Nonextensive Thermostatistics and the H-Theorem J. A. S. Lima1, R. Silva1, and A. R. Plastino2,3 1Departamento de F´ısica Te´orica e Experimental, Universidade Federal do Rio Grande do Norte, 59072-970, Natal-RN, Brazil 2Facultad de Ciencias Astronomicas y Geofisicas, Universidad Nacional de La Plata, C.C. 727, 1900 La Plata, Argentina 3Departament 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 on Tsallis entropy. It is also proved that the collisional equilibrium is given by Tsallis’ q-nonextensive velocity distribution. PACS numbers: 05.45.+b; 05.20.-y; 05.90.+m In 1988 Tsallis proposed a striking generalization of [4,5,6,7,8], such as the velocity distribution of galaxy the Boltzmann-Gibbs entropy functional given by [1], clusters [5], Landau damping in plasmas [7], superdif- fusion phenomena [8] and, more generally, systems ex- q Sq = −k pi lnq pi , (1) hibiting a nonextensive thermodynamic behaviour due, Xi for instance, to long range interactions [1,2]. It is worth to stress that some of the aforementioned developments where k is Boltzmann’s constant, p is the probability of i involve a quantitative agreement between experimental the i-th microstate, the parameter q is any real number, data and theoretical models based on Tsallis’ thermo- and the q-logarithmic function is defined as [2,3] statistics [4,5,6,7]. For instance, it was experimentally −1 1−q found that pure electron plasmas in Penning traps re- lnq f = (1 − q) (f − 1), (f > 0). (2) lax to metaestable states whose radial density profiles For future reference it is convenient to introduce the q- do not maximize Boltzmann-Gibbs entropy. However, exponential function eq(f), which is defined by Boghosian showed that the observed profiles are well de- scribed by Tsallis’ thermostatistics with q close to 1/2 1/1−q eq(f) = [1 + (1 − q)f] , (3) [4]. Beck’s recent treatment of fully developed turbulent flows constitutes another interesting example [6]. Based if 1 + (1 − q)f > 0 and by eq(f) = 0 otherwise. These on Tsallis formalism with q 6= 1, Beck calculated the q-functions satisfy, for f,g > 0, the identities (see [3] for probability density functions of velocity differences de- a thorough discussion) pending on distance and Reynolds number, as well as the concomitant scalling exponents, finding good agreement e (ln f)= f , q q with turbulence experiments. lnq f + lnq g = lnq fg + (q − 1)(lnq f)(lnq g) . (4) arXiv:cond-mat/0101030v2 3 Oct 2002 A large portion of the experimental evidence support- When q → 1 all the above expressions reproduce those ing Tsallis proposal involves the non-Maxwellian (power- verified by the usual elementary functions and Tsallis’ en- law) velocity q-distribution associated with Tsallis’ gen- tropy reduces to the standard logarithmic one, namely: eralized canonical ensemble approach to the classical N- body problem. This equilibrium q-distribution may be S1 = −k i pi ln pi. The most distinctive trait of Sq is its pseudoadditivity.P Given two systems A and B in- derived through a simple nonextensive generalization of dependent in the sense of factorizability of the (joint) the Maxwell ansatz [9]. Alternatively, it can be obtained microstate probabilities, the Tsallis entropy of the com- maximizing Tsallis entropy under the constraints im- posed by normalization and the energy mean value [10], posite system A⊕B verifies Sq(A⊕B)= Sq(A)+Sq(B)+ a procedure closely related to Jaynes information theory (1 − q)Sq(A)Sq(B). Hence, |1 − q| quantifies the lack of formulation of statistical mechanics [11,12]. So far, most extensivity of Sq. theoretical studies on Tsallis thermostatistics have been The q-thermostatistics associated with Sq [1,2] is nowa- days being hailed as the possible basis of a theoreti- developed on the basis of the maximum entropy principle cal framework appropriate to dealwith nonextensive set- [1,13,14,15]. As widely known, this approach to statis- tings. There is a growing body of evidence suggesting tical ensembles was, in the case of standard statistical mechanics, historically the latest one to appear. Even that Sq provides a convenient frame for the thermosta- tistical analysis of many physical systems and processes when Gibbs’ introduced his ensemble approach, the ki- 1 netic foundations of statistical mechanics were already himself accepted that the hypothesis of molecular chaos well developed. 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].