Brazilian Journal of Physics, vol. 00, no. 0000, 01, 2009 1 arXiv:1001.2855v1 [physics.data-an] 16 Jan 2010 Symbolic Sequences and Tsallis Entropy H. V. Ribeiro, ∗ E. K. Lenzi, and R. S. Mendes Departamento de F´ısica, Universidade Estadual de Maring´ a, Av. Colombo 5790, 87020-900, Maring´ a, PR, Brazil G. A. Mendes and L. R. da Silva Departamento de F´ısica, Universidade Federal do Rio Grande do Norte 59072-970 Natal, RN, Brazil We address this work to investigate symbolic sequences with long-range correlations by using computational simulation. We analyze sequences with two, three and four symbols that could be repeated l times, with the probability distribution p ( l ) ∝ 1 /l µ . For these sequences, we verified that the usual entropy increases more slowly when the symbols are correlated and the Tsallis entropy exhibits, for a suitable choice of q , a linear behavior. We also study the chain as a random walk-like process and observe a nonusual diffusive behavior depending on the values of the parameter µ . Keywords: Symbolic sequences, long-range correlations, Tsallis entropy, non-usual diffusion. 1. INTRODUCTION Two basic assumptions of the statistical mechanics are: the “equal a priori probabilities” and ergodicity. When these assumptions do not hold, we need other suitable tools to study systems which exhibit a nonusual behav- ior. A typical situation can be found by analyzing sys- tems which have an intermediate regime between periodic and chaotic [1]. This kind of system commonly shows a power law spectra and appears in several fields of sci- ence. Aspects of nonusual behavior have been explored, for instance, in biology[2], nuclear physics [3], financial market [4], music [5] and linguistics [6]. In this context, there are also works that search for correlations in DNA sequences [7–11] by using entropic indexes [12–16]. To provide a possible description for these systems which are not conveniently explained by the usual formalism, Tsallis[17] proposes an extension of the Boltzmann-Gibbs entropy. Many systems have been investigated by using this approach, e.g., long-range Hamiltonian systems like the HMF model [18], the generalized Lennard-Jonnes gas [19], self-gravitating systems [20] and anomalous diffusion [21]. In this direction, to try to clarify in a more direct way basic aspects related to Tsallis entropy, it may be con- venient to consider specifc models with a kind of long- range behavior. Considering this, the aim of this work is to explore the nonusual behavior of a symbolic model with an adjustable long-range behavior. More precisely, we investigate one dimensional symbolic sequences with long-range correlations which are generated by using the numerical experiment presented in Ref. [22]. The proce- dure uses two random numbers to obtain a lattice with N sites which represent the symbolic sequence. One of them, x , has a uniform distribution in the interval [0,1] and the first moment of this distribution is 〈 y 〉 = A/ ( µ − 2). By comparing the asymptotic limit of Eq.(2), p ( y ) ∼ 1 /y $^{µ}$, with the asymptotic limit of the L´ evy distributions, p ( y ) ∼ 1 /y 1+ $^{η}$, the relation between µ and η is µ = 1 + η . Note also that 〈 y 〉 diverges for µ → 2. This fact indicates that, when µ is close to two, N$_{y}$ may assume large values and fill a large part of the symbolic sequence with the same symbol. On the other hand, when µ is far from two ( µ ≫ 2), large values of N$_{y}$ become very rare and consequently the sequence has more alternated symbols. For the sequences generated with the procedure de- scribed above, we may obtain the probability distribution function of the variable y , p ( y ), and show that, depending on the values of the µ , it can be asymptotically related to a L´ evy distribution (for y ≥ 0). In fact, after some calculations, one can show that p ( y ) is given by where A and µ are real parameters. We go through the symbolic sequence drawing x and filling N$_{y}$ = [ y ] + 1 sites with the same value z , where [ y ] denotes the inte- ger part of y and z is a signal generator that can have one of four distinct values (0, 1, 2, 3) with the same sta- tistical weight. A typical example obtained within this procedure is and the other emerges from the expression y = A [ 1 (1 − x ) 1 / ( µ − 1) − 1 ] , (1) p ( y ) = ( µ − 1) A µ − 1 ( A + y ) µ , (2) ∗ Electronic address: hvr@dfi.uem.br