TY - JOUR

T1 - The maximization of Tsallis entropy with complete deformed functions and the problem of constraints

AU - Oikonomou, Thomas

AU - Bagci, G. Baris

N1 - Funding Information:
We thank U. Tirnakli for a careful reading of the manuscript. G.B.B. was supported by TUBITAK (Turkish Agency) under the Research Project number 108T013 and by Faculty of Science at Ege University under the project number 2009Fen076 .

PY - 2010/5/3

Y1 - 2010/5/3

N2 - By only requiring the q deformed logarithms (q exponentials) to possess arguments chosen from the entire set of positive real numbers (all real numbers), we show that the q-logarithm (q exponential) can be written in such a way that its argument varies between 0 and 1 (among negative real numbers) for 1 ≤ q < 2, while the interval 0 < q ≤ 1 corresponds to any real argument greater than 1 (positive real numbers). These two distinct intervals of the nonextensivity index q, also the expressions of the deformed functions associated with them, are related to one another through the relation (2 - q), which is so far used to obtain the ordinary stationary distributions from the corresponding escort distributions, and vice versa in an almost ad hoc manner. This shows that the escort distributions are only a means of extending the interval of validity of the deformed functions to the one of ordinary, undeformed ones. Moreover, we show that, since the Tsallis entropy is written in terms of the q-logarithm and its argument, being the inverse of microstate probabilities, takes values equal to or greater than 1, the resulting stationary solution is uniquely described by the one obtained from the ordinary constraint. Finally, we observe that even the escort stationary distributions can be obtained through the use of the ordinary averaging procedure if the argument of the q-exponential lies in (- ∞, 0]. However, this case corresponds to, although related, a different entropy expression than the Tsallis entropy.

AB - By only requiring the q deformed logarithms (q exponentials) to possess arguments chosen from the entire set of positive real numbers (all real numbers), we show that the q-logarithm (q exponential) can be written in such a way that its argument varies between 0 and 1 (among negative real numbers) for 1 ≤ q < 2, while the interval 0 < q ≤ 1 corresponds to any real argument greater than 1 (positive real numbers). These two distinct intervals of the nonextensivity index q, also the expressions of the deformed functions associated with them, are related to one another through the relation (2 - q), which is so far used to obtain the ordinary stationary distributions from the corresponding escort distributions, and vice versa in an almost ad hoc manner. This shows that the escort distributions are only a means of extending the interval of validity of the deformed functions to the one of ordinary, undeformed ones. Moreover, we show that, since the Tsallis entropy is written in terms of the q-logarithm and its argument, being the inverse of microstate probabilities, takes values equal to or greater than 1, the resulting stationary solution is uniquely described by the one obtained from the ordinary constraint. Finally, we observe that even the escort stationary distributions can be obtained through the use of the ordinary averaging procedure if the argument of the q-exponential lies in (- ∞, 0]. However, this case corresponds to, although related, a different entropy expression than the Tsallis entropy.

KW - Escort average

KW - Ordinary average

KW - Tsallis entropy

KW - q-exponential

KW - q-logarithm

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U2 - 10.1016/j.physleta.2010.03.038

DO - 10.1016/j.physleta.2010.03.038

M3 - Article

AN - SCOPUS:77950595479

VL - 374

SP - 2225

EP - 2229

JO - Physics Letters, Section A: General, Atomic and Solid State Physics

JF - Physics Letters, Section A: General, Atomic and Solid State Physics

SN - 0375-9601

IS - 22

ER -