Equivalence of the four versions of Tsallis’s statistics

Abstract

In spite of its undeniable success, there are still open questions regarding Tsallis' non-extensive statistical formalism, whose founding stone was laid in 1988 in JSTAT. Some of them are concerned with the so-called normalization problem of just how to evaluate expectation values. The Jaynes' MaxEnt approach for deriving statistical mechanics is based on the adoption of (1) a specific entropic functional form S and (2) physically appropriate constraints. The literature on non-extensive thermostatistics has considered, in its historical evolution, four possible choices for the evaluation of expectation values: (i) 1988 Tsallis-original (TO), (ii) Curado-Tsallis (CT), (iii) Tsallis-MendesPlastino (TMP), and (iv) the same as (iii), but using centered operators as constraints (OLM).The 1988 was promptly abandoned and replaced, mostly with versions ii) and iii). We will here (a) show that the 1988 is as good as any of the others, (b) demonstrate that the four cases can be easily derived from just one (any) of them, i.e., the probability distribution function in each of these four instances may be evaluated with a unique formula, and (c) numerically analyze some consequences that emerge from these four choices.PACS: 05.30.Jp

Classes of N-Dimensional Nonlinear Fokker-Planck Equations Associated to Tsallis Entropy

“…Equations (4), (7) and (9)]. The distributions obtained from the extremization of S q [P ] under the normalization constraint of Equation (3) and one of these definitions for the internal energy are essentially equivalent, being related to one another by redefining the index q and the associated Lagrange multiplier [4,8,9]. Next we shall prove the H-theorem considering a general definition for the internal energy,…”

“…yielding a stationary distribution similar to the one in Equation (8), with both q and the corresponding Lagrange multiplier β (3) redefined in terms of the quantities appearing in Equation (8) [4,[7][8][9].…”

Generalized Fisher information matrix in nonextensive systems with spatial correlation

“…There are four possible MEMs at the moment: original method [24], un-normalized method [28], normalized method [25], and the optimal Lagrange multiplier (OLM) method [29]. The four methods are equivalent in the sense that distributions derived in them are easily transformed each other [30]. A comparison among the four MEMs is made in Ref.…”

Equivalence among different formalisms in the Tsallis entropy framework

“…(See also [14] for an alternative discussion on the equivalence among these different formalisms). In a previous tentative to establish a correspondence among the various formulations described above, it has been introduced the escort entropy [6,15] …”

“…This entropy is derived from S q [p] by using the transformation 14) where distributions p (3) (v) and p (E) (v) came, respectively, from the entropy S q [p] constrained by the escort mean value U (3) q and from the entropy S (E) q (p), withq = 1/q, constrained by linear mean value U (E)…”