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Some remarks about the Tsallis’ Formalism L. Velazquez∗ Departamento de Física, Universidad de Pinar del Río Martí 270, esq. 27 de Noviembre, Pinar del Río, Cuba. 2 F. Guzmań† 0 Departamento de Física Nuclear 0 Instituto Superior de Ciencias y Tecnologías Nucleares 2 Quinta de los Molinos. Ave Carlos III y Luaces, Plaza n Ciudad de La Habana, Cuba. a (February 1, 2008) J 7 mostatisticsis veryatractive,since itallowsus to obtain ] h In the present paper theconditions for thevalidity of the directlytheprobabilisticdistributionfunctionofthegen- c Tsallis’ Statistics are analyzed. The same has been done fol- eralized canonical ensemble at the thermodynamic equi- e lowing the analogy with the traditional case: starting from librium, as well as to develop the dynamical study of m themicrocanonicaldescriptionofthesystemsandtakinginto systems in non-equilibrium processes. - account their self-similarity scaling properties in thethermo- t The main dificulty for this kind of description is to a dynamiclimit, itis analyzed thenecessary conditionsfor the determine the necesary conditions for the application of t equivalenceofmicrocanonicalensemblewiththeTsallis’ gen- s each specific entropic form. For example, in the Tsallis’ . eralization of the canonical ensemble. It is shown that the t Statistics, the theory is not be able to determine uni- a Tsallis’ Statistics is appropriate for the macroscopic descrip- vocally the value of the entropy index, q, so that, it m tion of systems with potential scaling laws of the asymptotic is needed the experiment or the computational simula- accessiblestatesdensityofthemicrocanonicalensemble. Our - d analysis shows many details of the Tsallis’ formalism: the tion in order to precise it (see for example in the refs. n q-expectation values, the generalized Legendre’s transforma- [12,32,13]). Similar arguments can be applied for other o tions between the thermodynamic potentials, as well as the formulations of the Thermodynamics based on a para- c conditions for its validity, having a priori the possibility to metricinformationentropicform. Thatisthereasonwhy [ estimatethevalueoftheentropicindexwithoutthenecessity we consider that the statistical description of nonexten- 5 of appealing to the computational simulations or the experi- sivesystemsshouldstartfromthemicroscopiccharacter- v ment. On the other hand, the definition of physical temper- istics of them. 1 ature received a modification which differs from the Toral’s Followingthetraditionalanalysis,thederivationofthe 4 result. Forthecaseoffinitesystems,wehavegeneralized the Thermostatistics from the microscopic properties of the 4 microcanonical thermostatistics of D. H. E. Gross with the systems could be performed by considerating the micro- 7 generalizationofthecurvaturetensorforthiskindofdescrip- canonicalensemble. ForthecaseoftheTsallis’Statistics 0 1 tion. this is not a new idea. 0 In1994A.PlastinoandA.R.Plastino[8]hadproposed PACS numbers: 05.20.Gg; 05.20.-y / one way to justify the q-generalized canonical ensemble t a withsimilarargumentsemployedbyGibbshimselfinde- m rivinghiscanonicalensemble. Itisbasedontheconsider- - I. INTRODUCTION ationofaclosedsystemcomposedbyasubsystemweakly d interacting with a finite thermal bath. They showedthat n the macroscopic characteristics of the subsystem are de- o In the last years many researchers have been work- scribedbytheTsallis’potentialdistribution,relatingthe c ing in the justification of the Tsallis’ Formalism. Many v: of them have pretended to do it in the context of the entropy index, q, with the finiteness of the last one. In this approach the Tsallis ad-hoc cut-off condition comes i Information Theory [1–3] without appealing to the mi- X in a natural fashion. croscopic properties of the systems. Through the years r manyfunctionalformsoftheinformationentropysimilar Another attempt was made by S. Abe and A. K. Ra- a jagopal [9]: a closed system composed by a subsystem totheShanonn-Boltzmann-Gibbs’havebeenproposedin weakly interacting with a very large thermal bath, this order to generalize the traditionalThermodynamics (see time analyzing the behavior of the systems around the for example in refs. [4–7]). This way to derive the Ther- equilibrium,consideringthisasastateinwhichthemost probableconfigurationsaregiven. Theyshowedthatthe Tsallis’ canonical ensemble can be obtained if the entropy counting rule is modified, introducing the Tsallis’ ∗[email protected] generalization of the logarithmic function for arbitrary †[email protected] entropicindex[11],showinginthiswaythepossibilityof 1 the nonuniqueness of the canonical ensemble theory. methodology used by D. H. E. Gross in deriving his Mi- SofarithasbeensaidthattheTsallis’Statisticsallows crocanonical Thermostatistics [18]throughthetechnique to extend the Thermodynamics to the study of systems of the steepest descend method. that are anomalous from the traditional point of view, In the ref. [19] was shown that the Boltzmann-Gibbs’ systems with long-rangecorrelationsdue to the presence Statistics can be applied to the macroscopic study of oflong-rangeinteractions,withadynamicsofnonmarko- thepseudoextensivesystems,thosewithexponential self- vian stochastic processes, where the entropic index gives similarity scaling laws [17,19] in the ThL, using an ad- a measure of the non extensivity degree of a system, an equate selection of the representation of the motion in- intrinsic characteristic of the same [10]. The identifi- tegrals space [17], ℑ . The previous analysis suggests N cation of this parameter with the finiteness of a thermal thatapossibleapplicationof Tsallis’formalismcouldbe bath is limited, since this argument is non-applicable on found for those systems with an scaling behavior weaker manyothercontextsinwhichtheTsallis’Statisticsisex- than the exponential. pected to work: astrophysical systems [14,15], turbulent In this analysis the following potential self-similarity fluids and non-screened plasma [16], etc. scaling laws will be considered: TheAbe-Rajogopal’sanalysissuggeststhatthereisan arbitrarinessintheselectionoftheentropycountingrule, N →N(α)=αN which determines the form of the distribution. In their I →I(α)=αχI ⇒Wasym(α)=ακWasym(1), workstheydonotestablishacriteriumthatallowstode- a→a(α)=απaa  fine the selectionofthe entropycountingruleunivocally.  (1) In the Boltzmann-Gibbs’Statistics the entropy counting rule is supported by means of the scaling behavior whereW istheaccessiblevolumeofthemicrocanon- asym of the microcanonical states density and the fundamen- ical ensemble in the system configurational space in the tal macroscopic observables, the integrals of motion and ThL, I are the system integrals of motion of the macro- external parameters, with the increasing of the system scopic description in a specific representation R of ℑ , I N degrees of freedom, and its Thermodynamic Formalism, a is a certain set of parameters, α is the scaling param- based on the Legendre’s Transformations between the eter, χ , π and κ are real constants characterizing the a Thermodynamic potentials, by the equivalence between scaling transformations. The nomenclature W (α) asym the microcanonical and the canonical ensembles in the represents: Thermodynamic Limit (ThL). In the ref. [17] it was addressed the problem of gen- W (α)=W [I(α),N(α),a(α)]. (2) asym asym eralizing the extensive postulates in order to extend theThermostatisticsforsomeHamiltoniannon-extensive This kind of self-similarity scaling laws demands an systems. Our proposition was that this derivation could entropy counting rule different from the logarithmic. It becarriedouttakingintoconsiderationtheself-similarity issupposedthattheTsallis’generalizationofexponential scaling properties of the systems with the increasing of and logarithmic functions [11]: theirdegreesoffreedomandanalyzingthe conditionsfor the equivalence of the microcanonical ensemble with the 1 x1−q−1 generalizedcanonicalensembleintheThL.Thelastargu- eq(x)=[1+(1−q)x]1−q lnq(x)= 1−q ≡e−q1(x) menthasamostgeneralcharacterthantheGibbs’,since (3) it does not demand the separability of one subsystem from the whole system. The Gibbs’ argument is in dis- are more convenient to deal with it. agreementwiththelong-rangecorrelationsofthenonex- Let us consider a finite Hamiltonian system with this tensive systems. The consideration of the self-similarity kind of scaling behavior in the ThL. We postulate that scaling properties of the systems allows us to precise the the Generalized Boltzmann’s Principle [17] adopts counting rule for the generalized Boltzmann’s entropy the following form: [17], as well as the equivalence of the microcanonicalen- semblewiththegeneralizedcanonicalonedeterminesthe (S ) =ln W. (4) necesary conditions for the applicability of the general- B q q ized canonical description in the ThL. The accessiblevolumeofthe microcanonicalensemble in the system configurational space, W, is given by: II. THE LEGENDRE’S FORMALISM W (I,N,a)=Ω(I,N,a)δI =δI δ[I−I (X;a)]dX, o o N Z In this section the analysis ofthe necessaryconditions (5) for the equivalence of the microcanonical ensemble with the Tsallis’ canonical one will be performed in analogy where δI is a suitable constant volume element which o with our previous work [19], which was motived by the makes W dimensionless. The corresponding information 2 entropy for the q-generalized Boltzmann’s entropy, the motion space [17], M , that is, the set M is composed c c Eq.(4), is the Tsallis’ nonextensive entropy (TNE) [20]: by those representations R satisfying the restriction: I S =− pqln p . (6) χ≡0, (15) q k q k Xk in the scaling transformation given in Eq.(1). In these In the thermodynamic equilibrium the TNE leads to cases, when the ThL is invoked, the main contribution theq-exponentialgeneralizationoftheBoltzmann-Gibbs’ totheintegralofthe Eq.(13)willcomefromthemaxima Distributions: oftheq-exponentialfunctionargument. Theequivalence between the microcanonical and the canonical ensemble 1 ω (X;β,N,a)= e [−β·I (X;a)], (7) will only take place when there is only one sharp peak. q Zq(β,N,a) q N Thus, the argument of the q-exponential function leads to assume the nonlinear generalizationof the Legendre’s where Z (β,a,N) is the partition function [21]For this q Formalism [26,27]given by: ensemble, the q-Generalized Laplace’s Transformation is given by: P (β,N,a)=Max c β·I−(S ) (I,N,a) . (16) q q B q dI h i Z (β,N,a)= e (−β·I)W (I,N,a) . (8) e q Z q δIo We recognized immediately the formalism of the nor- malized q-expectations values [26]. The maximization The Laplace’s Transformation establishes the connec- leads to the relation: tion between the fundamental potentials of both ensem- ∇(S ) bles, the q-generalizedPlanck’s potential: β = B q (1−(1−q)β·I). (17) 1+(1−q)(S ) B q P (β,N,a)=−ln [Z (β,N,a)], (9) q q q Using the identity: and the generalized Boltzmann’s entropy defined by the Eq.(4): ∇(S ) ∇S = B q , (18) B 1+(1−q)(S ) dI B q e [−P (β,N,a)]= e (−β·I)e (S ) (I,N,a) . q q Z q qh B q iδIo where S is the usual Boltzmann’s entropy, the Eq.(17) B (10) can be rewritten as: The q-logarithmic function satisfies the subadditivity re- β =∇SB(1−(1−q)β·I). (19) lation: Finally it is arrived to the relation: ln (xy)=ln (x)+ln (y)+(1−q)ln (x)ln (y), (11) q q q q q ∇S β = B . (20) and therefore: (1+(1−q)I·∇SB) e (x)e (y)=e [x+y+(1−q)xy]. (12) This is a very interesting result because it allows to q q q limit the values of the entropy index. If this formalism The last identity allows us to rewrite the Eq.(10) as: isarbitrarilyappliedto apseudoextensivesystem(see in ref. [19]), then I·∇S will not bound in the ThL and β B e [−P (β,N,a)]= e −c β·I+(S ) (I,N,a) dI , willvanishtrivially. Theonlypossibilityinthiscaseisto q q Z qh q B q iδIo impose the restriction q ≡ 1, that is, the Tsallis’ Statis- tics with an arbitrary entropy index can not be applied (13) to the pseudoextensive systems. There are many exam- ples in the literature in which the Tsallis’ Statistics has where: beenappliedindiscriminatelywithoutmindingifthesys- c =1+(1−q)(S ) . (14) tems are extensive or not, i.e., gases [27,28], blackbody q B q radiation [29], and others. In the Tsallis’ case, the linear form of the Legendre’s In some cases, the authors of these works have intro- Transformation is violated and therefore, the ordinary ducedsomeartificialmodificationstotheoriginalTsallis’ Legendre’s Formalism does not establish the correspon- formalism in order to obtain the same results as those dence between the two ensembles. In order to preserve of the classical Thermodynamics, i.e., the q-dependent the homogeneous scaling in the q-exponential function Boltzmann’s constant (see for example in ref. [31]). The argument,itmustbe demandedthe scalinginvarianceof aboveresultsindicatethenonapplicabilityoftheTsallis’ the set of admissible representations of the integrals of Statistics for these kind of systems. It must be pointed 3 out that this conclusion is supported with a great accu- e [−P (β,N,a)]≃ e −P (β,N,a) × q q q q racyby directexperimentalmensurementstrying to find h i R nonextensive effects in some ordinary extensive systems e 1 dI (scyostsemmics [b3a3c,3k4g]r,oguansdesb[l3a5c]k).body radiation [32], fermion ×eq(cid:20)−2(I −IM)µ· (−Kq)µν(cid:12)I=IM ·(I−IM)ν(cid:21)δIo. (cid:12) The Tsallis’ formalism introduces a correlation to the (cid:12) (23) canonical intensive parameters of the Boltzmann-Gibbs’ Probabilistic Distribution Function. This result differs The maximum will be stable if all the eingenvalues of from the one obtained by Toral [30], who applied to the theq-curvaturetensorarenegativeandverylarge. Inthis microcanonicalensemble the physical definitions of tem- case, in the q-generalized canonical ensemble there will perature and pressure introduced by S. Abe in the ref. besmallfluctuationsoftheintegralsofmotionaroundits [28]: q-expectation values. The integration of Eq.(23) yields: e [−P (β,N,a)]≃e −P (β,N,a) × q q q q kkPTTppp1hhhyyysss == 11++((11−−11qq))SSqq ∂∂∂∂EVSSqq,. (21) × 1 h e iΓ(cid:16)21−−qq(cid:17) . (24) WensheemnbtlheeasesudmefiinngittihonesgeanreeraaplipzleieddBtoolttzhmeamninc’rsoPcarinnocnipiclae,l δIodet21 (cid:18)12−πq (−Kq)µν(cid:12)I=IM(cid:19)Γ(cid:16)21−−qq + 21n(cid:17) the Eq.(4), the physical temperature coincides with the (cid:12) (cid:12) usual Boltzmann’s relation: Denoting K−1 by: q 1 ∂ kTphys = ∂ESB. K−1 = 1 Γ(cid:16)12−−qq(cid:17) , q theIteinstreoapsyytcoousnhtoiwng, trhualetothfitshreegsuenlterdaoleizsendoBtodletzpmenadnno’ns δIodet21 (cid:18)12−πq (−Kq)µν(cid:12)(cid:12)I=IM(cid:19)Γ(cid:16)21−−qq + 21n(cid:17) Principle [17], but on separability of a closed system in (cid:12) (25) subsystems weakly correlated among them, and the ad- and rewriting Eq.(24) again: ditivity of the integrals of motion and the macroscopic parameters. It must be recalled that these exigencies e [−P (β,N,a)]≃e −P (β,N,a)+ q q q q are only valid for the extensive systems, but, it is not h e the case that we are studying here. Our result comes in ln K−1 −(1−q)ln K−1 P (β,N,a) , (26) fashion as consequence of the system scaling laws in the q q q q q i (cid:0) (cid:1) (cid:0) (cid:1) thermodynamic limit. e it is finally arrivedto the condition: An important second condition must be satisfied for the validity of the Legendre’s transformation, the sta- R(q;β,N,a)= (1−q)ln K−1 ≪1. (27) bility of the maximum. This condition leads to the q- q q (cid:12) (cid:0) (cid:1)(cid:12) generalization of the Microcanonical Thermostatistics of Thelastconditionc(cid:12)ouldbe considered(cid:12)asanoptimiza- D. H. E. Gross [18]. In this approach, the stability of tion problem, since the entropic index is an independent the Legendre’s formalism is supported by the concavity variable in the functional dependency of the physical of the entropy, the negative definition of the quadratic quantities. The specific value of q could be chosen in formsofthecurvaturetensor[18,19]. IntheTsallis’case, order to minimize the function R(q;β,N,a) for all the the curvature tensor must be modified as: possible values of the integrals of motion . In this way, 1 ∂ ∂ the problem of the determination of the entropic index (Kq)µν = 1−(1−q)P (cid:20)(2−q)∂Iµ∂Iν (SB)q+ could be solved in the frame of the microcanonical the- q ory without appealing to the computational simulation e or the experiment. Thus,theq-generalizedPlanck’spotentialcouldbeob- ∂ ∂ +(1−q)(cid:18)βµ∂Iν (SB)q +βν∂Iµ (SB)q(cid:19)(cid:21). (22) tained by means of the generalized Legendre’s transformation: Taking into consideration that the scaling behavior of P (β,N,a)≃c β·I −(S ) (I,N,a), (28) thefunctions(S ) andP areidentical,whichisderived q q B q B q q from the Eq.(16), it is easy to see that the curvature e where the canonical parameters β hold the Eq.(20). tensor is scaling invariant. Using the above definition Thus, the q-generalization of the Boltzmann’s entropy and developing the Taylor’s power expansion up to the will be equivalent with the Tsallis’ entropy in the ThL: secondordertermintheq-exponentialargument,wecan approximate the Eq.(13) as: (S ) ≃S . (29) B q q 4 If the uniquenees of the maximum is not guarantized, [5] B. D. Sharma and D. P . Mittal, J. Math. Sci. 10 , 28 that is, any of the eingenvaluesof the q-curvaturetensor (1975). isnonnegativeinaspecific regionofthe integralsofmo- [6] S. 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