We determine a general link between two different solutions of the MaxEnt
variational problem, namely, the ones that correspond to using either Shannon's
or Tsallis' entropies in the concomitant variational problem. It is shown that
the two variations lead to equivalent solutions that take different appearances
but contain the same information. These solutions are linked by our
transformation

A. Plastino

Adare

Ananos

Bagci

Bakar

Birkhoff

Borges

Borland

Bosyk

Burlaga

Caruso

Celikoglu

Chevreuil

Curado

DeVoe

Douglas

E. Rufeil Fiori

Ferri

Flego

Furuichi

Gershenfeld

Hilhorst

Jauregui

Jaynes

Katz

Khachatryan

Lyra

Martínez

Pathria

Pickup

Plastino

Reif

Robinson

Sakurai

Shao

Tirnakli

Tsallis

Umarov

English

arXiv

Via a first-order linear differential equation, we determine a general link between two different solutions of the MaxEnt variational problem, namely, the ones that correspond to using either Shannon’s or Tsallis’ entropies in the concomitant variational problem. It is shown that the two variations lead to equivalent solutions that have different appearances but contain the same information. These solutions are linked by our transformation. However, the so-called collision entropy (Tsallis’ one with q=2) does not have a Shannon counterpart.Fil: Rufeil Fiori, Elena. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaFil: Plastino, Ángel Luis. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - La Plata. Instituto de Física La Plata. Universidad Nacional de La Plata. Facultad de Ciencias Exactas. Instituto de Física La Plata; Argentina. Universidad de las Islas Baleares; Españ

Rufeil Fiori, Elena

Plastino, Ángel Luis

LAReferencia - Red Federada de Repositorios Institucionales de Publicaciones Científicas Latinoamericanas

arXiv:1201.4507v2  [math.ST]  28 Jan 2012A Shannon-Tsallis transformationE. Rufeil Fiori1Facultad de Matemática, Astronomía y Física, Universidad Nacional de CórdobaCiudad Universitaria, 5000 Córdoba, ArgentinaA. Plastino2, 32National University La Plata & CONICET IFLP-CCT, C.C. 727 - 1900 LaPlata, Argentina3 Universitat de les Illes Balears and IFISC-CSIC, 07122 Palma de Mallorca,SpainAbstractWe determine a general link between two different solutions of the MaxEnt vari-ational problem, namely, the ones that correspond to using either Shannon’s orTsallis’ entropies in the concomitant variational problem. It is shown that the twovariations lead to equivalent solutions that take different appearances but containthe same information. These solutions are linked by our transformation.Key words: Shannon entropy, Tsallis entropy, MaxEnt.PACS: 89.70.Cf, 05.90.+m, 89.75.Da, 89.75.-k1 IntroductionNonextensive statistical mechanics (NEXT) [1,2,3], a generalization of the or-thodox Boltzmann-Gibbs (BG) one, is actively investigated and applied inmany areas of scientific endeavor. NEXT is based on a nonadditive (thoughextensive [4]) entropic information measure, that is characterized by a realindex q (with q = 1 recovering the standard BG entropy). It has been usedwith regards to variegated systems such as cold atoms in dissipative opti-cal lattices [5], dusty plasmas [6], trapped ions [7], spinglasses [8], turbu-lence in the heliosheath [9], self-organized criticality [10], high-energy experi-ments at LHC/CMS/CERN [11] and RHIC/PHENIX/Brookhaven [12], low-Email address: angeloplastino@gmail.com (A. Plastino2, 3 ).Preprint submitted to Elsevier 16 August 2017dimensional dissipative maps [13], finance [14], galaxies [15], Fokker-Planckequation’s applications [16], etc.A typical NEXT feature is that it can be expressed by recourse to general-izations à la q of standard mathematical concepts [17]. Included are, for in-stance, the logarithm [q-logarithm] and exponential functions (usually denotedas eq(x), with eq=1(x) = ex), addition and multiplication, Fourier transform(FT) and the Central Limit Theorem (CLT) [18]. The q-Fourier transform Fqexhibits the nice property of transforming q-Gaussians into q-Gaussians [18].Recently, plane waves, and the representation of the Dirac delta into planewaves have been also generalized [19,20,21,22].Our central interest here resides in the q-exponential function, regarded asthe MaxEnt variational solution [3] if the pertinent information measure isTsallis’ one. We will show that there is a transform procedure that converts anyShannon-MaxEnt solution [23,24] into a q-exponential, without modificationof the associated Lagrange multipliers, that carry with them all the physics ofthe problem at hand. Why? Because of the Legendre transform properties ofthe MaxEnt solutions [see for instance [23,24,25,26,27,28,29,30] and referencestherein].Accordingly, we are here proving that the physics of a given problem can bediscussed in equivalent fashion by recourse to either Shannon’s measure orTsallis’ one, indistinctly.2 The central ideaWe wish to connect orthodox exponentials with q-exponentials. Let us considerthe Shannon-MaxEnt solution for a constraint given by the average value ofthe variable in question, that we call u:p(u)du = exp [−µ − λu] du = e−µ e−λudu,〈u〉 = K,∫p(u)du = 1, (1)where the two Lagrange multipliers µ and λ correspond, respectively, to nor-malization and conservation of the u−mean value < u >= K.Consider now a second variable x such that dx/du = g(x), with g(x) a functionwe will wish to determine below. Assume that in the second variable we expressthe Tsallis-MaxEnt solution, with the same constraints, but employing theabove mentioned q-exponential functions (eq(x) = [1 + (1 − q)x]1/(1−q). The2support of this function is sometimes finite, depending on the q−value. Seemore details in, for instance, [2]).p(x)dx = C eq(−λ x) dx; C = normalization const. (2)We wantp(x)dx = p(u)du.This entails:C eq(−λ x) dx = p(x)dx = e−µ e−λu(x)(dx/g(x)), (3)that is, given that du = g(x)−1 dx,C g(x) eq(−λ x) = e−µ exp [−λ∫dx g(x)−1]. (4)Remember now thatd eq(x)dx= eq(x)q, (5)so that, taking the logarithm of Eq. (4) we findlnC + ln g(x) + ln [eq(−λ x)] = −µ − λ∫dx g(x)−1. (6)Now, we derive w.r.t. x Eq. (6) and haveg′(x)g(x)− λ eq(−λ x)q−1 = −λ g(x)−1, (7)which leads to a differential equation for our desired g(x):g′(x)− λ eq(−λ x)q−1 g(x) + λ = 0. (8)Solving this equation we establish the link we are looking for.3 The differential equationFor simplicity we now setP (x) = −λ eq(−λ x)q−1; Q(x) = −λ, (9)3and recast (8) asg′(x) + P (x) g(x) = Q(x). (10)Introduce now the integrating factor II = exp [∫dt P (t)], (11)and multiply (10) by itg′(x)I + P (x)I g(x) = Q(x)I. (12)Note thatddx[g(x)I] = g′(x)I + P (x)I g(x), (13)so that (11) becomesddx[g(x)I] = Q(x)I. (14)Integrating this we have nowg(x)I = Q(x)∫Idx+ c. (15)Finally, we can formally express our “solution" function g(x) asg(x) =∫Q(x)Idx+ cI. (16)Thus, for the differential equationg′(x)− λeq(−λx)(q−1)g(x) + λ = 0, (17)we obtain the solutiong(x)= expλx∫eq(−λx′)(q−1)dx′ ,−λx∫exp−λx′∫eq(−λx′′)(q−1)dx′′ dx′ + c , (18)4where c is an integration constant. Now, usingλx∫eq(−λx′)(q−1)dx′ = ln(eq(−λx)(−1)), (19)and− λx∫exp−λx′∫eq(−λx′′)(q−1)dx′′ dx′ =eq(−λx)(2−q)2− q, (20)we finally arrive atg(x)= eq(−λx)(−1)[eq(−λx)(2−q)2− q+ c], (21)which provides us with the Tsallis-Shannon JacobianJ(x) =1g(x). (22)Consider now the instance q → 1. One obviously ought to have g(x) = 1. Wefaceg(x)→ eλx[e−λx + c](23)= 1 + c eλx, (24)which entails c = 0 andg(x)=eq(−λx)(1−q)2− q=1− (1− q)λ x2− q. (25)3.1 Expansion near q=1Let us now take q = 1− ǫ in the solution g(x) = g(x; q) (Eq.(25)),g(x; q = 1− ǫ) =1− ǫλx1 + ǫ. (26)Thus, a first-order expansion in ǫ gives:5g(x; q = 1− ǫ) = 1− (1 + λx) ǫ+O(ǫ2). (27)3.2 The case q = 2Now, let us look at the q = 2− ǫ scenario for our solution g(x) = g(x; q);g(x; q = 2− ǫ) =1 + (1− ǫ)λxǫ. (28)That, in the limit of ǫ → 0 we haveg(x; q = 2− ǫ) ≍(1 + λx)ǫ, (29)→+∞ as ǫ → 0+ (i.e., q → 2−) and (30)→−∞ as ǫ → 0− (i.e., q → 2+). (31)There is a divergence in the first term, in the form of 1/ǫ.We are then in a position to state that, for λx > −1, and given the definitionof the q-exponential (see [2]), our “inverse-Jacobian" function g(x) is positivefor q < 2 and negative for q > 2, while diverging at q = 2. We conclude thatthe transform we are studying is not valid only in the isolated case q = 2 andchanges sign there.4 Arbitrary constraintWe generalize now the preceding considerations to the case of a generalizedconstraint < h(x) >, with h ∈ L2. The concomitant Shannon MaxEnt solutionis [24]p(u)du= e−µ e−λh(u)du, (32)〈h(u)〉=K, (33)∫p(u)du=1. (34)while the Tsallis-MaxEnt solution with the same constraint becomesp(x)dx=C eq(−λ h(x)) dx; C = normalization const. (35)6Assume that u(x) exists and call, as before, dx/du = g(x). We have, as ourcornerstone the relationp(x)dx = p(u)du, (36)entailingC eq(−λ h(x)) dx = e−µ e−λh(u(x))(dx/g(x)), (37)so that, taking the logarithm to this equation we findlnC + ln [g(x)] + ln [eq(−λ h(x))] = −µ − λ h(u(x)). (38)Taking derivatives w.r.t. x yieldsg′(x)g(x)− λ eq(−λ h(x))q−1h′(x) = −λ1g(x)h′(x), (39)which leads to a differential equation for our desired transformation functiong(x)g′(x)− λ eq(−λ h(x))q−1 h′(x)g(x) + λh′(x) = 0, (40)quite similar in shape to Eq. (17), being thus solved in similar fashion. UsingEq. (16) we encounter∫P =∫eq(−λh(x))q−1(−λh′(x))dx, (41)= ln(eq(−λh(x))), (42)e−∫P = eq(−λh(x))−1, (43)∫Qe∫P =∫eq(−λh(x))(−λh′(x))dx, (44)=eq(−λh(x))2−q2− q, (45)that leads tog(x) = eq(−λh(x))−1[eq(−λh(x))2−q2− q+ c]. (46)It is clear that for q = 1 we have g(x) = 1 if c=0. Thus,g(x) =1− (1− q)λh(x)2− q. (47)7Also, the regime-change at q = 2 discussed above does not change.4.1 Special case h(x) = x2For the special case of a variance constraint we haveg′(x)− λ eq(−λ x2)q−1 2xg(x) = −λ2x, (48)whose solution isg(x) =1− (1− q)λx22− q. (49)Let us take now q = 1 − ǫ in the solution g(x) = g(x; q). Then, an expansionnear ǫ = 0 yieldsg(x; q = 1− ǫ) = 1−(1 + λx2)ǫ+O(ǫ2). (50)5 Generalization to M constraintsNow, we generalize to the case of M constraints of the form:〈hi(u)〉 = Ki; i = 1, . . . ,M. (51)Lets use vector notation and call h = (h1, h2, . . . , hM) and λ = (λ1, λ2, . . . , λM)its associated Lagrange multipliers. Then we have for the Shannon MaxEntsolutionp(u)du = e−µ e−λ·h(u)du, (52)while the Tsallis-MaxEnt solution with the same constraints becomesp(x)dx=C eq(−λ · h(x)) dx; C = normalization const. (53)Assume that u(x) exists and call, as before, dx/du = g(x). Then, followingthe steps of the previous section and solving the corresponding differentialequation, we obtaing(x) = eq(−λ · h(x))−1[eq(−λ · h(x))2−q2− q+ c], (54)8where c = 0 in order to get g(x) = 1 in the limit q → 1. Thus,g(x) =1− (1− q)λ · h(x)2− q. (55)6 ConclusionsWe have shown here that, from a MaxEnt practitioner view-point, one canindistinctly employ Shannon’s logarithmic entropy S or Tsallis’ power-lawone Sq (for any q except q = 2). The physics described is the same. To choosebetween S and Sq is just a matter of convenience in the sense of getting simplerexpressions in one case than in the other.The link between the two concomitant probability distributions PShannon(x)and PTsallis(x) is given by the Jacobian J = 1/g, where g is the simple functiong(x) =1− (1− q)λh(x)2− q, (56)with λ the pertinent Lagrange multiplier, and h(x) ∈ L2 an arbitrary functionwhose mean value < h > constitutes MaxEnt’s informational input.7 Appendix: The four different Tsallis’ treatmentsA savvy Tsallis practitioner may wonder what happens with the four differentways of computing q−mean values that one finds in Tsallis’ literature (see[31] and references therein). In addition to the normal expectation values wehave employed above, one also encounters, for a quantity A(x) averaging waysthat, themselves, depend upon q (see below). This transforms our cornerstone-equality pShannon(u)du = pTsallis(x)dx into something much more complicated.However, there is a way out, following the discoveries reported in [31].Bernoulli published in the Ars Conjectandi the first formal attempt to dealwith probabilities already in 1713 and Laplace further formalized the subjectin his Théorie analytique des Probabilités of 1820. In the intervening centuriesProbability Theory (PT) has grown into a rich, powerful, and extremely usefulbranch of Mathematics. Contemporary Physics heavily relies on PT for a largepart of its basic structure, Statistical Mechanics [32,33,34], of course, being amost conspicuous example. One of PT basic definitions is that of the mean9value of an observable A (a measurable quantity). Let A stand for the linearoperator or dynamical variable associated with A. Then,〈A〉 =∫dx p(x)A(x). (57)This was the averging procedure that Tsallis used in his first, pioneering 1988paper [1], and the one discussed in the preceding Sections. It is well knownthat, in some specific cases, it becomes necessary to use “weighted" meanvalues, of the form〈A〉 =∫dx f [p(x)]A(x), (58)with f an analytical function of p. This happens, for instance, when there is aset of states characterized by a distribution with a recognizable maximum anda large tail that contains low but finite probabilities. One faces then the needof making a pragmatical (usually of experimental origin) decision regarding f[31]. In the first stage of NEXT-development, its pioneer practitioners madethe pragmatic choice of using “weighted" mean values, of rather unfamiliarappearance for many physicists. Why? The reasons were of theoretical origin.It was at the time believed that, using the familiar linear, unbiased meanvalues, one was unable to get rid of the Lagrange multiplier associated toprobability-normalization. Since the Tsallis’ formalism yields, in the limit q →1, the orthodox Jaynes-Shannon treatment, the natural choice was to constructweighted expectation values (EVs) using the index q,〈A〉q =∫dx p(x)qA(x), (59)the so-called Curado-Tsallis unbiased mean values (MV) [35]. As shown in [3],employing the Curado-Tsallis (CT) mean values allowed one to obtain an an-alytical expression for the partition function out of the concomitant MaxEntprocess [3]. This EV choice leads to a non extensive formalism endowed withinteresting features: i) the above mentioned property of its partition func-tion Z, ii) a numerical treatment that is relatively simple, and iii) properresults in the limit q → 1. It has, unfortunately, the drawback of exhibitingun-normalized mean values, i.e., 〈〈1〉〉q 6= 1. The latter problem was circum-vented in the subsequent work of Tsallis-Mendes-Plastino (TMP) [36], that“normalized" the CT treatment by employing mean values of the form〈A〉q =∫dxp(x)qXqA(x); Xq =∫dx p(x)q. (60)Most NEXT works employ the TMP procedure. However, the concomitanttreatment is not at all simple. Numerical complications often ensue, which has10encouraged the development of different, alternative approach called the OLMone [37], that preserves the main TMP-idea (the Xq normalization sum) butis numerically simpler. Now, despite appearances, the four Tsallis’ treatmentsare equivalent, as shown in [31]. By equivalence we mean that if one knowsthe probability treatment Pi; i = 1, 2, 3, 4 obtained by anyone of the fourtreatments, there is a unique, automatic way to write down Pj; j 6= i. Moreprecisely, any Pi is a q-exponential, and they all possess the same information-amount if the pair q, β is appropriately “translated” from a version to the other[31]. Indeed,Pi = Z−1 expq∗(−β∗x),Z =∑iexpq∗(−β∗x). (61)Then, as made explicit in [31], given any of the four possible Pi’s q∗, β∗,one can get the q, β values appropriate for the q-exponential of any j 6= i.Such “dictionary” allows one to translate the results obtained in the precedingSections to any other of the three remaining averaging procedures.Acknowledgments This work was partially supported by: i) the projectPIP1177 of CONICET (Argentina) and ii) the project FIS2008-00781/FIS(MICINN) and FEDER (EU) (Spain, EU).References[1] C. Tsallis, J. Stat. Phys. 52 (1988) 479.[2] M. Gell-Mann, C. Tsallis (Eds.), Nonextensive Entropy. InterdisciplinaryApplications (Oxford University Press, New York, 2004); C. Tsallis, Introductionto Nonextensive Statistical Mechanics: Approaching a Complex World (Springer,New York, 2009).[3] A.R. Plastino, A. Plastino, Phys. Lett A 177 (1993) 177.[4] C. Tsallis, M. Gell-Mann, Y. Sato, Proc. Natl. Acad. Sci. USA 102 (2005) 15377;F. Caruso, C. Tsallis, Phys. Rev. E 78 (2008) 021102.[5] P. Douglas, S. Bergamini, F. Renzoni, Phys. Rev. Lett. 96 (2006) 110601; G.B.Bagci, U. Tirnakli, Chaos 19 (2009) 033113.[6] B. Liu, J. Goree, Phys. Rev. Lett. 100 (2008) 055003.[7] R.G. DeVoe, Phys. Rev. Lett. 102 (2009) 063001.[8] R.M. Pickup, R. Cywinski, C. Pappas, B. Farago, P. Fouquet, Phys. Rev. Lett.102 (2009) 097202.11[9] L.F. Burlaga, N.F. Ness, Astrophys. J. 703 (2009) 311.[10] F. Caruso, A. Pluchino, V. Latora, S. Vinciguerra, A. Rapisarda, Phys. Rev. E75 (2007) 055101(R); B. Bakar, U. Tirnakli, Phys. Rev. E 79 (2009) 040103(R);A. Celikoglu, U. Tirnakli, S.M.D. Queiros, Phys. Rev. E 82 (2010) 021124.[11] V. Khachatryan, et al., CMS Collaboration, J. High Energy Phys. 1002 (2010)041; V. Khachatryan, et al., CMS Collaboration, Phys. Rev. Lett. 105 (2010)022002.[12] Adare, et al., PHENIX Collaboration, Phys. Rev. D 83 (2011) 052004; M. Shao,L. Yi, Z.B. Tang, H.F. Chen, C. Li, Z.B. Xu, J. Phys. G 37 (8) (2010) 085104.[13] M.L. Lyra, C. Tsallis, Phys. Rev. Lett. 80 (1998) 53; E.P. Borges, C. Tsallis,G.F.J. Ananos, P.M.C. de Oliveira, Phys. Rev. Lett. 89 (2002) 254103; G.F.J.Ananos, C. Tsallis, Phys. Rev. Lett. 93 (2004) 020601; U. Tirnakli, C. Beck,C. Tsallis, Phys. Rev. E 75 (2007) 040106(R); U. Tirnakli, C. Tsallis, C. Beck,Phys. Rev. E 79 (2009) 056209.[14] L. Borland, Phys. Rev. Lett. 89 (2002) 098701.[15] A.R. Plastino, A. Plastino, Phys. Lett A 174 (1993) 834.[16] A.R. Plastino, A. Plastino, Physica A 222 (1995) 347.[17] E.P. Borges, Physica A 340 (2004) 95.[18] S. Umarov, C. Tsallis, S. Steinberg, Milan J. Math. 76 (2008) 307; S. Umarov,C. Tsallis, M. Gell-Mann, S. Steinberg, J. Math. Phys. 51 (2010) 033502.[19] M. Jauregui, C. Tsallis, J. Math. Phys. 51 (2010) 063304.[20] A. Chevreuil, A. Plastino, C. Vignat, J. Math. Phys. 51 (2010) 093502.[21] A. Plastino and M.C. Rocca, J. Math. Phys 52 (2011) 103503.[22] H.J. Hilhorst, J. Stat. Mech. (2010) P10023.[23] E. T. Jaynes, Phys. Rev. 106 (1957) 620.[24] A. Katz, Principles of Statistical Mechanics: The Information Theory Approach(Freeman and Co., San Francisco, 1967).[25] A. Plastino, F. Olivares, S. Flego, M. Casas, Entropy 13 (2011) 184.[26] S.P. Flego, A. Plastino, and A.R. Plastino, Physica A 390 (2011) 2276.[27] S.P. Flego, A. Plastino, A.R. Plastino, J. Math. Phys. 52 (2011) 082103.[28] S.P. Flego, A. Plastino, A.R. Plastino, Ann. Phys. 326 (2011) 2533.[29] S. P. Flego, A. Plastino, A. R. Plastino, ENTROPY 13 (2011) 2049-2058.[30] A. Plastino, A. R. Plastino, Phys. Lett. A 226 (1997) 257.[31] G. L. Ferri, S. Martinez, A. Plastino, J. Stat. Mech. (2005) P04009.12[32] R.K. Pathria, Statistical Mechanics (Pergamon Press, Exeter, 1993).[33] F. Reif, Statistical and thermal physics (McGraw-Hill, NY, 1965).[34] J. J. Sakurai, Modern quantum mechanics (Benjamin, Menlo Park, Ca., 1985).[35] E.M.F. Curado, C. Tsallis, J. Phys. A 24 (1991) L69.[36] C. Tsallis, R.S. Mendes, A.R. Plastino, Physica A, 261 (1998) 534.[37] S. Martínez, F. Nicolás, F. Pennini, A. Plastino, Physica A 286 (2000) 489.13

Physica A Statistical Mechanics and its Applications

A Shannon-Tsallis transformation

CONICET Digital

arXiv:1201.4507v2  [math.ST]  28 Jan 2012A Shannon-Tsallis transformationE. Rufeil Fiori1Facultad de Matemática, Astronomía y Física, Universidad Nacional de CórdobaCiudad Universitaria, 5000 Córdoba, ArgentinaA. Plastino2, 32National University La Plata & CONICET IFLP-CCT, C.C. 727 - 1900 LaPlata, Argentina3 Universitat de les Illes Balears and IFISC-CSIC, 07122 Palma de Mallorca,SpainAbstractWe determine a general link between two different solutions of the MaxEnt vari-ational problem, namely, the ones that correspond to using either Shannon’s orTsallis’ entropies in the concomitant variational problem. It is shown that the twovariations lead to equivalent solutions that take different appearances but containthe same information. These solutions are linked by our transformation.Key words: Shannon entropy, Tsallis entropy, MaxEnt.PACS: 89.70.Cf, 05.90.+m, 89.75.Da, 89.75.-k1 IntroductionNonextensive statistical mechanics (NEXT) [1,2,3], a generalization of the or-thodox Boltzmann-Gibbs (BG) one, is actively investigated and applied inmany areas of scientific endeavor. NEXT is based on a nonadditive (thoughextensive [4]) entropic information measure, that is characterized by a realindex q (with q = 1 recovering the standard BG entropy). It has been usedwith regards to variegated systems such as cold atoms in dissipative opti-cal lattices [5], dusty plasmas [6], trapped ions [7], spinglasses [8], turbu-lence in the heliosheath [9], self-organized criticality [10], high-energy experi-ments at LHC/CMS/CERN [11] and RHIC/PHENIX/Brookhaven [12], low-Email address: angeloplastino@gmail.com (A. Plastino2, 3 ).Preprint submitted to Elsevier 16 August 2017dimensional dissipative maps [13], finance [14], galaxies [15], Fokker-Planckequation’s applications [16], etc.A typical NEXT feature is that it can be expressed by recourse to general-izations à la q of standard mathematical concepts [17]. Included are, for in-stance, the logarithm [q-logarithm] and exponential functions (usually denotedas eq(x), with eq=1(x) = ex), addition and multiplication, Fourier transform(FT) and the Central Limit Theorem (CLT) [18]. The q-Fourier transform Fqexhibits the nice property of transforming q-Gaussians into q-Gaussians [18].Recently, plane waves, and the representation of the Dirac delta into planewaves have been also generalized [19,20,21,22].Our central interest here resides in the q-exponential function, regarded asthe MaxEnt variational solution [3] if the pertinent information measure isTsallis’ one. We will show that there is a transform procedure that converts anyShannon-MaxEnt solution [23,24] into a q-exponential, without modificationof the associated Lagrange multipliers, that carry with them all the physics ofthe problem at hand. Why? Because of the Legendre transform properties ofthe MaxEnt solutions [see for instance [23,24,25,26,27,28,29,30] and referencestherein].Accordingly, we are here proving that the physics of a given problem can bediscussed in equivalent fashion by recourse to either Shannon’s measure orTsallis’ one, indistinctly.2 The central ideaWe wish to connect orthodox exponentials with q-exponentials. Let us considerthe Shannon-MaxEnt solution for a constraint given by the average value ofthe variable in question, that we call u:p(u)du = exp [−µ − λu] du = e−µ e−λudu,〈u〉 = K,∫p(u)du = 1, (1)where the two Lagrange multipliers µ and λ correspond, respectively, to nor-malization and conservation of the u−mean value < u >= K.Consider now a second variable x such that dx/du = g(x), with g(x) a functionwe will wish to determine below. Assume that in the second variable we expressthe Tsallis-MaxEnt solution, with the same constraints, but employing theabove mentioned q-exponential functions (eq(x) = [1 + (1 − q)x]1/(1−q). The2support of this function is sometimes finite, depending on the q−value. Seemore details in, for instance, [2]).p(x)dx = C eq(−λ x) dx; C = normalization const. (2)We wantp(x)dx = p(u)du.This entails:C eq(−λ x) dx = p(x)dx = e−µ e−λu(x)(dx/g(x)), (3)that is, given that du = g(x)−1 dx,C g(x) eq(−λ x) = e−µ exp [−λ∫dx g(x)−1]. (4)Remember now thatd eq(x)dx= eq(x)q, (5)so that, taking the logarithm of Eq. (4) we findlnC + ln g(x) + ln [eq(−λ x)] = −µ − λ∫dx g(x)−1. (6)Now, we derive w.r.t. x Eq. (6) and haveg′(x)g(x)− λ eq(−λ x)q−1 = −λ g(x)−1, (7)which leads to a differential equation for our desired g(x):g′(x)− λ eq(−λ x)q−1 g(x) + λ = 0. (8)Solving this equation we establish the link we are looking for.3 The differential equationFor simplicity we now setP (x) = −λ eq(−λ x)q−1; Q(x) = −λ, (9)3and recast (8) asg′(x) + P (x) g(x) = Q(x). (10)Introduce now the integrating factor II = exp [∫dt P (t)], (11)and multiply (10) by itg′(x)I + P (x)I g(x) = Q(x)I. (12)Note thatddx[g(x)I] = g′(x)I + P (x)I g(x), (13)so that (11) becomesddx[g(x)I] = Q(x)I. (14)Integrating this we have nowg(x)I = Q(x)∫Idx+ c. (15)Finally, we can formally express our “solution" function g(x) asg(x) =∫Q(x)Idx+ cI. (16)Thus, for the differential equationg′(x)− λeq(−λx)(q−1)g(x) + λ = 0, (17)we obtain the solutiong(x)= expλx∫eq(−λx′)(q−1)dx′ ,−λx∫exp−λx′∫eq(−λx′′)(q−1)dx′′ dx′ + c , (18)4where c is an integration constant. Now, usingλx∫eq(−λx′)(q−1)dx′ = ln(eq(−λx)(−1)), (19)and− λx∫exp−λx′∫eq(−λx′′)(q−1)dx′′ dx′ = eq(−λx)(2−q)2− q, (20)we finally arrive atg(x)= eq(−λx)(−1)[eq(−λx)(2−q)2− q+ c], (21)which provides us with the Tsallis-Shannon JacobianJ(x) =1g(x). (22)Consider now the instance q → 1. One obviously ought to have g(x) = 1. Wefaceg(x)→ eλx[e−λx + c](23)= 1 + c eλx, (24)which entails c = 0 andg(x)=eq(−λx)(1−q)2− q=1− (1− q)λ x2− q. (25)3.1 Expansion near q=1Let us now take q = 1− ǫ in the solution g(x) = g(x; q) (Eq.(25)),g(x; q = 1− ǫ) =1− ǫλx1 + ǫ. (26)Thus, a first-order expansion in ǫ gives:5g(x; q = 1− ǫ) = 1− (1 + λx) ǫ+O(ǫ2). (27)3.2 The case q = 2Now, let us look at the q = 2− ǫ scenario for our solution g(x) = g(x; q);g(x; q = 2− ǫ) =1 + (1− ǫ)λxǫ. (28)That, in the limit of ǫ→ 0 we haveg(x; q = 2− ǫ) ≍(1 + λx)ǫ, (29)→+∞ as ǫ→ 0+ (i.e., q → 2−) and (30)→−∞ as ǫ→ 0− (i.e., q → 2+). (31)There is a divergence in the first term, in the form of 1/ǫ.We are then in a position to state that, for λx > −1, and given the definitionof the q-exponential (see [2]), our “inverse-Jacobian" function g(x) is positivefor q < 2 and negative for q > 2, while diverging at q = 2. We conclude thatthe transform we are studying is not valid only in the isolated case q = 2 andchanges sign there.4 Arbitrary constraintWe generalize now the preceding considerations to the case of a generalizedconstraint < h(x) >, with h ∈ L2. The concomitant Shannon MaxEnt solutionis [24]p(u)du= e−µ e−λh(u)du, (32)〈h(u)〉=K, (33)∫p(u)du=1. (34)while the Tsallis-MaxEnt solution with the same constraint becomesp(x)dx=C eq(−λ h(x)) dx; C = normalization const. (35)6Assume that u(x) exists and call, as before, dx/du = g(x). We have, as ourcornerstone the relationp(x)dx = p(u)du, (36)entailingC eq(−λ h(x)) dx = e−µ e−λh(u(x))(dx/g(x)), (37)so that, taking the logarithm to this equation we findlnC + ln [g(x)] + ln [eq(−λ h(x))] = −µ − λ h(u(x)). (38)Taking derivatives w.r.t. x yieldsg′(x)g(x)− λ eq(−λ h(x))q−1h′(x) = −λ1g(x)h′(x), (39)which leads to a differential equation for our desired transformation functiong(x)g′(x)− λ eq(−λ h(x))q−1 h′(x)g(x) + λh′(x) = 0, (40)quite similar in shape to Eq. (17), being thus solved in similar fashion. UsingEq. (16) we encounter∫P =∫eq(−λh(x))q−1(−λh′(x))dx, (41)= ln(eq(−λh(x))), (42)e−∫P = eq(−λh(x))−1, (43)∫Qe∫P =∫eq(−λh(x))(−λh′(x))dx, (44)=eq(−λh(x))2−q2− q, (45)that leads tog(x) = eq(−λh(x))−1[eq(−λh(x))2−q2− q+ c]. (46)It is clear that for q = 1 we have g(x) = 1 if c=0. Thus,g(x) =1− (1− q)λh(x)2− q. (47)7Also, the regime-change at q = 2 discussed above does not change.4.1 Special case h(x) = x2For the special case of a variance constraint we haveg′(x)− λ eq(−λ x2)q−1 2xg(x) = −λ2x, (48)whose solution isg(x) =1− (1− q)λx22− q. (49)Let us take now q = 1 − ǫ in the solution g(x) = g(x; q). Then, an expansionnear ǫ = 0 yieldsg(x; q = 1− ǫ) = 1−(1 + λx2)ǫ+O(ǫ2). (50)5 Generalization to M constraintsNow, we generalize to the case of M constraints of the form:〈hi(u)〉 = Ki; i = 1, . . . ,M. (51)Lets use vector notation and call h = (h1, h2, . . . , hM) and λ = (λ1, λ2, . . . , λM)its associated Lagrange multipliers. Then we have for the Shannon MaxEntsolutionp(u)du = e−µ e−λ·h(u)du, (52)while the Tsallis-MaxEnt solution with the same constraints becomesp(x)dx=C eq(−λ · h(x)) dx; C = normalization const. (53)Assume that u(x) exists and call, as before, dx/du = g(x). Then, followingthe steps of the previous section and solving the corresponding differentialequation, we obtaing(x) = eq(−λ · h(x))−1[eq(−λ · h(x))2−q2− q+ c], (54)8where c = 0 in order to get g(x) = 1 in the limit q → 1. Thus,g(x) =1− (1− q)λ · h(x)2− q. (55)6 ConclusionsWe have shown here that, from a MaxEnt practitioner view-point, one canindistinctly employ Shannon’s logarithmic entropy S or Tsallis’ power-lawone Sq (for any q except q = 2). The physics described is the same. To choosebetween S and Sq is just a matter of convenience in the sense of getting simplerexpressions in one case than in the other.The link between the two concomitant probability distributions PShannon(x)and PTsallis(x) is given by the Jacobian J = 1/g, where g is the simple functiong(x) =1− (1− q)λh(x)2− q, (56)with λ the pertinent Lagrange multiplier, and h(x) ∈ L2 an arbitrary functionwhose mean value < h > constitutes MaxEnt’s informational input.7 Appendix: The four different Tsallis’ treatmentsA savvy Tsallis practitioner may wonder what happens with the four differentways of computing q−mean values that one finds in Tsallis’ literature (see[31] and references therein). In addition to the normal expectation values wehave employed above, one also encounters, for a quantity A(x) averaging waysthat, themselves, depend upon q (see below). This transforms our cornerstone-equality pShannon(u)du = pTsallis(x)dx into something much more complicated.However, there is a way out, following the discoveries reported in [31].Bernoulli published in the Ars Conjectandi the first formal attempt to dealwith probabilities already in 1713 and Laplace further formalized the subjectin his Théorie analytique des Probabilités of 1820. In the intervening centuriesProbability Theory (PT) has grown into a rich, powerful, and extremely usefulbranch of Mathematics. Contemporary Physics heavily relies on PT for a largepart of its basic structure, Statistical Mechanics [32,33,34], of course, being amost conspicuous example. One of PT basic definitions is that of the mean9value of an observable A (a measurable quantity). Let A stand for the linearoperator or dynamical variable associated with A. Then,〈A〉 =∫dx p(x)A(x). (57)This was the averging procedure that Tsallis used in his first, pioneering 1988paper [1], and the one discussed in the preceding Sections. It is well knownthat, in some specific cases, it becomes necessary to use “weighted" meanvalues, of the form〈A〉 =∫dx f [p(x)]A(x), (58)with f an analytical function of p. This happens, for instance, when there is aset of states characterized by a distribution with a recognizable maximum anda large tail that contains low but finite probabilities. One faces then the needof making a pragmatical (usually of experimental origin) decision regarding f[31]. In the first stage of NEXT-development, its pioneer practitioners madethe pragmatic choice of using “weighted" mean values, of rather unfamiliarappearance for many physicists. Why? The reasons were of theoretical origin.It was at the time believed that, using the familiar linear, unbiased meanvalues, one was unable to get rid of the Lagrange multiplier associated toprobability-normalization. Since the Tsallis’ formalism yields, in the limit q →1, the orthodox Jaynes-Shannon treatment, the natural choice was to constructweighted expectation values (EVs) using the index q,〈A〉q =∫dx p(x)qA(x), (59)the so-called Curado-Tsallis unbiased mean values (MV) [35]. As shown in [3],employing the Curado-Tsallis (CT) mean values allowed one to obtain an an-alytical expression for the partition function out of the concomitant MaxEntprocess [3]. This EV choice leads to a non extensive formalism endowed withinteresting features: i) the above mentioned property of its partition func-tion Z, ii) a numerical treatment that is relatively simple, and iii) properresults in the limit q → 1. It has, unfortunately, the drawback of exhibitingun-normalized mean values, i.e., 〈〈1〉〉q 6= 1. The latter problem was circum-vented in the subsequent work of Tsallis-Mendes-Plastino (TMP) [36], that“normalized" the CT treatment by employing mean values of the form〈A〉q =∫dxp(x)qXqA(x); Xq =∫dx p(x)q. (60)Most NEXT works employ the TMP procedure. However, the concomitanttreatment is not at all simple. Numerical complications often ensue, which has10encouraged the development of different, alternative approach called the OLMone [37], that preserves the main TMP-idea (the Xq normalization sum) butis numerically simpler. Now, despite appearances, the four Tsallis’ treatmentsare equivalent, as shown in [31]. By equivalence we mean that if one knowsthe probability treatment Pi; i = 1, 2, 3, 4 obtained by anyone of the fourtreatments, there is a unique, automatic way to write down Pj; j 6= i. Moreprecisely, any Pi is a q-exponential, and they all possess the same information-amount if the pair q, β is appropriately “translated” from a version to the other[31]. Indeed,Pi = Z−1 expq∗(−β∗x),Z =∑iexpq∗(−β∗x). (61)Then, as made explicit in [31], given any of the four possible Pi’s q∗, β∗,one can get the q, β values appropriate for the q-exponential of any j 6= i.Such “dictionary” allows one to translate the results obtained in the precedingSections to any other of the three remaining averaging procedures.Acknowledgments This work was partially supported by: i) the projectPIP1177 of CONICET (Argentina) and ii) the project FIS2008-00781/FIS(MICINN) and FEDER (EU) (Spain, EU).References[1] C. Tsallis, J. Stat. Phys. 52 (1988) 479.[2] M. Gell-Mann, C. Tsallis (Eds.), Nonextensive Entropy. InterdisciplinaryApplications (Oxford University Press, New York, 2004); C. Tsallis, Introductionto Nonextensive Statistical Mechanics: Approaching a Complex World (Springer,New York, 2009).[3] A.R. Plastino, A. Plastino, Phys. Lett A 177 (1993) 177.[4] C. Tsallis, M. Gell-Mann, Y. Sato, Proc. Natl. Acad. Sci. USA 102 (2005) 15377;F. Caruso, C. Tsallis, Phys. Rev. E 78 (2008) 021102.[5] P. Douglas, S. Bergamini, F. Renzoni, Phys. Rev. Lett. 96 (2006) 110601; G.B.Bagci, U. Tirnakli, Chaos 19 (2009) 033113.[6] B. Liu, J. Goree, Phys. Rev. Lett. 100 (2008) 055003.[7] R.G. DeVoe, Phys. Rev. Lett. 102 (2009) 063001.[8] R.M. Pickup, R. Cywinski, C. Pappas, B. Farago, P. Fouquet, Phys. Rev. Lett.102 (2009) 097202.11[9] L.F. Burlaga, N.F. Ness, Astrophys. J. 703 (2009) 311.[10] F. Caruso, A. Pluchino, V. Latora, S. Vinciguerra, A. Rapisarda, Phys. Rev. E75 (2007) 055101(R); B. Bakar, U. Tirnakli, Phys. Rev. E 79 (2009) 040103(R);A. Celikoglu, U. Tirnakli, S.M.D. Queiros, Phys. Rev. E 82 (2010) 021124.[11] V. Khachatryan, et al., CMS Collaboration, J. High Energy Phys. 1002 (2010)041; V. Khachatryan, et al., CMS Collaboration, Phys. Rev. Lett. 105 (2010)022002.[12] Adare, et al., PHENIX Collaboration, Phys. Rev. D 83 (2011) 052004; M. Shao,L. Yi, Z.B. Tang, H.F. Chen, C. Li, Z.B. Xu, J. Phys. G 37 (8) (2010) 085104.[13] M.L. Lyra, C. Tsallis, Phys. Rev. Lett. 80 (1998) 53; E.P. Borges, C. Tsallis,G.F.J. Ananos, P.M.C. de Oliveira, Phys. Rev. Lett. 89 (2002) 254103; G.F.J.Ananos, C. Tsallis, Phys. Rev. Lett. 93 (2004) 020601; U. Tirnakli, C. Beck,C. Tsallis, Phys. Rev. E 75 (2007) 040106(R); U. Tirnakli, C. Tsallis, C. Beck,Phys. Rev. E 79 (2009) 056209.[14] L. Borland, Phys. Rev. Lett. 89 (2002) 098701.[15] A.R. Plastino, A. Plastino, Phys. Lett A 174 (1993) 834.[16] A.R. Plastino, A. Plastino, Physica A 222 (1995) 347.[17] E.P. Borges, Physica A 340 (2004) 95.[18] S. Umarov, C. Tsallis, S. Steinberg, Milan J. Math. 76 (2008) 307; S. Umarov,C. Tsallis, M. Gell-Mann, S. Steinberg, J. Math. Phys. 51 (2010) 033502.[19] M. Jauregui, C. Tsallis, J. Math. Phys. 51 (2010) 063304.[20] A. Chevreuil, A. Plastino, C. Vignat, J. Math. Phys. 51 (2010) 093502.[21] A. Plastino and M.C. Rocca, J. Math. Phys 52 (2011) 103503.[22] H.J. Hilhorst, J. Stat. Mech. (2010) P10023.[23] E. T. Jaynes, Phys. Rev. 106 (1957) 620.[24] A. Katz, Principles of Statistical Mechanics: The Information Theory Approach(Freeman and Co., San Francisco, 1967).[25] A. Plastino, F. Olivares, S. Flego, M. Casas, Entropy 13 (2011) 184.[26] S.P. Flego, A. Plastino, and A.R. Plastino, Physica A 390 (2011) 2276.[27] S.P. Flego, A. Plastino, A.R. Plastino, J. Math. Phys. 52 (2011) 082103.[28] S.P. Flego, A. Plastino, A.R. Plastino, Ann. Phys. 326 (2011) 2533.[29] S. P. Flego, A. Plastino, A. R. Plastino, ENTROPY 13 (2011) 2049-2058.[30] A. Plastino, A. R. Plastino, Phys. Lett. A 226 (1997) 257.[31] G. L. Ferri, S. Martinez, A. Plastino, J. Stat. Mech. (2005) P04009.12[32] R.K. Pathria, Statistical Mechanics (Pergamon Press, Exeter, 1993).[33] F. Reif, Statistical and thermal physics (McGraw-Hill, NY, 1965).[34] J. J. Sakurai, Modern quantum mechanics (Benjamin, Menlo Park, Ca., 1985).[35] E.M.F. Curado, C. Tsallis, J. Phys. A 24 (1991) L69.[36] C. Tsallis, R.S. Mendes, A.R. Plastino, Physica A, 261 (1998) 534.[37] S. Martínez, F. Nicolás, F. Pennini, A. Plastino, Physica A 286 (2000) 489.13

Crossref

A Shannon–Tsallis transformation

Via a first-order linear differential equation, we determine a general link between two different solutions of the MaxEnt variational problem, namely, the ones that correspond to using either Shannon's or Tsallis' entropies in the concomitant variational problem. It is shown that the two variations lead to equivalent solutions that have different appearances but contain the same information. These solutions are linked by our transformation. However, the so-called collision entropy (Tsallis' one with q=2) does not have a Shannon counterpart. © 2013 Elsevier B.V. All rights reserved.This work was partially supported by (i) the project PIP1177 of CONICET (Argentina) and (ii) the project FIS2008-00781/FIS (MICINN) and FEDER (EU) (Spain, EU).Peer Reviewe

Rufeil Fiori, E.

Plastino, A.

Digital.CSIC

Via a first-order linear differential equation, we determine a general link between two different solutions of the MaxEnt variational problem, namely, the ones that correspond to using either Shannon’s or Tsallis’ entropies in the concomitant variational problem. It is shown that the two variations lead to equivalent solutions that have different appearances but contain the same information. These solutions are linked by our transformation. However, the so-called collision entropy (Tsallis’ one with q=2) does not have a Shannon counterpart.Instituto de Física La PlataConsejo Nacional de Investigaciones Científicas y Técnica

El Servicio de Difusión de la Creación Intelectual

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A Shannon-Tsallis transformation
E. Rufeil Fiori1
Facultad de Matemática, Astronomía y Física, Universidad Nacional de Córdoba
Ciudad Universitaria, 5000 Córdoba, Argentina
A. Plastino2, 3
2National University La Plata & CONICET IFLP-CCT, C.C. 727 - 1900 La
Plata, Argentina
3 Universitat de les Illes Balears and IFISC-CSIC, 07122 Palma de Mallorca,
Spain
Abstract
We determine a general link between two different solutions of the MaxEnt vari-
ational problem, namely, the ones that correspond to using either Shannon’s or
Tsallis’ entropies in the concomitant variational problem. It is shown that the two
variations lead to equivalent solutions that take different appearances but contain
the same information. These solutions are linked by our transformation.
Key words: Shannon entropy, Tsallis entropy, MaxEnt.
PACS: 89.70.Cf, 05.90.+m, 89.75.Da, 89.75.-k
1 Introduction
Nonextensive statistical mechanics (NEXT) [1,2,3], a generalization of the or-
thodox Boltzmann-Gibbs (BG) one, is actively investigated and applied in
many areas of scientific endeavor. NEXT is based on a nonadditive (though
extensive [4]) entropic information measure, that is characterized by a real
index q (with q = 1 recovering the standard BG entropy). It has been used
with regards to variegated systems such as cold atoms in dissipative opti-
cal lattices [5], dusty plasmas [6], trapped ions [7], spinglasses [8], turbu-
lence in the heliosheath [9], self-organized criticality [10], high-energy experi-
ments at LHC/CMS/CERN [11] and RHIC/PHENIX/Brookhaven [12], low-
Email address: angeloplastino@gmail.com (A. Plastino2, 3 ).
Preprint submitted to Elsevier 16 August 2017
dimensional dissipative maps [13], finance [14], galaxies [15], Fokker-Planck
equation’s applications [16], etc.
A typical NEXT feature is that it can be expressed by recourse to general-
izations à la q of standard mathematical concepts [17]. Included are, for in-
stance, the logarithm [q-logarithm] and exponential functions (usually denoted
as eq(x), with eq=1(x) = e
x), addition and multiplication, Fourier transform
(FT) and the Central Limit Theorem (CLT) [18]. The q-Fourier transform Fq
exhibits the nice property of transforming q-Gaussians into q-Gaussians [18].
Recently, plane waves, and the representation of the Dirac delta into plane
waves have been also generalized [19,20,21,22].
Our central interest here resides in the q-exponential function, regarded as
the MaxEnt variational solution [3] if the pertinent information measure is
Tsallis’ one. We will show that there is a transform procedure that converts any
Shannon-MaxEnt solution [23,24] into a q-exponential, without modification
of the associated Lagrange multipliers, that carry with them all the physics of
the problem at hand. Why? Because of the Legendre transform properties of
the MaxEnt solutions [see for instance [23,24,25,26,27,28,29,30] and references
therein].
Accordingly, we are here proving that the physics of a given problem can be
discussed in equivalent fashion by recourse to either Shannon’s measure or
Tsallis’ one, indistinctly.
2 The central idea
We wish to connect orthodox exponentials with q-exponentials. Let us consider
the Shannon-MaxEnt solution for a constraint given by the average value of
the variable in question, that we call u:
p(u)du = exp [−µ − λu] du = e−µ e−λudu,
〈u〉 = K,∫
p(u)du = 1, (1)
where the two Lagrange multipliers µ and λ correspond, respectively, to nor-
malization and conservation of the u−mean value < u >= K.
Consider now a second variable x such that dx/du = g(x), with g(x) a function
we will wish to determine below. Assume that in the second variable we express
the Tsallis-MaxEnt solution, with the same constraints, but employing the
above mentioned q-exponential functions (eq(x) = [1 + (1 − q)x]
1/(1−q). The
2
support of this function is sometimes finite, depending on the q−value. See
more details in, for instance, [2]).
p(x)dx = C eq(−λ x) dx; C = normalization const. (2)
We want
p(x)dx = p(u)du.
This entails:
C eq(−λ x) dx = p(x)dx = e
−µ e−λu(x)(dx/g(x)), (3)
that is, given that du = g(x)−1 dx,
C g(x) eq(−λ x) = e
−µ exp [−λ
∫
dx g(x)−1]. (4)
Remember now that
d eq(x)
dx
= eq(x)
q, (5)
so that, taking the logarithm of Eq. (4) we find
lnC + ln g(x) + ln [eq(−λ x)] = −µ − λ
∫
dx g(x)−1. (6)
Now, we derive w.r.t. x Eq. (6) and have
g′(x)
g(x)
− λ eq(−λ x)
q−1 = −λ g(x)−1, (7)
which leads to a differential equation for our desired g(x):
g′(x)− λ eq(−λ x)
q−1 g(x) + λ = 0. (8)
Solving this equation we establish the link we are looking for.
3 The differential equation
For simplicity we now set
P (x) = −λ eq(−λ x)
q−1; Q(x) = −λ, (9)
3
and recast (8) as
g′(x) + P (x) g(x) = Q(x). (10)
Introduce now the integrating factor I
I = exp [
∫
dt P (t)], (11)
and multiply (10) by it
g′(x)I + P (x)I g(x) = Q(x)I. (12)
Note that
d
dx
[g(x)I] = g′(x)I + P (x)I g(x), (13)
so that (11) becomes
d
dx
[g(x)I] = Q(x)I. (14)
Integrating this we have now
g(x)I = Q(x)
∫
Idx+ c. (15)
Finally, we can formally express our “solution" function g(x) as
g(x) =
∫
Q(x)Idx+ c
I
. (16)
Thus, for the differential equation
g′(x)− λeq(−λx)
(q−1)g(x) + λ = 0, (17)
we obtain the solution
g(x)= exp

λ
x∫
eq(−λx
′)(q−1)dx′

 ,

−λ
x∫
exp

−λ
x′∫
eq(−λx
′′)(q−1)dx′′

 dx′ + c

 , (18)
4
where c is an integration constant. Now, using
λ
x∫
eq(−λx
′)(q−1)dx′ = ln
(
eq(−λx)
(−1)
)
, (19)
and
− λ
x∫
exp

−λ
x′∫
eq(−λx
′′)(q−1)dx′′

 dx′ = eq(−λx)(2−q)
2− q
, (20)
we finally arrive at
g(x)= eq(−λx)
(−1)
[
eq(−λx)
(2−q)
2− q
+ c
]
, (21)
which provides us with the Tsallis-Shannon Jacobian
J(x) =
1
g(x)
. (22)
Consider now the instance q → 1. One obviously ought to have g(x) = 1. We
face
g(x)→ eλx
[
e−λx + c
]
(23)
= 1 + c eλx, (24)
which entails c = 0 and
g(x)=
eq(−λx)
(1−q)
2− q
=
1− (1− q)λ x
2− q
. (25)
3.1 Expansion near q=1
Let us now take q = 1− ǫ in the solution g(x) = g(x; q) (Eq.(25)),
g(x; q = 1− ǫ) =
1− ǫλx
1 + ǫ
. (26)
Thus, a first-order expansion in ǫ gives:
5
g(x; q = 1− ǫ) = 1− (1 + λx) ǫ+O(ǫ2). (27)
3.2 The case q = 2
Now, let us look at the q = 2− ǫ scenario for our solution g(x) = g(x; q);
g(x; q = 2− ǫ) =
1 + (1− ǫ)λx
ǫ
. (28)
That, in the limit of ǫ→ 0 we have
g(x; q = 2− ǫ) ≍
(1 + λx)
ǫ
, (29)
→+∞ as ǫ→ 0+ (i.e., q → 2−) and (30)
→−∞ as ǫ→ 0− (i.e., q → 2+). (31)
There is a divergence in the first term, in the form of 1/ǫ.
We are then in a position to state that, for λx > −1, and given the definition
of the q-exponential (see [2]), our “inverse-Jacobian" function g(x) is positive
for q < 2 and negative for q > 2, while diverging at q = 2. We conclude that
the transform we are studying is not valid only in the isolated case q = 2 and
changes sign there.
4 Arbitrary constraint
We generalize now the preceding considerations to the case of a generalized
constraint < h(x) >, with h ∈ L2. The concomitant Shannon MaxEnt solution
is [24]
p(u)du= e−µ e−λh(u)du, (32)
〈h(u)〉=K, (33)∫
p(u)du=1. (34)
while the Tsallis-MaxEnt solution with the same constraint becomes
p(x)dx=C eq(−λ h(x)) dx; C = normalization const. (35)
6
Assume that u(x) exists and call, as before, dx/du = g(x). We have, as our
cornerstone the relation
p(x)dx = p(u)du, (36)
entailing
C eq(−λ h(x)) dx = e
−µ e−λh(u(x))(dx/g(x)), (37)
so that, taking the logarithm to this equation we find
lnC + ln [g(x)] + ln [eq(−λ h(x))] = −µ − λ h(u(x)). (38)
Taking derivatives w.r.t. x yields
g′(x)
g(x)
− λ eq(−λ h(x))
q−1h′(x) = −λ
1
g(x)
h′(x), (39)
which leads to a differential equation for our desired transformation function
g(x)
g′(x)− λ eq(−λ h(x))
q−1 h′(x)g(x) + λh′(x) = 0, (40)
quite similar in shape to Eq. (17), being thus solved in similar fashion. Using
Eq. (16) we encounter
∫
P =
∫
eq(−λh(x))
q−1(−λh′(x))dx, (41)
= ln(eq(−λh(x))), (42)
e−
∫
P = eq(−λh(x))
−1, (43)∫
Qe
∫
P =
∫
eq(−λh(x))(−λh
′(x))dx, (44)
=
eq(−λh(x))
2−q
2− q
, (45)
that leads to
g(x) = eq(−λh(x))
−1
[
eq(−λh(x))
2−q
2− q
+ c
]
. (46)
It is clear that for q = 1 we have g(x) = 1 if c=0. Thus,
g(x) =
1− (1− q)λh(x)
2− q
. (47)
7
Also, the regime-change at q = 2 discussed above does not change.
4.1 Special case h(x) = x2
For the special case of a variance constraint we have
g′(x)− λ eq(−λ x
2)q−1 2xg(x) = −λ2x, (48)
whose solution is
g(x) =
1− (1− q)λx2
2− q
. (49)
Let us take now q = 1 − ǫ in the solution g(x) = g(x; q). Then, an expansion
near ǫ = 0 yields
g(x; q = 1− ǫ) = 1−
(
1 + λx2
)
ǫ+O(ǫ2). (50)
5 Generalization to M constraints
Now, we generalize to the case of M constraints of the form:
〈hi(u)〉 = Ki; i = 1, . . . ,M. (51)
Lets use vector notation and call h = (h1, h2, . . . , hM) and λ = (λ1, λ2, . . . , λM)
its associated Lagrange multipliers. Then we have for the Shannon MaxEnt
solution
p(u)du = e−µ e−λ·h(u)du, (52)
while the Tsallis-MaxEnt solution with the same constraints becomes
p(x)dx=C eq(−λ · h(x)) dx; C = normalization const. (53)
Assume that u(x) exists and call, as before, dx/du = g(x). Then, following
the steps of the previous section and solving the corresponding differential
equation, we obtain
g(x) = eq(−λ · h(x))
−1
[
eq(−λ · h(x))
2−q
2− q
+ c
]
, (54)
8
where c = 0 in order to get g(x) = 1 in the limit q → 1. Thus,
g(x) =
1− (1− q)λ · h(x)
2− q
. (55)
6 Conclusions
We have shown here that, from a MaxEnt practitioner view-point, one can
indistinctly employ Shannon’s logarithmic entropy S or Tsallis’ power-law
one Sq (for any q except q = 2). The physics described is the same. To choose
between S and Sq is just a matter of convenience in the sense of getting simpler
expressions in one case than in the other.
The link between the two concomitant probability distributions PShannon(x)
and PTsallis(x) is given by the Jacobian J = 1/g, where g is the simple function
g(x) =
1− (1− q)λh(x)
2− q
, (56)
with λ the pertinent Lagrange multiplier, and h(x) ∈ L2 an arbitrary function
whose mean value < h > constitutes MaxEnt’s informational input.
7 Appendix: The four different Tsallis’ treatments
A savvy Tsallis practitioner may wonder what happens with the four different
ways of computing q−mean values that one finds in Tsallis’ literature (see
[31] and references therein). In addition to the normal expectation values we
have employed above, one also encounters, for a quantity A(x) averaging ways
that, themselves, depend upon q (see below). This transforms our cornerstone-
equality pShannon(u)du = pTsallis(x)dx into something much more complicated.
However, there is a way out, following the discoveries reported in [31].
Bernoulli published in the Ars Conjectandi the first formal attempt to deal
with probabilities already in 1713 and Laplace further formalized the subject
in his Théorie analytique des Probabilités of 1820. In the intervening centuries
Probability Theory (PT) has grown into a rich, powerful, and extremely useful
branch of Mathematics. Contemporary Physics heavily relies on PT for a large
part of its basic structure, Statistical Mechanics [32,33,34], of course, being a
most conspicuous example. One of PT basic definitions is that of the mean
9
value of an observable A (a measurable quantity). Let A stand for the linear
operator or dynamical variable associated with A. Then,
〈A〉 =
∫
dx p(x)A(x). (57)
This was the averging procedure that Tsallis used in his first, pioneering 1988
paper [1], and the one discussed in the preceding Sections. It is well known
that, in some specific cases, it becomes necessary to use “weighted" mean
values, of the form
〈A〉 =
∫
dx f [p(x)]A(x), (58)
with f an analytical function of p. This happens, for instance, when there is a
set of states characterized by a distribution with a recognizable maximum and
a large tail that contains low but finite probabilities. One faces then the need
of making a pragmatical (usually of experimental origin) decision regarding f
[31]. In the first stage of NEXT-development, its pioneer practitioners made
the pragmatic choice of using “weighted" mean values, of rather unfamiliar
appearance for many physicists. Why? The reasons were of theoretical origin.
It was at the time believed that, using the familiar linear, unbiased mean
values, one was unable to get rid of the Lagrange multiplier associated to
probability-normalization. Since the Tsallis’ formalism yields, in the limit q →
1, the orthodox Jaynes-Shannon treatment, the natural choice was to construct
weighted expectation values (EVs) using the index q,
〈A〉q =
∫
dx p(x)qA(x), (59)
the so-called Curado-Tsallis unbiased mean values (MV) [35]. As shown in [3],
employing the Curado-Tsallis (CT) mean values allowed one to obtain an an-
alytical expression for the partition function out of the concomitant MaxEnt
process [3]. This EV choice leads to a non extensive formalism endowed with
interesting features: i) the above mentioned property of its partition func-
tion Z, ii) a numerical treatment that is relatively simple, and iii) proper
results in the limit q → 1. It has, unfortunately, the drawback of exhibiting
un-normalized mean values, i.e., 〈〈1〉〉q 6= 1. The latter problem was circum-
vented in the subsequent work of Tsallis-Mendes-Plastino (TMP) [36], that
“normalized" the CT treatment by employing mean values of the form
〈A〉q =
∫
dx
p(x)q
Xq
A(x); Xq =
∫
dx p(x)q. (60)
Most NEXT works employ the TMP procedure. However, the concomitant
treatment is not at all simple. Numerical complications often ensue, which has
10
encouraged the development of different, alternative approach called the OLM
one [37], that preserves the main TMP-idea (the Xq normalization sum) but
is numerically simpler. Now, despite appearances, the four Tsallis’ treatments
are equivalent, as shown in [31]. By equivalence we mean that if one knows
the probability treatment Pi; i = 1, 2, 3, 4 obtained by anyone of the four
treatments, there is a unique, automatic way to write down Pj; j 6= i. More
precisely, any Pi is a q-exponential, and they all possess the same information-
amount if the pair q, β is appropriately “translated” from a version to the other
[31]. Indeed,
Pi = Z
−1 expq∗(−β
∗x),
Z =
∑
i
expq∗(−β
∗x). (61)
Then, as made explicit in [31], given any of the four possible Pi’s q
∗, β∗,
one can get the q, β values appropriate for the q-exponential of any j 6= i.
Such “dictionary” allows one to translate the results obtained in the preceding
Sections to any other of the three remaining averaging procedures.
Acknowledgments This work was partially supported by: i) the project
PIP1177 of CONICET (Argentina) and ii) the project FIS2008-00781/FIS
(MICINN) and FEDER (EU) (Spain, EU).
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A Shannon-Tsallis transformation

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