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Article

On the Nature of the Tsallis–Fourier Transform

La Plata Physics Institute-CCT-Conicet-Exact Sciences Faculty, Universidad Nacional (UNLP), C.C. 727, 1900 La Plata, Argentina
*
Author to whom correspondence should be addressed.
Mathematics 2015, 3(3), 644-652; https://doi.org/10.3390/math3030644
Submission received: 2 May 2015 / Revised: 11 July 2015 / Accepted: 13 July 2015 / Published: 21 July 2015
(This article belongs to the Special Issue Mathematical physics)

Abstract

:
By recourse to tempered ultradistributions, we show here that the effect of a q-Fourier transform (qFT) is to map equivalence classes of functions into other classes in a one-to-one fashion. This suggests that Tsallis’ q-statistics may revolve around equivalence classes of distributions and not individual ones, as orthodox statistics does. We solve here the qFT’s non-invertibility issue, but discover a problem that remains open.

1. Introduction

Non-extensive statistical mechanics (NEXT) [1,2,3], a well-known generalization of the Boltzmann–Gibbs (BG) one, is used in many scientific and technological endeavors. NEXT’s central concept is that of a non-additive (though extensive [4]) entropic information measure characterized by the real index q (with q = 1 recovering the standard BG entropy). Applications include cold atoms in dissipative optical lattices [5], dusty plasmas [6], trapped ions [7], spin glasses [8], turbulence in the heliosphere [9], self-organized criticality [10], high-energy experiments at LHC/CMS/CERN [11] and RHIC/PHENIX/Brookhaven [12], low-dimensional dissipative maps [13], finance [14],galaxies [15], Fokker–Planck equation’s studies [16], EEG’s [17], complex signals [18], Vlasov–Poissonequations [19], etc.
So-called q-Fourier transforms (qFT) were developed by Tsallis et al. [20]. They constitute a central piece in Tsallis’ q-machinery. However (see [21] and the references therein), qFT seems not to be an invertible transformation.
The imaginary q-exponential function, a protagonist of Tsallis’ statistics, is defined as:
e q ( i x ) = 1 + i ( 1 q ) x 1 1 q ,
with:
e q ( i x ) exp ( i x ) whenever q 1 .
However, the function:
e q b ( i x ) = 1 + i ( 1 q ) x 1 1 q + b ,
with < b < 0 , b a real number, also fulfills:
e q b ( i x ) exp ( i x ) whenever q 1 .
Accordingly, there is a class of functions e q b ( i x ) , labeled by b, that tend to the ordinary exponential in the limit q 1 . This fact profoundly affects the workings of the q-Fourier transform.
The same happens with the q-logarithm defined as:
ln q ( x ) = x 1 q 1 1 q ln ( x ) whenever q 1
The function:
ln q c ( x ) = x 1 q 1 + c ( 1 q ) 1 1 q ln ( x ) whenever q 1
where < c < 0 is a real number. Moreover:
ln q b [ e q b ( i x ) ] = i x .
Recall that Schwartz space 𝒮 is the space of functions, all of whose derivatives are rapidlydecreasing [22]. This space has the important property that the Fourier transform is an automorphism on this space. This property enables one, by duality, to define the Fourier transform for elements in the dual space of 𝒮, called the space of tempered distributions [22].
In the present communication, we reconcile the Tsallis et al. developments [20] with the non-invertibility issue and show, by recourse to tempered ultradistributions (a generalization and extension to the complex plane of Schwartz’ tempered distributions) that the qFT does indeed map, in a one-to-one fashion, classes of functions into other classes, not isolated functional instances. Thus, such an issue can be resolved by appealing to a higher order of mathematical perspective. Section 2 recapitulates the findings of our work in [21], related to an extension of the Tsallis et al. environment [23], which becomes just a particular case (real line) of a more encompassing theory (complex plane) [21]. Section 3, the core of our presentation, specializes the developments of [21] for an important situation. We analyze there the particular instantiation of the theory that leads to the scenario investigated by bothTsallis et al. [20]. The results thus obtained, our main contribution here, are illustrated in Section 5 by an important example. Finally, the mathematical problem that remains open is discussed in Section 5. Conclusions are drawn in Section 6.

2. Reviewing an Alternative qFT Definition

In [21] (see also [23]), we introduced an alternative qFT-definition that, by generalizing and extending the original one, overcomes the non-invertibility problem afflicting the one of [20]. We briefly review that alternative version in this section. Our protagonists in such an endeavor were tempered ultradistributions [22,23,24,25], which constitute a generalization of the distribution set for which the test functions are members of a Schwartz space 𝒮, a function space in which its members possess derivatives that are rapidly decreasing. 𝒮 exhibits a notable property: the Fourier transform is an automorphism on 𝒮, a property that allows, by duality, defining the Fourier transform for elements in the dual space of 𝒮. This dual is the space of tempered distributions [22].
In physics, it is not uncommon to face functions that grow exponentially in space or time. In such circumstances, Schwartz’ space of tempered distributions is too restrictive. One needs ultradistributions to generate appropriate answers [26]. They are continuous linear functionals defined on the space of entire functions rapidly decreasing on straight lines parallel to the real axis [26]. Now, following [23], we appeal to the Heaviside step function H:
H ( x ) = 1 f o r x 0 0 f o r x < 0 .
These step functions allow us to properly define the q-Fourier transform in a different fashion as that of Tsallis et al. [20] by recourse [23] to the space 𝒰 of tempered ultradistributions (see [24])
As stated above, a tempered ultradistribution is a continuous linear functional defined on the space 1 of entire functions rapidly decreasing on straight lines parallel to the real axis. Let Ω be the space of functions of the real variable x that are parametrized by a real parameter q:
Ω = { f q ( x ) / f q ( x ) Ω + Ω } ,
where:
Ω + = f q ( x ) / f q + ( x ) { 1 + i ( 1 q ) k x [ f q + ( x ) ] ( q 1 ) } 1 1 q 1 [ + ] ; f q ( x ) 0 ; f q ( x ) | x | p g ( x ) e a x ; p , a + ; k ; ( k ) 0 1 q , q < 2 ,
and:
Ω = f q ( x ) / f q ( x ) { 1 + i ( 1 q ) k x [ f q ( x ) ] ( q 1 ) } 1 1 q 1 [ ] ; f q ( x ) 0 ; f q ( x ) | x | p g ( x ) e a x ; p , a + ; k ; ( k ) 0 1 q , q < 2 ,
Here, 1 is the space of functions integrable in the Lebesgue sense, g ( x ) is bounded, continuous and positive-definite, f q + ( x ) is the restriction of f q ( x ) to + and f q ( x ) is the restriction of f q ( x ) to .
Our q-Fourier transform is now defined ( ( k ) is the imaginary part of k) as:
F : Ω 𝒰 ( t h e s p a c e o f t e m p e r e d u l t r a d i s t r i b u t i o n s ) ,
where:
F ( f q ) ( k , q ) F ( k , q , q ) ,
with:
F ( k , q , q ) = [ H ( q 1 ) H ( q 2 ) ] × H [ ( k ) ] 0 f q ( x ) { 1 + i ( 1 q ) k x [ f q ( x ) ] ( q 1 ) } 1 1 q , d x H [ ( k ) ] 0 f q ( x ) { 1 + i ( 1 q ) k x [ f q ( x ) ] ( q 1 ) } 1 1 q d x .
The inverse transformation is ([21,23]):
f q ( x ) = 1 2 π Γ lim ϵ 0 + 1 2 F ( k , q , q ) δ ( q 1 ϵ ) d q e i k x d k .
As has been proven in [21,23], F is one-to-one from Ω to 𝒰
On the real axis:
F ( k , q ) = [ H ( q 1 ) H ( q 2 ) ] × f q ( x ) { 1 + i ( 1 q ) k x [ f q ( x ) ] ( q 1 ) } 1 1 q d x ,
for the real transform and:
f q ( x ) = 1 2 π lim ϵ 0 + 1 2 F ( k , q , q ) δ ( q 1 ϵ ) d q e i k x d k ,
for its inverse.

3. q-FT in the Limit q q

Our main result is to be presented now, by consideration of the limit q q .
This leads to the (restricted) scenario in which the Tsallis et al. non-invertibility issue raises its head.
Define the restricted (i.e., to the q = q situation) transform, i.e., the Tsallis et al. one,
F T : Ω 𝒰
as:
F T ( f q ) ( k ) = lim q q F ( f q ) ( k , q ) = F ( f q ) ( k , q ) q = q
Thus, according to Equation (14),
F T ( k , q ) = [ H ( q 1 ) H ( q 2 ) ] × H [ ( k ) ] 0 f q ( x ) { 1 + i ( 1 q ) k x [ f q ( x ) ] ( q 1 ) } 1 1 q , d x H [ ( k ) ] 0 f q ( x ) { 1 + i ( 1 q ) k x [ f q ( x ) ] ( q 1 ) } 1 1 q d x
It is seen in [21] that F T is not one-to-one from Ω to 𝒰.
The problem is best understood is we introduce a particularly important set Λ f q , crucial for our considerations. Let Λ f q be given by:
Λ f q = g q Ω / F T ( g q ) ( k ) = F T ( f q ) ( k ) ,
and:
Λ = Λ f q / f q Ω .
We define the equivalence relation:
g q ( x ) f q ( x ) g q Λ f q
and, subsequently, the special the version of the Tsallis et al. q-Fourier transform F U T S : [20]
F U T S : Λ 𝒰
as:
F U T S ( Λ f q ) ( k ) = F T ( f q ) ( k ) .
We see now that F U T S is an application from equivalence classes into equivalence classes and, as a consequence, one-to-one from Λ into 𝒰!
We realize now that the Tsallis et al. qFT is actually a (one-to-one) set-to-set transformation, which solves the non-invertibility issue that occupies our attention in this work.

4. Illustration

Illustrating our theory, we reconsider an important example. We focus our attention on the so-called Hilhorst function (see the pertinent details in [21] and the references therein):
f q ( x ) = λ x β ; x [ a , b ] ; 0 < a < b ; λ > 0 0 ; x outside [ a , b ] ,
with:
λ = q 1 2 q a q 2 q 1 b q 2 q 1 1 q β = 1 q 1 .
In [23], we evaluated the q-Fourier transform on this function and obtained:
F ( k , q , q ) = [ H ( q 1 ) H ( q 2 ) ] H [ ( k ) ] × H ( q 1 ) H q 1 + 1 β × ( q 1 ) λ β ( 2 q ) [ ( 1 q ) i k λ β ] 1 q 1 × a q 2 q 1 F 1 q 1 , 2 q ( q 1 ) [ 1 β ( q 1 ) ] , 1 q 1 + β ( 2 q ) 1 β ( q 1 ) ; 1 ( q 1 ) i k λ β ( q 1 ) a 1 β ( q 1 ) b q 2 q 1 F 1 q 1 , 2 q ( q 1 ) [ 1 β ( q 1 ) ] , 1 q 1 + β ( 2 q ) 1 β ( q 1 ) ; 1 ( q 1 ) i k λ β ( q 1 ) b 1 β ( q 1 ) + H q 1 + 1 β H ( q 2 ) λ β β 1 × a 1 β F 1 q 1 , β 1 β ( q 1 ) 1 , β q 2 β ( q 1 ) 1 ; ( q 1 ) i k λ β ( q 1 ) a 1 β ( q 1 ) b 1 β F 1 q 1 , β 1 β ( q 1 ) 1 , β q 2 β ( q 1 ) 1 ; ( q 1 ) i k λ β ( q 1 ) b 1 β ( q 1 ) .
Taking q = q in Equation (28), we have for F U T S :
F U T S ( Λ f q ) ( k ) = H [ ( k ) ] H ( q 1 ) H ( q 2 ) 1 + ( 1 q ) i k λ 1 1 q ,
and, on the real axis,
F U T S ( Λ f q ) ( k ) = H ( q 1 ) H ( q 2 ) 1 + ( 1 q ) i ( k + i 0 ) λ 1 1 q = H ( q 1 ) H ( q 2 ) 1 + ( 1 q ) i k λ 1 1 q .
Now, from Equations (29) and (30), we see that the Tsallis et al. q-Fourier transform is one-to-one. However, it is a transformation from Λ into 𝒰, a class-to-class one.
This fact reconciles the viewpoints of Tsallis et al. and those of [21], and this is achieved via a rigorous definition of the qFT and of its domain and image.
As a second example, we consider:
f ( x ) = H ( x ) .
In this case, the equivalence class is made up of just one function:
F U T S ( k , q ) = [ H ( q 1 ) H ( q 2 ) ] H [ ( k ) ] 0 1 + ( 1 q ) i k x 1 1 q d x .
Evaluating the integral, we have:
F U T S ( k , q ) = [ H ( q 1 ) H ( q 2 ) ] H [ ( k ) ] Γ 2 q q 1 Γ 1 q 1 ( 1 q ) i k 1 ,
and, finally,
F U T S ( k , q ) = [ H ( q 1 ) H ( q 2 ) ] i 2 q H [ ( k ) ] k .

5. An Open Problem

Let ^ be the equivalence relation defined by:
g q ^ f q lim q 1 g q = lim q 1 f q ,
where f q , g q Ω . Let:
Ξ f q = { g q Ω / g q ^ f q } ,
and let:
Ξ = { Ξ f q / f q Ω } .
If f q = f 1 q + ( q 1 ) f 2 q , g q = g 1 q + ( q 1 ) g 2 q and lim q 1 f 1 q = lim q 1 g 1 q , then f q , g q Ξ f q . However, since f 2 q , g 2 q Ω are different in general, one does not have f q g q . As a simple example of this, take f 1 q ( x ) = f 2 q ( x ) = x q 1 and g 1 q = g 2 q = x 2 ( q 1 ) . Accordingly, Λ f q Ξ f q .
One would, of course, be interested in finding out which are the mathematical properties of the functions f q and g q that generate the belonging g q Λ f q . In other words, what are the mathematical properties that the functions f q and g q need to have so that F T ( f q ) = F T ( g q ) . Note that one has:
F T ( f q ) = F T ( g q ) F ( f ) = F ( g ) f = g .
That is lim q 1 f q = lim q 1 g q . This is a problem that we were unable to solve and that we would like to be considered by the mathematical community

6. Conclusions

We have shown here an important original result: the q-generalization advanced by Tsallis et al. in [20] is to be properly regarded as a transformation between classes of equivalence and, thus, one-to-one, a finding of this paper that solves the qFT’s non-invertibility issue [21].
Our present findings may indicate that Tsallis’ q-statistics revolves around equivalence classes of distributions and not individual ones, as orthodox statistics does. In Section 5, we have seen, however, that an open problem remains that should be addressed in the future.

Author Contributions

A. Plastino and Mario C. Rocca contributed equally to this work.

Conflicts of Interest

The authors declare no conflict of interest.

References

  1. Gell-Mann, M.; Tsallis, C. (Eds.) Nonextensive Entropy: Interdisciplinary Applications; Oxford University Press: New York, NY, USA, 2004.
  2. Tsallis, C. Introduction to Nonextensive Statistical Mechanics: Approaching a Complex World; Springer: New York, NY, USA, 2009. [Google Scholar]
  3. Plastino, A.R.; Plastino, A. Tsallis Entropy, Erhenfest Theorem and Information Theory. Phys. Lett. A 1993, 177, 177–179. [Google Scholar] [CrossRef]
  4. Tsallis, C.; Gell-Mann, M.; Sato, Y. Asymptotically scale-invariant occupancy of phase space makes the entropy Sq extensive. Proc. Natl. Acad. Sci. USA 2005, 102, 15377–15382. [Google Scholar] [CrossRef] [PubMed]
  5. Douglas, P.; Bergamini, S.; Renzoni, F. Tunable Tsallis Distributions in Dissipative Optical Lattices. Phys. Rev. Lett. 2006, 96, 110601. [Google Scholar] [CrossRef]
  6. Liu, B.; Goree, J. Superdiffusion and Non-Gaussian Statistics in a Driven-Dissipative 2D Dusty Plasma. Phys. Rev. Lett. 2008, 100. [Google Scholar] [CrossRef]
  7. DeVoe, R.G. Power-Law Distributions for a Trapped Ion Interacting with a Classical Buffer Gas. Phys. Rev. Lett. 2009, 102, 063001. [Google Scholar] [CrossRef]
  8. Pickup, R.M.; Cywinski, R.; Pappas, C.; Farago, B.; Fouquet, P. Generalized Spin-Glass Relaxation. Phys. Rev. Lett. 2009, 102, 097202. [Google Scholar] [CrossRef]
  9. Burlaga, L.F.; Ness, N.F. Compressible turbulence observed in the helioheath by Voyager 2. Astrophys. J. 2009, 703, 311. [Google Scholar] [CrossRef]
  10. Caruso, F.; Pluchino, A.; Latora, V.; Vinciguerra, S.; Rapisarda, A. Analysis of self-organized criticality in the Olami-Feder-Christensen model and in real earthquakes. Phys. Rev. E 2007, 75, 055101(R). [Google Scholar] [CrossRef]
  11. The CMS Collaboration. Transverse-Momentum and Pseudorapidity Distributions of Charged Hadrons in pp Collisions at s=7 TeV. Phys. Rev. Lett. 2010, 105, 022002. [Google Scholar] [CrossRef]
  12. Adare, A.; Afanasiev, S.; Aidala, C.; Ajitanand, N.N.; Akiba, Y.; Al-Bataineh, H.; Alexander, J.; Aoki, K.; Aphecetche, L.; Armendariz, R.; et al. Measurement of neutral mesons in p+p collisions at s=200 GeV and scaling properties of hadron. Phys. Rev. D 2011, 83, 052004. [Google Scholar] [CrossRef]
  13. Lyra, M.L.; Tsallis, C. Nonextensivity and Multifractality in Low- Dimensional Dissipative Systems. Phys. Rev. Lett. 1998, 80, 53. [Google Scholar] [CrossRef]
  14. Borland, L. Option Pricing Formulas Based on a Non-Gaussian Stock Price Model. Phys. Rev. Lett. 2002, 89, 098701. [Google Scholar] [CrossRef]
  15. Plastino, A.R.; Plastino, A. Tsallis Entropy and Stellar Polytropes. Phys. Lett A 1993, 174, 384–386. [Google Scholar] [CrossRef]
  16. Plastino, A.R.; Plastino, A. Non-extensive statistical mechanics and gen- eralized Fokker-Planck equation. Phys. A 1995, 222, 347–354. [Google Scholar] [CrossRef]
  17. Capurro, A.; Diambra, L.; Lorenzo, D.; Macadar, O.; Martin, M.T.; Mostaccio, C.; Plastino, A. Tsallis entropy and cortical dynamics: The analysis of EEG signals. Phys. A 1998, 257, 149–155. [Google Scholar] [CrossRef]
  18. Martin, M.T.; Plastino, A.R.; Plastino, A. Tsallis-like information mea- sures and the analysis of complex signals. Phys. A 2000, 275, 262–271. [Google Scholar] [CrossRef]
  19. Plastino, A.R.; Plastino, A. Non-extensive solutions to the Vlasov equation. Braz. J. Phys. 1999, 29, 79. [Google Scholar]
  20. Umarov, S.; Tsallis, C.; Steinberg, S. On a q-Central Limit Theorem Consistent with Nonextensive Statistical Mechanics. Milan J. Math. 2008, 76, 307. [Google Scholar] [CrossRef]
  21. Plastino, A.; Rocca, M.C. Inversion of Tsallis’ q-Fourier Transform and the complex-plane generalization. Phys. A 2012, 391, 4740–4747. [Google Scholar] [CrossRef]
  22. Sebastiao e Silva, J. Les fonctions analytiques comme ultra-distributions dans le calcul operationnel. Math. Ann. 1958, 136, 58–96. [Google Scholar] [CrossRef]
  23. Plastino, A.; Rocca, M.C. q-Fourier Transform and its inversion- problem. Milan J. Math. 2012, 80, 243–249. [Google Scholar] [CrossRef]
  24. Hasumi, M. Note on the n-dimensional tempered ultra-distributions. Tôhoku Math. J. 1961, 13, 94–104. [Google Scholar] [CrossRef]
  25. Schwartz, L. Théorie des Distributions; Hermann: Paris, France, 1966. [Google Scholar]
  26. Bollini, C.G.; Escobar, T.; Rocca, M.C. Convolution of ultradistri-butions and Âŕeld theory. Int. J. Theor. Phys. 2003, 43. [Google Scholar] [CrossRef]

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Plastino, A.; Rocca, M.C. On the Nature of the Tsallis–Fourier Transform. Mathematics 2015, 3, 644-652. https://doi.org/10.3390/math3030644

AMA Style

Plastino A, Rocca MC. On the Nature of the Tsallis–Fourier Transform. Mathematics. 2015; 3(3):644-652. https://doi.org/10.3390/math3030644

Chicago/Turabian Style

Plastino, A., and Mario C. Rocca. 2015. "On the Nature of the Tsallis–Fourier Transform" Mathematics 3, no. 3: 644-652. https://doi.org/10.3390/math3030644

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