Nonextensivity and Multifractality in Low-Dimensional Dissipative Systems

Abstract

Power-law sensitivity to initial conditions at the edge of chaos provides a natural relation between the scaling properties of the dynamics attractor and its degree of nonextensivity as prescribed in the generalized statistics recently introduced by one of us (C.T.) and characterized by the entropic index q. We show that general scaling arguments imply that 1/(1 − q) = 1/α min − 1/α max , where α min and α max are the extremes of the multifractal singularity spectrum f (α) of the attractor. This relation is numerically checked to hold in standard one-dimensional dissipative maps. The above result sheds light on a longstanding puzzle concerning the relation between the entropic index q and the underlying microscopic dynamics. 05.45.+b; 05.20.-y; 05.90.+m Typeset using REVT E X

Non-Gaussian velocity distributions in optical lattices

“…At this stage, it is interesting to note that the Tsallis function has been introduced in the context of non-extensive thermodynamics [22,23]. The large amount of literature in this context allows one to find many papers dealing with problems already addressed in laser cooling; in particular anomalous diffusion in the presence of external forces [24,25,26], multiplicative noise problems, and the relation to the edge of chaos in mixed phase space dynamics [27,28]. It is known that Sisyphus cooling can give rise to anomalous diffusion [29,30], in particular for shallow optical potentials, where an atom can travel over many wavelengths before being trapped again.…”

Multiplicative duality, q-triplet and (μ,ν,q)-relation derived from the one-to-one correspondence between the (μ

“…where b stands for a natural scale for the partitions, α F is the Feigenbaum universal scaling factor and z represents the nonlinearity of the map at the vicinity of its extremal point [18].…”

Multifractal spectrum of phase space related to generalized thermostatistics

“…Above formalism is based on the proposition that the phase space is related to a monofractal set determined by single dimension d. However, the considerations [4,5,6] show that a complex system behaviour can be determined by the phase space geometry, being much more complicated, in particular multifractal. In this connection, we aim to generalize the Tsallis thermostatistics onto the multifractal phase space with a spectrum f (d).…”