fractals 3 (1995) 541. [25] S. Denisov, Phys. Lett. A 235 (1997) 447. [26] C.R. da Silva, H.R. da Cruz, M.L. Lyra, Braz. J. Phys. 29 (1999) 144 and references therein. [27] U. Tirnakli, Phys. Rev. E 66 (2002) 066212. [28] E. Mayoral, A. Robledo, Physica A 340 (2004) 219.