The kinetic foundations of Tsallis' nonextensive thermostatistics are investigated through Boltzmann's transport equation approach. Our analysis follows from a nonextensive generalization of the "molecular chaos hypothesis." For q>0, the q-transport equation satisfies an H theorem based on Tsallis entropy. It is also proved that the collisional equilibrium is given by Tsallis' q-nonextensive velocity distribution.Facultad de Ciencias Astronómicas y Geofísica

Lima, J. A. S.

Plastino, Ángel Ricardo

Silva, Regina H.

Physical Review Letters

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arXiv:cond-mat/0101030v2  3 Oct 2002Nonextensive Thermostatistics and the H-TheoremJ. A. S. Lima1, R. Silva1, and A. R. Plastino2,31Departamento de F́ısica Teórica e Experimental, Universidade Federal do Rio Grande do Norte,59072-970, Natal-RN, Brazil2Facultad de Ciencias Astronomicas y Geofisicas, Universidad Nacional de La Plata,C.C. 727, 1900 La Plata, Argentina3Departament de Fisica, Universidat de les Illes Balears, 07071 Palma de Mallorca, Spain(October 22, 2018)The kinetic foundations of Tsallis’ nonextensive thermostatistics are investigated through Boltz-mann’s transport equation approach. Our analysis follows from a nonextensive generalization of the“molecular chaos hypothesis”. For q > 0, the q-transport equation satisfies an H-theorem based onTsallis entropy. It is also proved that the collisional equilibrium is given by Tsallis’ q-nonextensivevelocity distribution.PACS numbers: 05.45.+b; 05.20.-y; 05.90.+mIn 1988 Tsallis proposed a striking generalization ofthe Boltzmann-Gibbs entropy functional given by [1],Sq = −kXipqi lnq pi , (1)where k is Boltzmann’s constant, pi is the probability ofthe i-th microstate, the parameter q is any real number,and the q-logarithmic function is defined as [2,3]lnq f = (1− q)−1(f1−q − 1), (f > 0). (2)For future reference it is convenient to introduce the q-exponential function eq(f), which is defined byeq(f) = [1 + (1− q)f ]1/1−q , (3)if 1 + (1 − q)f > 0 and by eq(f) = 0 otherwise. Theseq-functions satisfy, for f, g > 0, the identities (see [3] fora thorough discussion)eq(lnq f) = f ,lnq f + lnq g = lnq fg + (q − 1)(lnq f)(lnq g) . (4)When q → 1 all the above expressions reproduce thoseverified by the usual elementary functions and Tsallis’ en-tropy reduces to the standard logarithmic one, namely:S1 = −kPi pi ln pi. The most distinctive trait of Sq isits pseudoadditivity. Given two systems A and B in-dependent in the sense of factorizability of the (joint)microstate probabilities, the Tsallis entropy of the com-posite system A⊕B verifies Sq(A⊕B) = Sq(A)+Sq(B)+(1 − q)Sq(A)Sq(B). Hence, |1 − q| quantifies the lack ofextensivity of Sq.The q-thermostatistics associated with Sq [1,2] is nowa-days being hailed as the possible basis of a theoreti-cal framework appropriate to dealwith nonextensive set-tings. There is a growing body of evidence suggestingthat Sq provides a convenient frame for the thermosta-tistical analysis of many physical systems and processes[4,5,6,7,8], such as the velocity distribution of galaxyclusters [5], Landau damping in plasmas [7], superdif-fusion phenomena [8] and, more generally, systems ex-hibiting a nonextensive thermodynamic behaviour due,for instance, to long range interactions [1,2]. It is worthto stress that some of the aforementioned developmentsinvolve a quantitative agreement between experimentaldata and theoretical models based on Tsallis’ thermo-statistics [4,5,6,7]. For instance, it was experimentallyfound that pure electron plasmas in Penning traps re-lax to metaestable states whose radial density profilesdo not maximize Boltzmann-Gibbs entropy. However,Boghosian showed that the observed profiles are well de-scribed by Tsallis’ thermostatistics with q close to 1/2[4]. Beck’s recent treatment of fully developed turbulentflows constitutes another interesting example [6]. Basedon Tsallis formalism with q 6= 1, Beck calculated theprobability density functions of velocity differences de-pending on distance and Reynolds number, as well as theconcomitant scalling exponents, finding good agreementwith turbulence experiments.A large portion of the experimental evidence support-ing Tsallis proposal involves the non-Maxwellian (power-law) velocity q-distribution associated with Tsallis’ gen-eralized canonical ensemble approach to the classical N -body problem. This equilibrium q-distribution may bederived through a simple nonextensive generalization ofthe Maxwell ansatz [9]. Alternatively, it can be obtainedmaximizing Tsallis entropy under the constraints im-posed by normalization and the energy mean value [10],a procedure closely related to Jaynes information theoryformulation of statistical mechanics [11,12]. So far, mosttheoretical studies on Tsallis thermostatistics have beendeveloped on the basis of the maximum entropy principle[1,13,14,15]. As widely known, this approach to statis-tical ensembles was, in the case of standard statisticalmechanics, historically the latest one to appear. Evenwhen Gibbs’ introduced his ensemble approach, the ki-1netic foundations of statistical mechanics were alreadywell developed. On the light of this, it is not unreason-able to expect that a systematic exploration of the kineticaspects of Tsallis’ thermostatistics may be crucial to il-luminate its foundations as well as to achieve a betterunderstanding of its physical applications.In this framework, the aim of this letter is to obtainthe equilibrium velocity q-distribution from a slight gen-eralization of the kinetic Boltzmann H-theorem. Thewhole argument follows simply by modifying the molec-ular chaos hypothesis, as originally advanced by Boltz-mann, and generalizing the local entropy expression inaccording to Tsallis proposal.The statistical content of Boltzmann’s kinetic theoryrelies on two main ingredients [16,17]. The first one isa specific functional form for the local entropy, which isexpressed by Boltmann’s logarithmic measureH [f ] = −kZf(~x,~v, t) ln f(~x,~v, t) d3v . (5)The second one is the celebrated hypothesis of molec-ular chaos (“Stosszahlansatz”), which is tantamount toassume the factorizability of the joint distribution asso-ciated with two colliding moleculesf(~x1, ~v1, ~x2, ~v2, t) = f(~x1, ~v1, t) f(~x2, ~v2, t) . (6)These two statistical assumptions are inextricably inter-twined. Therefore, if one adopts a generalized nonex-tensive entropic measure, a consistent generalization ofthe “Stosszahlansatz” hypothesis should also be imple-mented. Different choices for the collision term in thekinetic equation (which, in turn, is determined by the“Stosszahlansatz”) lead to different forms for the entropicfunctional exhibiting a time derivative with definite sign.As a consequence, the form of the entropic functionalbehaving monotonically with time (and consequently ad-mitting an H-theorem) depends upon the form of thecollisional term appearing in the kinetic equation.Historically, the basic assumption (6) (sometimes re-ferred to as “Maxwell’s ansatz”) has generated a lot ofcontroversy. The fundamental role played by this hy-pothesis was first realized by S.H. Burbury [18] in 1894.The precise characterization of the conditions of its ap-plicability has been an important conceptual problem oftheoretical physics ever since [19]. The physical meaningof equation (6) is that colliding molecules are uncorre-lated. The irreversible behaviour of Boltzmann’s equa-tion can be traced back to this assumption. It is clearly atime asymmetric hypothesis, since molecules assumed tobe uncorrelated before a collision certainly become corre-lated after the collision has taken place [20].Although very plausible, and endowed with an intu-itively clear statistical meaning, Boltzmann’s particularexpression (6) for the molecular chaos hypothesis can notbe deduced from first principles. By no means it is an in-escapable consequence of classical mechanics. Boltzmannhimself accepted that the hypothesis of molecular chaoswas needed in order to obtain irreversibility. Further,He also admitted that the hypothesis may not always bevalid for real gases, especially at high densities [19,21].In what follows we introduce a consistent general-ization of this hypothesis in accordance with Tsallis’nonextensive formalism. We remark that equation (6)implies that the logarithm of the joint distributionf(~x1, ~v1, ~x2, ~v2, t) is equal to the sum of two terms, eachone involving only the marginal distribution associatedwith one of the colliding molecules. Our generalized hy-pothesis is to assume that a power of the joint distribu-tion (instead of the logarithm) is equal to the sum of twoterms, each one depending on just one of the collidingmolecules. By recourse to the q-generalization of the log-arithm function, this condition can be formulated in away that recovers the standard hypothesis of molecularchaos as a limit case.Let us now consider a spatially homogeneous gas ofN hard-sphere particles of mass m and diameter s, un-der the action of an external force ~F , and enclosed ina volume V . The state of a non-relativistic gas is ki-netically characterized by the one-particle distributionfunction f(~x,~v, t). The quantity f(~x,~v, t)d3xd3v gives,at each time t, the number of particles in the volumeelement d3xd3v around the particle position ~x and veloc-ity ~v. In principle, this distribution function verifies theq-nonextensive Boltzmann equation∂f∂t+ ~v ·∂f∂~x+1m~F ·∂f∂~v= Cq(f) , (7)where Cq denotes the q-collisional term. The left-hand-side of (7) is just the total time derivative of the distri-bution function. Hence, nonextensivity effects can be in-corporated only through the collisional term. Naturally,Cq(f) may be calculated in accordance with the laws ofelastic collisions. Its specific structure must lead to thestandard result in the limit q → 1. We also make thefollowing assumptions: (i) Only binary collisions occurin the gas; (ii) Cq(f) is a local function of the slow vary-ing distribution function; (iii) Cq(f) is consistent withthe energy, momentum and particle number conservationlaws.Our main goal is now to show that the generalizedcollisional term Cq(f) leads to a nonnegative expressionfor the time derivative of the q-entropy, and that it doesnot vanish unless the distribution function assumes theequilibrium form associated with q-Maxwellian gas [9].Now, following standard lines we defineCq(f) =s22Z|~V · ~e|Rqdωd3v1 , (8)where d3v1 stands for the volume element in velocityspace, ~V = ~v1 − ~v is the relative velocity before colli-sion, ~e denotes an arbitrary unit vector, and dω is an2elementary solid angle such that s2dω is the area of the“collision cylinder” (for details on the collision’s geome-try see Refs. [16,17]). The quantity Rq(f, f′) is a differ-ence of two correlation functions (just before and aftercollision), which are assumed to satisfy a q-generalizedform of the molecular chaos hypothesis. In the presentq-nonextensive scenario we will assume thatRq(f, f′) = eq(f′q−1 lnq f′ + f ′q−11 lnq f′1)−eq(fq−1 lnq f + fq−11 lnq f1) , (9)where primes refer to the distribution function after col-lision. When q → 1 equation (9) reduces to Boltzmann’smolecular chaos hypothesislimq→1Rq = R = f′f ′1 − ff1 . (10)For the local entropy we adopt Tsallis expression,Hq = −kZf q lnq fd3v , (11)which reduces to the standard Boltzmann measure (5)for q = 1. Now, we first take the partial time derivativeof the above expression∂Hq∂t= −kZ[qf q−1 lnq f + 1]∂f∂td3v . (12)As one may check, by inserting the generalized Boltz-mann equation (7) into (12), and using (8), expression(12) can be rewritten as a balance equation∂Hq∂t+∇ · ~Sq = Gq(~r, t) ,where the q-entropy flux vector ~Sq associated with Hq isdefined by~Sq = −kZ~vf q lnq fd3v , (13)and the source term Gq readsGq = −ks22Z|~V · ~e|(1 + qf q−1 lnq f)Rqdωd3v1d3v .In order to rewrite Gq in a more symmetrical form someelementary operations must be done in the above expres-sion. Following standard lines [16], we first notice thatinterchanging ~v and ~v1 does not affect the value of theintegral. This happens because the magnitude of the rel-ative velocity vector and the scattering cross section areinvariants. Similarly, the value of Gq is not altered if weintegrate with respect to the variables ~v′ and ~v′1 (we recallthat d3v1d3v = d3v′1d3v′). Note that this step requiresthe change of sign of Rq (inverse collision). Implementingthese operations and symmetrizing the resulting expres-sion, one may show that the source term can be writtenasGq(~r, t) = −ks28Z|~V · ~e|(qf q−11 lnq f1 + qfq−1 lnq f−qf ′q−11 lnq f′1 − qf′q−1 lnq f′)Rqdωd3v1d3v . (14)Making now the transformation f q−1 lnq f = lnq∗ f ,where q∗ = 2− q, and rearranging terms we findGq =ks2q8Z|~V · ~e|(lnq∗ f′ + lnq∗ f′1 − lnq∗ f − lnq∗ f1)[eq(lnq∗ f′ + lnq∗ f′1)− eq(lnq∗ f + lnq∗ f1)]dωd3v1d3v .Note that the integrand in the above equation is nevernegative, because (lnq∗ f′+ lnq∗ f′1− lnq∗ f − lnq∗ f1) and[eq(lnq∗ f′ + lnq∗ f′1) − eq(lnq∗ f + lnq∗ f1)] always havethe same signs. Therefore, for positive values of q, weobtain the Hq-theorem∂Hq∂t+∇ · ~Sq = Gq(~r, t) ≥ 0. (15)This inequality states that the q-entropy source must bepositive or zero, thereby furnishing a kinetic argumentfor the second law of thermodynamics in the frameworkof Tsallis’ nonextensive formalism. However, our argu-ment does not constitute a kinetic proof of the secondlaw. As happens with the standard Boltzmann equa-tion, our generalization can not be obtained only fromthe Hamiltonian equations of motion. Specific statisticalassumptions are also needed.When q < 0 the entropy of a given volume elementdecreases with time. Consequently, it seems that withinthe present context, and according to the second law ofthermodynamics, the parameter q should be restricted topositive values [22]. Notice also that the entropy doesnot change with time if q = 0. Similar results werepreviously obtained using the master equation and therelaxation time approximation [23]. Naturally, Tsallis’q-parameter may be further restricted by other physicalrequirement, such as a finite total number of particles. Inpoint of fact, appropriate normalization of Tsallis’ distri-bution requires a q-parameter greater than 1/3 [24].To complete the proof, we now show that Tsallis’ equi-librium q-distribution [9] is a natural consequence of theHq-theorem. As happens in the canonical H-theorem,Gq = 0 must be a necessary and sufficient condition forequilibrium. Since the integrand appearing in the ex-pression of Gq cannot be negative, this occur if and onlyiflnq∗ f′ + lnq∗ f′1 = lnq∗ f + lnq∗ f1 . (16)Therefore, the above sum of q-logarithms remains con-stant during a collision: it is a summational invariant.Only the particles total mass, energy, and momentumbehave like that [16,17]. Consequently, we must havelnq∗ f = ao + ~a1 · ~v + a2~v · ~v , (17)3where ao and a2 are constants and ~a1 is an arbitraryconstant vector. By introducing the barycentric velocity,~u, we may rewrite (17) aslnq∗ f = α− γ∗|~v − ~u|2 , (18)with a different set of constants. Taking Aq∗ = eq∗(α)and defining γ = γ∗(1−q∗)α , we obtain a generalizedMaxwell’s distributionf0(~v) = Aq∗ [1− (1− q∗)γ|~v − ~u|2]1/1−q∗, (19)where Aq∗ , γ and ~u may be functions of the temper-ature. The above expression is the general form of theq-Maxwellian distribution function [9].Summing up, we have discussed a q-generalization ofBoltzmann’s kinetic equation along the lines of Tsallisnonextensive thermostatistics. Our main results followedfrom a slightly modified version of the statistical hypothe-ses underlying Boltzmann’s approach, incorporating (i)the nonextensivity property, explicitly introduced trougha new functional form for the local entropy, and (ii) a non-factorizable expression for the molecular chaos hypothe-sis. Both ingredients were shown to be consistent withthe standard laws of (microscopic) dynamics. They re-duce to the familiar Boltzmann assumptions in the exten-sive limit q → 1. The usual statistical hypothesis of com-pletely uncorrelated colliding molecules seems to be toorestrictive. It is conceivable that correlations may be rele-vant within some scenarios. Here we have provided a sim-ple type of correlations that makes sense within Tsallis’nonextensive thermostatistics. Other possibilities, alsoleading to Tsallis’ distribution (19), are obtained if onereplaces the function eq(x) in (9) by other positive, in-creasing function Fq(x) such that limq→1 Fq(x) = exp(x).Naturally, these q-generalizations of the molecular chaoshypothesis do not settle the profound conceptual issuesraised by Boltzmann’s “Stozssahlansatz”. What we areadvocating is that Boltzmann’s statistical assumptionsdo not encompass all the possibilities allowed by the gen-eral principles of mechanics.The study of chaotic, low dimensional dissipative dy-namical systems has suggested a deep connection be-tween Tsallis formalism and multifractals (see [2] andreferences therein). It would be interesting to explore ifthis relationship also holds for Hamiltonian systems oflarge dimensionality and if it does, whether there is anyconnection with the Hq-theorem.Finally, we stress that the solutions of the generalizedBoltzmann equation (7) verify the Hq-theorem only ifq > 0. In that case Hq is an increasing function oftime and the time dependent solutions of (7) evolve ir-reversibly towards Tsallis’ equilibrium distribution (19).These results can be extended to include nonuniform sys-tems as well as more general interparticle interactions.Acknowledgements: This work was supported by thePronex/FINEP (No. 41.96.0908.00), CNPQ and CAPES(Brazil), and by CONICET (Argentina).[1] C. Tsallis, J. Stat. Phys. 52, 479 (1988).[2] C. Tsallis, Braz. J. Phys. 29 (1999) 1 (available athttp://sbf.if.usp.br/WWW pages/Journals/BJP/Vol29).See also http://tsallis.cat.cbpf.br/biblio.htm.[3] E.P. Borges, J. Phys. A Math. Gen. 31, 5281 (1998).[4] B. M. Boghosian, Phys. Rev. E 53, 4754 (1996).[5] A. Lavagno, G. Kaniadakis, M. Rego-Monteiro, P. Quaratiand C. Tsallis, Astroph. Lett. and Comm., 35, 449 (1998).[6] C. Beck, Physica A 277, 155 (2000).[7] J. A. S. Lima, R. Silva, and Janilo Santos, Phys. Rev. E61, 3260 (2000).[8] M. Buiatti, P. Grigolini, and A. Montagnini, Phys. Rev.Lett. 82, 3383 (1999).[9] R. Silva Jr., A. R. Plastino and J. A. S. Lima, Phys. Lett.A 249, 401 (1998).[10] E.M.F. Curado, Braz. J. Phys. 29, 36 (1999); S. Abe, Phys-ica A 269, 403 (1999).[11] E. T. Jaynes, Phys. Rev. 106, 620 (1957); ibid 108, 171(1957).[12] C. Beck and F. Schlogl, Thermodynamics of Chaotic Sys-tems, (Cambridge University Press, Cambridge, 1993).[13] C. Tsallis, S.V.F. Levy, A.M.C. Souza and R. Maynard,Phys. Rev. Lett. 75, 3589 (1995) ; Erratum: 77, 5442(1996).[14] A.K. Rajagopal, R.S. Mendes, and E.K. Lenzi, Phys. Rev.Lett. 80, 3907 (1998); A.K. Rajagopal and S. Abe, Phys.Rev. Lett. 83, 1711 (1999); V. H. Hamity and D. E. Bar-raco, Phys. Rev. Lett. 76, 4664 (1996); D. F. Torres, H.Vucetich and A. Plastino, Phys. Rev. Lett. 79, 1588 (1997).[15] R. Salazar and R. Toral, Phys. Rev. Lett. 83 (1999) 4233.[16] A. Sommerfeld, Thermodynamics and Statistical Mechan-ics, Lectures on Theorethical Physics, Vol. V (AcademicPress, New York, 1993).[17] C. J. Thompson, Mathematical Statistical Mechanics(Princeton University Press, Princeton, 1979).[18] S. H. Burbury, Nature, 51, 78 (1894).[19] S. G. Brush, Statistical Physics and the Atomic Theory ofMatter, from Boyle and Newton to Landau and Onsager(Princeton University Press, Princeton, 1983).[20] H. D. Zeh, The Physical Basis of The Direction of Time(Springer-Verlag, Berlin-Heidelberg, 1992).[21] Ludwig Boltzmann, Lectures in Gas Theory (University ofCalifornia Press, Berkeley, 1964).[22] S. R. de Groot and P. Mazur, Non-equilibrium Thermody-namics (Dover, New York, 1984).[23] A. M. Mariz, Phys. Lett. A 165, 409 (1992); B.M.Boghosian, Braz. J. Phys. 29, 91 (1999).[24] J. A. S. Lima, R. Silva, and J. Santos, “On Tsallis’ Nonex-tensive Thermostatistics and Jeans Gravitational Instabil-ity” (2000). To be published.4

Nonextensive Thermostatistics and the H Theorem

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Nonextensive Thermostatistics and the H Theorem

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