In defense of Tsallis' original probability distribution

TL;DR

This paper defends Tsallis' original q-probability distribution against recent criticisms, reaffirming its validity and addressing misconceptions in the ongoing debate about non-extensive statistical mechanics.

Contribution

It provides a rebuttal to recent attacks on Tsallis' distribution, reaffirming its correctness and defending its foundational role in non-extensive statistical physics.

Findings

The attack on Tsallis' distribution is unfounded.

Tsallis' original distribution remains valid.

The paper clarifies misconceptions about the distribution.

Abstract

Tsallis' pioneer q-probability distribution $P_i=\frac {[1+\beta(1-q)U_i]^{\frac {1} {q-1}}} {Z}$, $Z=\sum\limits_{i=1}^n [1+\beta(1-q)U_i]^{\frac {1} {q-1}}$ [J. of Stat. Phys., {\bf 52} (1988) 479] has been recently attacked in arXiv:1705.01752, in a Reply to our arXiv:1704.07493 publication. We show here that such an attack is groundless.

Equations

$$\left(\frac{q}{1-q}\right)P_i^{q-1}+{\lambda }_1{U}_i+{\lambda }_2=0,$$

$${\lambda }_1=\beta q{Z}^{1-q},$$

$${\lambda }_2=\frac{q}{q-1}{Z}^{1-q},$$

$$P_i=\frac{[1+\beta (1-q)U_i{]}^{\frac{1}{q-1}}}{Z},$$

$$Z=\sum _{i=1}^n[1+\beta (1-q)U_i{]}^{\frac{1}{q-1}}.$$

$$(P_i{)}_{OB}=[(\frac{1-q}{q})(\beta U_i+\gamma ){]}^{1/(q-1)},$$

$$\sum _i(P_i{)}_{OB}=1=\sum _i[(\frac{1-q}{q})(\beta U_i+\gamma ){]}^{1/(q-1)},$$

$$\beta =-{\lambda }_1;\gamma =-{\lambda }_2.$$

$$P_i=\frac{[1+\beta (1-q)(U_i-<U>){]}^{\frac{1}{q-1}}}{Z},$$

$$Z=\sum _{i=1}^n[1+\beta (1-q)(U_i-<U>){]}^{\frac{1}{q-1}}.$$

$$Z=e^{\beta <U>}\sum _{i=1}^n{e}^{-\beta U_i}.$$

$$P_i=\frac{[1+\beta (1-q)(U_i-<U>){]}^{\frac{1}{q-1}}}{\sum _{i=1}^n[1+\beta (1-q)(U_i-<U>){]}^{\frac{1}{q-1}}},$$

$$P_i=\frac{e^{-\beta (U_i-<U>)}}{\sum _{i=1}^n{e}^{-\beta (U_i-<U>)}},$$

$$P_i=\frac{e^{\beta <U>}{e}^{-\beta U_i}}{e^{\beta <U>}\sum _{i=1}^n{e}^{-\beta U_i}}.$$

$$P_i=\frac{e^{-\beta U_i}}{\sum _{i=1}^n{e}^{-\beta U_i}},$$

$$Z=\sum _{i=1}^n{e}^{-\beta U_i}.$$

Full text

$P_{i}=\frac{[1+\beta(1-q)U_{i}]^{\frac{1}{q-1}}}{Z}$

$Z=\sum\limits_{i=1}^{n}[1+\beta(1-q)U_{i}]^{\frac{1}{q-1}}$

[J. of Stat. Phys., 52 (1988) 479] has been recently attacked in ArXiv:1705.01752, in a Reply to our arXiv: 1704.07493 publication. We show here that such an attack is groundless.

Keywords: MaxEnt, Tsallis’ functional variation, q-statistics.

1 The Tsallis’probability distribution

We reply here to reference (arXiv:1705.01752) of Oikonomou and Bagci (OB by short) [1]. Under the pretense of replying to our reference [2], they question our variational procedure in such paper, but in so doing they are really attacking Tsallis’ original probability distribution (PD) [3]. Let us see first how we proceeded in [2]. The pertinent variational equation is (OB’s Eq. (1))

[TABLE]

and OB call this PD $P$ by the name PR1. Of course, Eq. (1.1) is the Tsallis’ Euler-Lagrange one of [3].

To make things transparent, we revisit now the procedure given in [2]. One first gives the Lagrange multipliers $\lambda_{1}$ and $\lambda_{2}$ a prescribed form in terms of a (thus far unknown) quantity $Z$:

[TABLE]

[TABLE]

and then has

[TABLE]

so that normalization demands for $Z$

[TABLE]

The ensuing PD is (curiously) called PR2 by OB [1].

It is obvious that PR1 and PR2 are one and the same PD! However, OB claim that they are different. OB try to validate such strange statement with a graph of three curves.

They introduce still a third PDF that they call OB, and hypothetically follows from their own variational equation (called by them Eq. (5)). In such Eq. (5) they inadvisedly FIX the energy-Lagrange multiplier as $\beta$, with disastrous consequences, as we will presently show. From their variational equation one obtains for the PD:

[TABLE]

so that OB’s normalization entails

[TABLE]

and one immediately appreciates the sad fact that $\gamma$ cannot be obtained in closed form. This makes normalization a difficult task, particularly in the continuum limit. OB preposterously claim that their $(P_{i})_{OB}$ is identical to PR1, which is patently absurd.

In order to get out of this conundrum OB state(see below their graph) that things are remedied by setting their Lagrange multipliers $\beta,\,\,\gamma$ equivalent to ours $\lambda_{1},\,\,\lambda_{2}$ via

[TABLE]

But then, PR1 becomes identical to $(P_{i})_{OB}$! These two PDFs cannot yield different results, as OB enthusiastically and with absolute confidence claim.

2 The Renyi probability distribution

It is asserted in [4] that Renyi’s probability distribution (PD) is

[TABLE]

with

[TABLE]

It is erroneously stated in [1] that, in the limit $q\rightarrow 1$, the above partition function $Z$ becomes

[TABLE]

This happens because the authors of [1] did not bother to take the limit of the complete PD. Indeed, from

[TABLE]

one deduces that for $q\rightarrow 1$ one has

[TABLE]

or, equivalently,

[TABLE]

Thus,

[TABLE]

and then

[TABLE]

We see that (2.3) from [1] is not correct.

3 Conclusion

In view of these considerations, one concludes that reference [1] has no logical support.