Generalized bit cumulants in nonextensive thermodynamics

Tsallis versus Renyi entropic form for systems with q-exponential behaviour: the case of dissipative maps

“…It was observed in [28] that the ratio C (T ) 2 /C 2 evaluated in the chaotic region of the map (9), gives a scaling factor referred to as 'slope' in this paper (see Figs.), whose value depends on values of q as well as the inflexion parameter z. Naturally, the slope tends to unity for q → 1.…”

Geometry of escort distributions

“…[7][8][9][10], in which statistical properties of chaotic dynamical systems are investigated. The ordinary bit-variance is recovered in the limit q → 1.…”

Generalized bit moments and cumulants based on discrete derivative

“…The generalization we aim at can be achieved in two ways: (i) only generalize the bit-moments and then use the standard relations between bit-moments and bit-cumulants to define generalized cumulants, which yields q-cumulants of Ref. [4]; (ii) generalize the moments and cumulants independently from their basic definition (see below), using the discrete deriva-tive. This procedure does not preserve standard relations between moments and cumulants.…”

“…In this spirit, the classical bit cumulants were generalized [4] within nonextensive thermodynamic approach. Bit statistics is a tool to describe the complicated probability distributions, such as generated by chaotic systems.…”

“…where from now on we imply ∆ = (1 − q), a generalization [4] of standard cumulants was considered by modifying the partition function in Eq. (3) based on [a i ] instead of a i .…”