Brazilian Journal of Physics, vol. 29, no. 1, March, 1999 79 Tsallis Entropy and the Vlasov-Poisson Equations 1;2;3 3;4 A. R. Plastino and A. Plastino 1 Faculty of Astronomy and Geophysics, National University La Plata, C. C. 727, 1900 La Plata, Argentina 2 CentroBrasileiro de Pesquisas F sicas CBPF Rua Xavier Sigaud 150, Rio de Janeiro, Brazil 3 Argentine Research Agency CONICET. 4 Physics Department, National University La Plata, C.C. 727, 1900 La Plata, Argentina Received 07 Decemb er, 1998 We revisit Tsallis Maximum Entropy Solutions to the Vlasov-Poisson Equation describing gravita- tional N -b o dy systems. We review their main characteristics and discuss their relationshi p with other applicatio ns of Tsallis statistics to systems with long range interactions. In the following con- siderations we shall b e dealing with a D -dimensional space so as to b e in a p osition to investigate p ossible dimensional dep endences of Tsallis' parameter q . The particular and imp ortant case of the Schuster solution is studied in detail, and the p ertinent Tsallis parameter q is given as a function of the space dimension. In the sp ecial case of three dimensional space we recover the value q =7=9, that has already app eared in many applications of Tsallis' formalism involving long range forces. I Intro duction R q 1 [f x] dx S = 1 q q 1 where q is a real parameter characterizing the entropy Foravarietyofphysical reasons, muchwork has b een functional S , and f x is a probability distribution de- devoted recently to the exploration of alternativeor q N ned for x 2 R . In the limiting case q ! 1 the stan- generalized information measures, based up on entropy R dard logarithmic entropy, S = f xlnf x dx,is functionals di erent from the standard Boltzmann- 1 recovered. From an Information Theory viewp oint, 1 Gibbs-Shannon-Jaynes entropy[1]. Despite its great constitutes an information measure distinct from that overall success, this ortho dox formalism is unable to of Shannon. deal with a varietyofinteresting physical problems The Tsallis entropy preserves or suitable generalizes such as the thermo dynamics of self-gravitating systems, the relevant features of the Boltzamnn-Gibbs entropy some anomalous di usion phenomena, L evy ights and [5, 6]. It has b een shown to b e compatible with the In- distributions, and turbulence, among others see [1] for formation Theory foundations of Statistical Mechanics a more detailed list. The approachofJaynes to Statis- given byJaynes [4] and with the dynamical thermostat- tical Mechanics [2, 3] is also compatible with the p os- ing approach to statistical ensembles [7]. Increasing at- sibility of building up a thermostatistics starting with tention is b eing devoted to exploring the mathematical a nonlogarithmicentropy, as demonstrated by Plastino prop erties of Tsallis formalism [8-14,30]. and Plastino [4]. Tsallis has intro duced a family of gen- eralized entropies [5], namely, The Tsallis formalism has b een already applied to 80 A. Plastino and A. R. Plastino astrophysical self-gravitating systems [16,17, 18], cos- The rst physical application of Tsallis entropywas mology [19,20], the solar neutrino problem [21], L evy concerned with self-gravitational systems [16]. It is well ights and distributions [22, 23, 24], phonon-electron known that the standard Boltzmann-Gibbs formalism thermalization in solids [25], turbulence phenomena is unable to provide a useful description of this typ e of [26, 27], low-dimensional dissipative systems [28,29], systems [33,34,35, 36]. Moreover, the connection b e- non linear Fokker-Planck equations [30], etc. The inter- tween Tsallis entropy and gravitation provided the rst ested reader is referred to [1] for additional references. hint of just which are the kind of problems where the generalized formalism might b e useful. Tsallis theory has also b een confronted with direct Although the standard Boltzmann-Gibbs formalism exp erimental and observational data. Tsallis MaxEnt characterized by the value q = 1 of Tsallis parameter distributions have b een shown to provide b etter de- has great diculties in dealing with gravitation in three scriptions than the ones given by the standard ther- dimensional space, it shows no problems in the case of mostatistics, for b oth the exp erimentally measured dis- one or two dimensions. On the other hand, it is p ossible tributions of pure electron plasmas [26], and the obser- to obtain physically acceptable answers in D =3,ifwe vational p eculiar velo city distribution of galaxy clusters employ an appropriate value of q<7=9. These facts [31]. Tsallis thermostatistics has also b een advanced as suggest that dimensionality plays a decisive role within a p ossible explanation for the solar neutrino problem this context. The aim of the presentwork is to clarify [21]. It is worth remarking that all these exp erimental some asp ects of the application of Tsallis formalism to evidences involve N -b o dy systems whose constituent 1 the Valsov-Poisson equations, and to generalize some particles interact according to the r gravitational or of the known results connecting Tsallis Entropy with Coulomb p otential. self-gravitating systems, in order to explore the p ossi- Other alternative measures of entropy or informa- ble dep endence of the Tsallis parameter q on the spatial tion have also b een successfully applied in di erent dimensionality D . areas of theoretical physics. For instance, R enyi in- formation is a very useful to ol in the study of chaotic dynamical systems [32]. However, it lacks a de nite II Gravitation and other Long concavity, crucial for the discussion of thermo dynami- Range Interactions. cal stability. Tsallis entropy, on the other hand, do es exhibit such an imp ortant mathematical prop erty, and One of the main features characterizing the thermo dy- was thus historically the rst one b esides the standard namics of self-gravitating systems is that the total en- Boltzmann-Gibbs employed to develop an entirely con- ergy,aswell as other variables usually regarded as addi- sistent Statistical Mechanics formalism [6, 4]. tive quantities, lose their extensivecharacter. This fact suggests that a non-additiveentropic measure, like the Active research conducted during the last few years one prop osed by Tsallis, may b e more useful in dealing is giving raise to a general picture of the physical sce- with gravitation than the standard Boltzamnn-Gibbs narios where Tsallis theory provides a b etter thermo- entropy. Indeed, non-extensivity is one of the main statistical description than the one given by the stan- prop erties of Tsallis entropy. Given two non-correlated dard Boltzmann-Gibbs formalism. These physical sys- systems A and B , with entropies S [A] and S [B ], the tems are characterized by the presence of long range q q Tsallis entropy S [A + B ] corresp onding two b oth sys- interactions, memory e ects, or a fractal phase space. q tems regarded as a single, unique, system is given by Summing up, they show nonextensive e ects. Brazilian Journal of Physics, vol. 29, no. 1, March, 1999 81 1 =D 1 N N = ; 3 S [A + B ]=S [A]+ S [B ]+1 q S [A]S [B ]: 2 q q q q q 1 =D In the limit q ! 1, yielding Boltzmann-Gibbs thermo- D b eing the spatial dimension. But this scaling with statistics, the standard additive b ehaviour of entropyis the numb er of particles is not applicable to the strict recovered. gravitational case, since the canonical ensemble for a In the case of gravitational physics, the non- given nite numb er of particles is meaningless. There extensive b ehaviour is due to the long range charac- is another fundamental feature of the thermo dynam- ter of the gravitational interaction. The early applica- ics of systems endowed with long range interactions tions of Tsallis' Thermostatistics to gravitational sys- that has not yet b een fully addressed by the simpli- tems stimulated the exploration, within Tsallis formal- ed mo dels discussed in [37, 38, 39]. This prop ertyis ism, of other many b o dy problems with long range the breaking of the translational symmetry of space that interaction showing non-extensive e ects [37,38,39]. is implicitly assumed in standard thermo dynamics [40]. These studies considered systems with a Lennard-Jones This prop erty of systems with long range interactions likeinterparticle p otential, characterized by a repulsive is illustrated by the p olytropic mo del discussed in [16] b ehaviour at short distances together with an attrac- and reviewed here. tive long range comp onent falling like r . Itisim- p ortant to realize that this kind of p otentials, while III The Vlasov-Poisson Equa- b eing useful in order to illuminate the thermo dynam- tions. ical consequences of long range forces, do di er in an Vlasov equation constitutes one of the most imp or- essential way from the gravitational interaction. The tant mathematical to ols employed to mo del astrophys- Gibbs canonical ensemble for a system of a N particles ical self-gravitating N -b o dy systems like galaxies and interacting via a Lennard-Jones-like p otential, enclosed galaxy clusters. Let us consider a system of N identi- within a b oxofvolume V ,iswell de ned and has a cal stars of mass m. Vlasov equation reads [33] convergent partition function. On the other hand, and due to the singularity at the origin of the gravitational @f @ @f @f p otential, the Gibbs canonical ensemble for a system of + v =0; 4 @t @ x @ x @ v N gravitationally interacting particles even if they are where f x; v dxdv denotes the numb er of stars in enclosed within a nite b ox has a divergent partition the 2D -dimensional volume element dxdv in p osition- function and thus is not well de ned.