q-distributions in complex systems: a brief review

q-distributions in complex systems: a brief review

S. Picoli Jr.; R. S. Mendes; L. C. Malacarne; R. P. B. Santos

Departamento de Física and National Institute of Science and Technology for Complex Systems, Universidade Estadual de Maringá, Avenida Colombo 5790, 87020-900 Maringá, PR, Brazil

ABSTRACT

The nonextensive statistical mechanics proposed by Tsallis is today an intense and growing research field. Probability distributions which emerges from the nonextensive formalism(q-distributions) have been applied to an impressive variety of problems. In particular, the role of q-distributions in the interdisciplinary field of complex systems has been expanding. Here, we make a brief review of q-exponential, q-Gaussian and q-Weibull distributions focusing some of their basic properties and recent applications. The richness of systems analyzed may indicate future directions in this field.

Keywords:q-exponential, q-Gaussian, q-Weibull, Nonextensive statistics

1. INTRODUCTION

Common characteristics of complex systems include long-range correlations, multifractality and non-Gaussian distributions with asymptotic power law behavior. Typically, such systems are not well described by approaches based on the usual statistical mechanics. In this scenario, a new formalism capable of providing a better description of complex systems is welcome. This is the case of the generalized (nonextensive) statistical mechanics proposed by Tsallis -nowadays, an intense and growing research field[1-4].

Concepts related with nonextensive statistical mechanics have found applications in a variety of disciplines including physics, chemistry, biology, mathematics, geography, economics, medicine, informatics, linguistics among others[5- 7]. Probability distributions which emerge from the nonextensive formalism -also called q-distributions -have been applied to an impressive variety of problems indiverse research areas including the interdisciplinary field of complex systems.

In the present work we focus on q-exponential, q-Gaussian and q-Weibull distributions. We summarized some of their basic properties and provide useful references of recent applications. The richness of systems analyzed may indicate future directions in this research line.

2. q-EXPONENTIAL DISTRIBUTION

The q-exponential distribution is given by the probability density function (pdf)

for 1 -(1 - q) x/x₀> 0. If p₀ =(2 - q)/x₀, eq. (1) is normalized.

In the limit q→ 1, eq. (1) recovers the usual exponential distribution in the same way in which the q-exponential function, defined as , recovers exponential function in the limit q → 1 . If q < 1, eq. (1) has a finite value for any finite real value of x since, by definition, p_qe(x)= 0 for 1- (1- q)x/a < 0. if q > 1, eq. (1) exhibits power law asymptotic behavior,

Note also that p_qe(x) 1+x for small x, independently of the q value. Figures 1a and 1b show p_qe(x) versus x for typical values of q.

The q-exponential distribution, for q > 1, corresponds to the Zipf-Mandelbrot law[8] and a Burr-type distribution[9]. In this sense, the q-exponential is a generalization of these distributions for q < 1. Thus, by choosing suitable values for q, q-exponentials may be used to represent both short and long tailed distributions. This feature also holds for the other q-distributions.

The cumulative distribution function (cdf) associated to eq. (1) is given by

defined for q< 2, with q' = 1/(2- q), x'₀= x₀/(2 - q) and p'₀= p₀x₀/(2 -q). Observe that R_qe(x) and p_qe(x) exhibit the same mathematical form.

It is possible to visualize q-exponential distributions as straight lines in graphs with appropriate scales. Applying the q-logarithm function, defined as ln_qx ≡ [x^(1-q) - 1]/(1- q), with ln₁x ≡ ln(x), in both sides of eq. (1), we have

A similar result holds for R_qe(x). Figure 1c shows ln_q p_qe(x) versus x for typical values of x₀.

The q-exponential function given by eq. (1) has been employed in a growing number of theoretical and empirical works on a large variety of themes. Examples include scale-free networks[10-14], dynamical systems[15-27], algebraic structures[28-31] among other topics in statistical physcics[32-36].

As specific examples of q-exponential distributions in complex systems, let us consider results on population of cities[37] and circulation of magazines[38]. Figure 2 shows the cumulative distribution of the population of cities in the USA and Brazil. Figure 3 shows the cumulative distribution of circulation of magazines in the USA and UK. In both cases - population of cities and circulation of magazines - the empirical data are consistent with a q-exponential distribution, with q 1.4.

q-exponential distributions have also been applied in the empirical study of stock markets[39-42], DNA sequences[43], family names[44], human behavior[45-47], geomagnetic records[48, 49], train delays[50], reaction kinetics[51], air networks[52], hydrological phenomena[53], fossil register[54], basketball[55], earthquakes[56-58], world track records[59], voting processes[60], internet[61], individual success[62], citations of scientific papers[63, 64], football[65], linguistics[66, 67] and solar neutrinos[68, 69].

3. q-GAUSSIAN DISTRIBUTION

The q-Gaussian distributionis specified by the pdf

for 1 - (1- q)(x/x₀)²> 0 and p_qg(x)= 0 otherwise. It is normalized if p₀=(2/x₀). In addition, eq. (5) presents unit variance if = 5 - 3q, with q< 5/3.

In the limit q→ 1, eq. (5) recovers the usual Gaussian distribution, so q 1 indicates a departure from Gaussian statistics. For q > 1, the tails of q-Gaussian decrease as power laws,

Figures 4a and 4b show p_qg(x) for typical values of q.

Applying the q-logarithm function in both sides of eq. (5), we have

Figure 4c shows ln_qp_qg(x) versus x² for typical values of x₀.

Recent works have been focused on the study of mathematical properties of q-Gaussian functions[70-78], including methods for generating random numbers which follow q-Gaussian distributions[79, 80]. q-Gaussians have been employed in the study of a wide range of themes including probabilistic models[81, 82], stellar plasmas[83], porous-medium equation[84], Bose-condensed gases[85-87], dynamical systems[88-90], polymeric networks[91], small-world networks[92], fingering processes[93], processes with stochastic volatility[94,95] and nonlinear diffusion[96, 97].

In order to illustrate a recent application of q-Gaussian distributions in complex systems, we mention here results on the dynamics of earthquakes[98]. Figure 5 shows the distribution of energy differences between successive earthquakes at the San Andreas Fault. The empirical data is consistent with a q-Gaussian distribution, with q= 1.75.

Other recent applications of q-Gaussian distribution include stock markets[99-107], DNA molecules[108], the solar wind[109-111], galaxies[112], optical lattices[113], cellular aggregates[114] and the atmosphere[115].

4. q-WEIBULL DISTRIBUTION

The q-Weibull distribution is given by the pdf

for 1 - (1- q)(x/x₀)^r> 0and p_qw(x)= 0 otherwise. Eq. (8) is normalized if p₀ = 2- q.

In the limits q→ 1, r → 1, and q→ 1, r → 1, eq. (8) recovers Weibull, q-exponential and exponential distributions, respectively. If q< 1, p_qw(x) has a finite limit since p_qw(x)= 0 for 1-(1 - q)(x/x₀)^r < 0. if q > 1, p_qw(x) exhibits power law behavior both for small and large values of x. More specifically,

with ξ = (1 - r) for small x and ξ = r[(2- q)/(q- 1)] + 1 for large x. Figures 6a, 6b and 6c show p_qw(x) versus x for typical values of q and r.

 

The cdf associated to p_qw(x) is given by

with q' = 1/(2- q), (x'₀)^r = x^r₀/(2- q) and p'₀= p₀/(2- q). Applying the q-logarithm function in both sides of the cdf R_qw, we have

Figure 6c shows ln_q'R_qW(x) versus x^r for typical values of x₀.

If p_qw(x) is normalized(p₀ = 2 - q), Eq. (11) reduces to ln_q'R_qw(x)= -(x/x₀)^r. In this case,

As specific example of q-Weibull distribution in complex systems, we now consider results on citations in scientific journals[116]. Figure 7 shows the distribution of the impact factor of scientific journals in comparison with a q-Weibull curve. The empirical data is consistent with a q-Weibull distribution, with q= 1.45 and r = 1.50.

Other recent works have been related to q-Weibull distributions. For example, new classes of generalized asymmetric distributions have been introduced which include q-Weibull as a special case[117, 118]. q-Weibull has also been applied in the study of fractal kinetics[119], dieletric breakdown in oxides[120], relaxation in heterogeneous systems[121], ciclone victims and highway lengths[55] among others.

5. BASIS FOR q-DISTRIBUTIONS

¿From the viewpoint of the principle of the maximum entropy, some q-distributions optimizes generalized entropies more general entropic measures than the standard Boltzmann-Gibbs entropy. A striking exampleis the q-entropy proposed by Tsallis[1]

where W is the total number of microstates of the system, p_i are the occupation probabilities and q isa real parameter. The standard Boltzmann-Gibbs entropy is recovered in the limit q→ 1.

The maximization of S_q subject to specific constraints generates occupation probabilities followinga q-exponential distribution. The q-exponential optimizes other generalized entropic measures such as the Renyi and normalized Tsallis entropies. However, only Tsallis entropy can provide an appropriate basis for the q-exponential distribution since it presents several properties essential for an entropy[122, 123]. Changing the constraints, the maximization of S_q also generates occupation probabilities following a q-Gaussian distribution.

Formally, q-distributions can arise when the exponential function of the original distribution is replaced by a q-exponential function. For example, this basic procedure applied in standard exponential, Gaussian and Weibull distributions leads to q-exponential, q-Gaussian and q-Weibull, respectively[55]. This viewpoint suggests the consideration of other q-distributions which could be obtained by simply replacing its exponential function by a q-exponential one.

q-distributions can also emerge from compound distributions[124]

where f(λ) is a Gamma function. For example, if p(x,λ) is a Weibull distribution, p_q(x) is given by a q-Weibull distribution[120]. Naturally, other forms for f(λ) may be considered to obtain alternative distributions. In a physical context, this scenario has been explored with success in superstatistics where nonequilibrium situations with local fluctuations of the environment are taken into account[125-127].

The generalized distributions considered here can also be obtained from the following ordinary differential equation:

In fact, if ρ is constant, the solution of eq. (15) is a qexponential; if ρ ∝ x, the solution is a q-Gaussian. If y is the cdf and ρ ∝ x^r, we have a q-Weibull. By considering further terms in eq. (15), other q-distributions can be obtained[128]. q-distributions can also emerge in other contexts. For instance, q-Gaussian arises from the non-linear diffusion (porous media) equation[84] and from a generalization of the central limit theorem[3]. Another example is the q-lognormal distribution which emerges from generalized cascades[28].

6. CONCLUSION

The present work presents a brief overview of recent applications of some q-distributions largely used in the context of Tsallis statistics. It illustrates how q-exponential, q-Gaussian and q-Weibull distributions have been applied in the study of a wide variety of systems in several fields.

The success of q-distributions in describing diverse systems is in part due to its ability of exhibit heavy-tails and model power law phenomena - a typical characteristic of complex systems. The positive and exciting results obtained with q-distributions also indicate possible applications of Tsallis nonextensive statistical mechanics. Naturally, further work may be necessary to explore possible relations between the analyzed systems and the present theory.

(Received on 26 March, 2009)

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Publication Dates

• Publication in this collection
  10 Sept 2009
• Date of issue
  Aug 2009

History

• Accepted
  26 Mar 2009
• Received
  0000