Nonextensive Thermostatistics and the H-Theorem

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On the light of this, it is not unreason- was needed in order to obtain irreversibility. Further, able to expect that a systematic exploration of the kinetic He also admitted that the hypothesis may not always be aspects of Tsallis’ thermostatistics may be crucial to il- valid for real gases, especially at high densities [19,21]. luminate its foundations as well as to achieve a better In what follows we introduce a consistent general- understanding of its physical applications. ization of this hypothesis in accordance with Tsallis’ In this framework, the aim of this letter is to obtain nonextensive formalism. We remark that equation (6) the equilibrium velocity q-distribution from a slight gen- implies that the logarithm of the joint distribution eralization of the kinetic Boltzmann H-theorem. The f(~x1, ~v1, ~x2, ~v2,t) is equal to the sum of two terms, each whole argument follows simply by modifying the molec- one involving only the marginal distribution associated ular chaos hypothesis, as originally advanced by Boltz- with one of the colliding molecules. Our generalized hy- mann, and generalizing the local entropy expression in pothesis is to assume that a power of the joint distribu- according to Tsallis proposal. tion (instead of the logarithm) is equal to the sum of two The statistical content of Boltzmann’s kinetic theory terms, each one depending on just one of the colliding relies on two main ingredients [16,17]. The first one is molecules. By recourse to the q-generalization of the log- a specific functional form for the local entropy, which is arithm function, this condition can be formulated in a expressed by Boltmann’s logarithmic measure way that recovers the standard hypothesis of molecular chaos as a limit case. H[f] = −k f(~x, ~v, t) ln f(~x, ~v, t) d3v . (5) Let us now consider a spatially homogeneous gas of Z N hard-sphere particles of mass m and diameter s, un- The second one is the celebrated hypothesis of molec- der the action of an external force F~ , and enclosed in ular chaos (“Stosszahlansatz”), which is tantamount to a volume V . The state of a non-relativistic gas is ki- assume the factorizability of the joint distribution asso- netically characterized by the one-particle distribution ciated with two colliding molecules function f(~x, ~v, t). The quantity f(~x, ~v, t)d3xd3v gives, at each time t, the number of particles in the volume f(~x , ~v , ~x , ~v ,t) = f(~x , ~v ,t) f(~x , ~v ,t) . (6) 1 1 2 2 1 1 2 2 element d3xd3v around the particle position ~x and veloc- These two statistical assumptions are inextricably inter- ity ~v. In principle, this distribution function verifies the twined. Therefore, if one adopts a generalized nonex- q-nonextensive Boltzmann equation tensive entropic measure, a consistent generalization of ∂f ∂f 1 ∂f the “Stosszahlansatz” hypothesis should also be imple- + ~v · + F~ · = Cq(f) , (7) mented. Different choices for the collision term in the ∂t ∂~x m ∂~v kinetic equation (which, in turn, is determined by the where Cq denotes the q-collisional term. The left-hand- “Stosszahlansatz”) lead to different forms for the entropic side of (7) is just the total time derivative of the distri- functional exhibiting a time derivative with definite sign. bution function. Hence, nonextensivity effects can be in- As a consequence, the form of the entropic functional corporated only through the collisional term. Naturally, behaving monotonically with time (and consequently ad- Cq(f) may be calculated in accordance with the laws of mitting an H-theorem) depends upon the form of the elastic collisions. Its specific structure must lead to the collisional term appearing in the kinetic equation. standard result in the limit q → 1. We also make the Historically, the basic assumption (6) (sometimes re- following assumptions: (i) Only binary collisions occur ferred to as “Maxwell’s ansatz”) has generated a lot of in the gas; (ii) Cq(f) is a local function of the slow vary- controversy. The fundamental role played by this hy- ing distribution function; (iii) Cq(f) is consistent with pothesis was first realized by S.H. Burbury [18] in 1894. the energy, momentum and particle number conservation The precise characterization of the conditions of its ap- laws. plicability has been an important conceptual problem of Our main goal is now to show that the generalized theoretical physics ever since [19]. The physical meaning collisional term Cq(f) leads to a nonnegative expression of equation (6) is that colliding molecules are uncorre- for the time derivative of the q-entropy, and that it does lated. The irreversible behaviour of Boltzmann’s equa- not vanish unless the distribution function assumes the tion can be traced back to this assumption. It is clearly a equilibrium form associated with q-Maxwellian gas [9]. time asymmetric hypothesis, since molecules assumed to Now, following standard lines we define be uncorrelated before a collision certainly become corre- lated after the collision has taken place [20]. s2 C (f)= |V~ · ~e|R dωd3v , (8) Although very plausible, and endowed with an intu- q 2 Z q 1 itively clear statistical meaning, Boltzmann’s particular 3 expression (6) for the molecular chaos hypothesis can not where d v1 stands for the volume element in velocity be deduced from first principles. By no means it is an in- space, V~ = ~v1 − ~v is the relative velocity before colli- escapable consequence of classical mechanics. Boltzmann sion, ~e denotes an arbitrary unit vector, and dω is an

2 2 2 ks − − elementary solid angle such that s dω is the area of the G (~r, t)= − |V~ · ~e|(qf q 1 ln f + qf q 1 ln f “collision cylinder” (for details on the collision’s geome- q 8 Z 1 q 1 q ′ ′ − ′ ′ − ′ try see Refs. [16,17]). The quantity Rq(f,f ) is a diﬀer- q 1 q 1 3 3 −qf 1 lnq f 1 − qf lnq f )Rqdωd v1d v . (14) ence of two correlation functions (just before and after q−1 collision), which are assumed to satisfy a q-generalized Making now the transformation f lnq f = lnq∗ f, form of the molecular chaos hypothesis. In the present where q∗ =2 − q, and rearranging terms we ﬁnd q-nonextensive scenario we will assume that 2 ks q ′ ′ ′ ′q−1 ′ ′q−1 ′ ~ ∗ ∗ ∗ ∗ Gq = |V · ~e|(lnq f + lnq f 1 − lnq f − lnq f1) Rq(f,f )= eq(f lnq f + f 1 lnq f1) 8 Z q−1 q−1 ′ ′ ∗ ∗ ∗ ∗ 3 3 −eq(f lnq f + f1 lnq f1) , (9) [eq(lnq f + lnq f 1) − eq(lnq f + lnq f1)]dωd v1d v . where primes refer to the distribution function after col- Note that the integrand in the above equation is never ′ ′ lision. When q → 1 equation (9) reduces to Boltzmann’s ∗ ∗ ∗ ∗ negative, because (lnq f + lnq f1 − lnq f − lnq f1) and ′ ′ molecular chaos hypothesis ∗ ∗ ∗ ∗ [eq(lnq f + lnq f 1) − eq(lnq f + lnq f1)] always have ′ ′ the same signs. Therefore, for positive values of q, we lim Rq = R = f f 1 − ff1 . (10) q→1 obtain the Hq-theorem

For the local entropy we adopt Tsallis expression, ∂H q + ∇ · S~ = G (~r, t) ≥ 0. (15) ∂t q q q 3 Hq = −k f lnq fd v , (11) Z This inequality states that the q-entropy source must be positive or zero, thereby furnishing a kinetic argument which reduces to the standard Boltzmann measure (5) for q = 1. Now, we first take the partial time derivative for the second law of thermodynamics in the framework of the above expression of Tsallis’ nonextensive formalism. However, our argument does not constitute a kinetic proof of the second ∂Hq − ∂f law. As happens with the standard Boltzmann equa- = −k [qf q 1 ln f + 1] d3v . (12) ∂t Z q ∂t tion, our generalization can not be obtained only from the Hamiltonian equations of motion. Specific statistical As one may check, by inserting the generalized Boltz- assumptions are also needed. mann equation (7) into (12), and using (8), expression When q < 0 the entropy of a given volume element (12) can be rewritten as a balance equation decreases with time. Consequently, it seems that within ∂H the present context, and according to the second law of q + ∇ · S~ = G (~r, t) , ∂t q q thermodynamics, the parameter q should be restricted to positive values [22]. Notice also that the entropy does ~ where the q-entropy flux vector Sq associated with Hq is not change with time if q = 0. Similar results were defined by previously obtained using the master equation and the relaxation time approximation [23]. Naturally, Tsallis’ S~ = −k ~vf q ln fd3v , (13) q-parameter may be further restricted by other physical q Z q requirement, such as a finite total number of particles. In and the source term Gq reads point of fact, appropriate normalization of Tsallis’ distribution requires a q-parameter greater than 1/3 [24]. 2 ks q−1 3 3 To complete the proof, we now show that Tsallis’ equi- Gq = − |V~ · ~e|(1 + qf lnq f)Rqdωd v1d v . 2 Z librium q-distribution [9] is a natural consequence of the Hq-theorem. As happens in the canonical H-theorem, In order to rewrite Gq in a more symmetrical form some elementary operations must be done in the above expres- Gq = 0 must be a necessary and sufficient condition for sion. Following standard lines [16], we first notice that equilibrium. Since the integrand appearing in the expression of Gq cannot be negative, this occur if and only interchanging ~v and ~v1 does not affect the value of the integral. This happens because the magnitude of the rel- if ′ ′ ative velocity vector and the scattering cross section are ∗ ∗ ∗ ∗ lnq f + lnq f 1 = lnq f + lnq f1 . (16) invariants. Similarly, the value of Gq is not altered if we ~′ ~′ integrate with respect to the variables v and v1 (we recall Therefore, the above sum of q-logarithms remains con- 3 3 3 ′ 3 ′ that d v1d v = d v1d v ). Note that this step requires stant during a collision: it is a summational invariant. the change of sign of Rq (inverse collision). Implementing Only the particles total mass, energy, and momentum these operations and symmetrizing the resulting expres- behave like that [16,17]. Consequently, we must have sion, one may show that the source term can be written as lnq∗ f = ao + ~a1 · ~v + a2~v · ~v , (17)

3 where ao and a2 are constants and ~a1 is an arbitrary constant vector. By introducing the barycentric velocity, ~u, we may rewrite (17) as ∗ 2 lnq∗ f = α − γ |~v − ~u| , (18) [1] C. Tsallis, J. Stat. Phys. 52, 479 (1988). with a different set of constants. Taking Aq∗ = eq∗ (α) ∗ [2] C. Tsallis, Braz. J. Phys. 29 (1999) 1 (available at γ and defining γ = (1−q∗)α , we obtain a generalized http://sbf.if.usp.br/WWW pages/Journals/BJP/Vol29). Maxwell’s distribution See also http://tsallis.cat.cbpf.br/biblio.htm. ∗ − ∗ [3] E.P. Borges, J. Phys. A Math. Gen. 31, 5281 (1998). ∗ 2 1/1 q f0(~v)= Aq [1 − (1 − q )γ|~v − ~u| ] , (19) [4] B. M. Boghosian, Phys. Rev. E 53, 4754 (1996). where Aq∗ , γ and ~u may be functions of the temper- [5] A. Lavagno, G. Kaniadakis, M. Rego-Monteiro, P. Quarati ature. The above expression is the general form of the and C. Tsallis, Astroph. Lett. and Comm., 35, 449 (1998). 277 q-Maxwellian distribution function [9]. [6] C. Beck, Physica A , 155 (2000). Summing up, we have discussed a q-generalization of [7] J. A. S. Lima, R. Silva, and Janilo Santos, Phys. Rev. E 61, 3260 (2000). Boltzmann’s kinetic equation along the lines of Tsallis [8] M. Buiatti, P. Grigolini, and A. Montagnini, Phys. Rev. nonextensive thermostatistics. Our main results followed Lett. 82, 3383 (1999). from a slightly modified version of the statistical hypothe- [9] R. Silva Jr., A. R. Plastino and J. A. S. Lima, Phys. Lett. ses underlying Boltzmann’s approach, incorporating (i) A 249, 401 (1998). the nonextensivity property, explicitly introduced trough [10] E.M.F. Curado, Braz. J. Phys. 29, 36 (1999); S. Abe, Phys- a new functional form for the local entropy, and (ii) a non- ica A 269, 403 (1999). factorizable expression for the molecular chaos hypothe- [11] E. T. Jaynes, Phys. Rev. 106, 620 (1957); ibid 108, 171 sis. Both ingredients were shown to be consistent with (1957). the standard laws of (microscopic) dynamics. They re- [12] C. Beck and F. Schlogl, Thermodynamics of Chaotic Sys- duce to the familiar Boltzmann assumptions in the exten- tems, (Cambridge University Press, Cambridge, 1993). sive limit q → 1. The usual statistical hypothesis of com- [13] C. Tsallis, S.V.F. Levy, A.M.C. Souza and R. Maynard, Phys. Rev. Lett. 75, 3589 (1995) ; Erratum: 77, 5442 pletely uncorrelated colliding molecules seems to be too (1996). restrictive. It is conceivable that correlations may be rele- [14] A.K. Rajagopal, R.S. Mendes, and E.K. Lenzi, Phys. Rev. vant within some scenarios. Here we have provided a sim- Lett. 80, 3907 (1998); A.K. Rajagopal and S. Abe, Phys. ple type of correlations that makes sense within Tsallis’ Rev. Lett. 83, 1711 (1999); V. H. Hamity and D. E. Bar- nonextensive thermostatistics. Other possibilities, also raco, Phys. Rev. Lett. 76, 4664 (1996); D. F. Torres, H. leading to Tsallis’ distribution (19), are obtained if one Vucetich and A. Plastino, Phys. Rev. Lett. 79, 1588 (1997). replaces the function eq(x) in (9) by other positive, in- [15] R. Salazar and R. Toral, Phys. Rev. Lett. 83 (1999) 4233. creasing function Fq(x) such that limq→1 Fq(x) = exp(x). [16] A. Sommerfeld, Thermodynamics and Statistical Mechan- Naturally, these q-generalizations of the molecular chaos ics, Lectures on Theorethical Physics, Vol. V (Academic hypothesis do not settle the profound conceptual issues Press, New York, 1993). raised by Boltzmann’s “Stozssahlansatz”. What we are [17] C. J. Thompson, Mathematical Statistical Mechanics (Princeton University Press, Princeton, 1979). advocating is that Boltzmann’s statistical assumptions [18] S. H. Burbury, Nature, 51, 78 (1894). do not encompass all the possibilities allowed by the gen- [19] S. G. Brush, Statistical Physics and the Atomic Theory of eral principles of mechanics. Matter, from Boyle and Newton to Landau and Onsager The study of chaotic, low dimensional dissipative dy- (Princeton University Press, Princeton, 1983). namical systems has suggested a deep connection be- [20] H. D. Zeh, The Physical Basis of The Direction of Time tween Tsallis formalism and multifractals (see [2] and (Springer-Verlag, Berlin-Heidelberg, 1992). references therein). It would be interesting to explore if [21] Ludwig Boltzmann, Lectures in Gas Theory (University of this relationship also holds for Hamiltonian systems of California Press, Berkeley, 1964). large dimensionality and if it does, whether there is any [22] S. R. de Groot and P. Mazur, Non-equilibrium Thermody- connection with the H -theorem. namics (Dover, New York, 1984). q 165 Finally, we stress that the solutions of the generalized [23] A. M. Mariz, Phys. Lett. A , 409 (1992); B.M. Boghosian, Braz. J. Phys. 29, 91 (1999). Boltzmann equation (7) verify the H -theorem only if q [24] J. A. S. Lima, R. Silva, and J. Santos, “On Tsallis’ Nonex- q > 0. In that case Hq is an increasing function of tensive Thermostatistics and Jeans Gravitational Instabil- time and the time dependent solutions of (7) evolve ir- ity” (2000). To be published. reversibly towards Tsallis’ equilibrium distribution (19). These results can be extended to include nonuniform systems as well as more general interparticle interactions. Acknowledgements: This work was supported by the Pronex/FINEP (No. 41.96.0908.00), CNPQ and CAPES (Brazil), and by CONICET (Argentina).