Self-similar motion for modeling anomalous diffusion and nonextensive statistical distributions

Abstract

We introduce a new universality class of one-dimensional iteration model giving rise to self-similar motion, in which the Feigenbaum constants are generalized as self-similar rates and can be predetermined. The curves of the mean-square displacement versus time generated here show that the motion is a kind of anomalous diffusion with the diffusion coefficient depending on the self-similar rates. In addition, it is found that the distribution of displacement agrees to a reliable precision with the q-Gaussian type distribution in some cases and bimodal distribution in some other cases. The results obtained show that the self-similar motion may be used to describe the anomalous diffusion and nonextensive statistical distributions.

Probabilistic model of N correlated binary random variables and non-extensive statistical mechanics

“…Statistics associated to Eq. (1) has been successfully applied to phenomena with the scale-invariant geometry, like in low-dimensional dissipative and conservative maps in the dynamical systems [9][10][11], anomalous diffusion [12,13], turbulent flows [14], Langevin dynamics with fluctuating temperature [15,16], spin-glasses [17], plasma [18] and to the financial systems [19][20][21]. By maximizing the entropy (1) with the constraints +∞ −∞ p(x)dx = 1 and…”

Canonical ensemble in non-extensive statistical mechanics

“…In the statistical mechanics it is postulated that in the equilibrium the probability of each microstate is the same and equal 1/W tot . Thus the most probable state of the composite system corresponds to the largest term in the sum (8). The most probable energy U of the system S corresponding to this largest term can be found from the condition…”

“…Statistical description of complex systems can be provided using the non-extensive statistical mechanics that generalizes the Boltzmann-Gibbs statistics [1][2][3]. The non-extensive statistical mechanics has been used to describe phenomena in various in high-energy physics [4], spin-glasses [5], cold atoms in optical lattices [6], trapped ions [7], anomalous diffusion [8,9], dusty plasmas [10], low-dimensional dissipative and conservative maps in the dynamical systems [11][12][13], turbulent flows [14], Langevin dynamics with fluctuating temperature [15,16]. Concepts related to the non-extensive statistical mechanics have found applications not only in physics but in chemistry, biology, mathematics, economics, and informatics as well [17][18][19].…”

New mathematics for the nonadditive Tsallis’ scenario

“…The non-extensive statistical mechanics of Tsallis' has been employed to fruitfully discuss phenomena in variegated fields. One may mention, for instance, high-energy physics [3]- [4], spin-glasses [5], cold atoms in optical lattices [6], trapped ions [7], anomalous diffusion [8], [9], dusty plasmas [10], low-dimensional dissipative and conservative maps in dynamical systems [11], [12], [13], turbulent flows [14], Levy flights [16], the QCD-based Nambu, Jona, Lasinio model of a many-body field theory [17], etc. Notions related to qstatistical mechanics have been found useful not only in physics but also in chemistry, biology, mathematics, economics, informatics, and quantum mechanics [18], [19], [20], [21].…”