4 Generalized thermodynamics and Tsallis entropy

Recently, Taruya & Sakagami (2002) have investigated the stability of polytropic spheres within the framework of generalized thermodynamics. It has been argued by Tsallis and co-workers that ordinary statistical mechanics and thermodynamics does not describe correctly systems with long-range interactions, which are in essence non extensive. A family of functionals of the form

$$S_{q}=-{1\over q-1}\int (f^q-f){\rm d}^{3}{\vec r}{\rm d}^{3}{\vec v},$$ (67)

known as Tsallis entropies, has been introduced to extend the classical Boltzmann-Gibbs statistical mechanics to these systems. These functionals are labeled by a parameter q. The Boltzmann entropy is recovered in the limit $q\rightarrow 1$. This new formalism has been applied in various domains of physics, astrophysics, fluid mechanics, biology, economy etc. and it extends the results obtained with ordinary statistical mechanics. In the case of self-gravitating systems, it was shown by Plastino & Plastino (1993) that the extremalization of the Tsallis entropy at fixed mass and energy yields a polytropic equation of state of the form (1). The parameter q is related to the index n of the polytrope by the relation

$$n={1\over q-1}+{3\over 2}, \qquad (q\ge 1).$$ (68)

Then, Taruya & Sakagami (2002) investigated the stability problem in the microcanonical ensemble by extending the analysis of Padmanabhan (1989) for the classical Antonov instability. Polytropic configurations are said to be stable if they correspond to (generalized) entropy maxima at fixed mass M and energy E. Taruya & Sakagami showed that, for n>5, an equilibrium exists only above a critical energy $E_{{\rm c}}$ depending on the index of the configuration. In the limit $n\rightarrow +\infty$, the Antonov result $E_{{\rm c}}=-0.335GM^{2}/R$ obtained with the Boltzmann entropy is recovered. Furthermore, they showed that the onset of instability coincides with the point of minimum energy in the series of equilibrium in agreement with standard turning point analysis (e.g., Katz 1978). For n<5, there is no critical value of energy and the polytropic configurations are stable. These results differ from those obtained in the present paper where it is found that the index marking the transition from stability to instability is n=3 (in agreement with standard theorems of stellar pulsations). The origin of this discrepency is certainly related to the inequivalence of statistical ensembles in extended thermodynamics, like in ordinary thermodynamics, for self-gravitating systems (see, e.g., Padmanabhan 1990; Chavanis 2002c; Chavanis et al. 2002). Taruya & Sakagami work in the microcanonical ensemble and maximize Tsallis entropy at fixed mass and energy. On the contrary, in our study, we keep the polytropic temperature $\Theta_{\gamma}=Km/k$ constant, which corresponds to the canonical description. It would be of interest to study the maximization of Tsallis free energy at fixed mass and temperature to see if it provides the same conditions of stability as our dynamical approach based on the Navier-Stokes equations. For the Boltzmann entropy, a clear connexion was found between dynamical stability and thermodynamical stability in the canonical ensemble (Semelin et al. 2001; Chavanis 2002a) and it is desirable to check whether this connexion is preserved by Tsallis generalized thermodynamics. This requires in particular to define properly the notion of temperature and free energy in the generalized sense.

In Figs. 8-11, we have represented the equilibrium phase diagram (for different values of n) obtained in the framework of extended thermodynamics, interpreting the parameter $\eta$ as a generalized inverse temperature. The temperature-energy curve is defined in a parametric form by the equations

$$\eta=-\alpha^{n+1\over n-1}\theta'(\alpha),$$ (69)

$$\Lambda\equiv -\!{ER\over GM^{2}}=-{1\over n-5}\biggl\lbrack ... ...)+\!{n-2\over n+1}{u(\alpha)\over v(\alpha)}\biggr\rbrack\cdot$$ (70)

The expression (70) for the energy has been derived by Taruya & Sakagami (2002). For n<3, there is no turning point in the diagram, so the polytropes are always stable. For 3<n<5, the inverse temperature $\eta$ presents a maximum but not the energy. Therefore, the polytropes are always stable in the microcanonical ensemble but they are unstable in the canonical ensemble after the turning point. It can be noted that the unstable region has a negative specific heat $C\sim {{\rm d}\Lambda\over {\rm d}\eta}<0$ (in the generalized sense). For n>5, the temperature and the energy both present an infinite number of extrema. For $n\sim 6$, there are crossing points at which two solutions have the same value of temperature and energy but a different density contrast (to our knowledge, this is the first time such a situation is reported). The polytropes are unstable in the canonical ensemble after the first turning point of temperature and they are unstable in the microcanonical ensemble after the first turning point of energy. These results are strikingly similar to those obtained with the classical Boltzmann entropy $(n=\infty)$.

Figure 8: Phase diagram for n=2.

Figure 9: Phase diagram for n=4.

Figure 10: Phase diagram for n=6.

Figure 11: Phase diagram for n=10.

It is interesting that the results obtained for isothermal gas spheres described by ordinary thermodynamics (Boltzmann entropy) can be extended to polytropic configurations by introducing a generalized functional (Tsallis entropy). This makes this new formalism of interest. However, we would like to make two comments. (i) First of all, Tsallis entropy does not describe polytropic gaseous stars although it yields an equation of state of the polytropic form. Indeed, Tsallis entropy predicts in addition a distribution function of the form

$$f({\vec r},{\vec v})=A\biggl \lbrack \Phi_{0}-\Phi({\vec r})-{1\over 2}v^{2}\biggr\rbrack^{1/(q-1)},$$ (71)

whereas in polytropic stars, the distribution function corresponds to a local thermodynamical equilibrium (L.T.E) characterized by a Maxwellian distribution of velocities

$$f({\vec r},{\vec v})=\biggl ({m\over 2\pi kT({\vec r})}\biggr )^{3/2}\rho({\vec r}){\rm e}^{-{mv^{2}\over 2kT({\bf r})}},$$ (72)

with a local temperature $T({\vec r})$ (recall that, for convective equilibrium, the entropy is uniform not the temperature). In view of these remarks, the connexion between generalized thermodynamical stability (in the canonical ensemble) and dynamical stability with respect to Navier-Stokes equations is not obvious. It is however expected to be true since the condition of dynamical stability is consistent with the turning point argument of thermodynamical stability. (ii) Tsallis entropy could describe a particular class of stellar systems, apparently not observed in nature, known as stellar polytropes (see Binney & Tremaine 1987). These systems have the same structure as polytropic gas spheres in physical space but not in phase space (the equivalence only occurs for a polytropic index $n= \infty$, i.e. for an isothermal configuration). This distinction is important because isolated stellar polytropes are always stable with respect to the Vlasov (or collisionless Boltzmann) equation (see Binney & Tremaine 1987) while isolated polytropic stars are unstable with respect to the Navier-Stokes equations for $n\ge 3$. Generalized thermodynamics could be developed (at least formally) for this particular class of stellar systems.

However, generalized thermodynamics is often presented as a way of solving the "problems'' associated with the use of the Boltzmann entropy in systems with long-range interactions and we would like to criticize this interpretation. For self-gravitating systems, the "problems'' usually reported are the absence of global entropy maximum (resulting in the "infinite mass problem'' and the "gravothermal catastrophe''), the negative specific heats, the inequivalence of statistical ensembles and the absence of a thermodynamical limit. It should be noted that the Tsallis entropy displays the same phenomena, at least for values of q<9/7(i.e. n>5) considered by Taruya & Sakagami (2002). On the other hand, these relatively unusual phenomena should not cause surprise if one recognizes that self-gravitating systems are relatively exceptional among N-body systems. For example the "infinite mass problem'' (see, e.g., Binney & Tremaine 1987) is just a mathematical curiosity. There is no justification in maximizing entropy in an infinite domain even if an entropy maximum exists. Due to kinetic effects, the relaxation is incomplete and the ergodic hypothesis which sustains the statistical analysis is necessarily restricted to a finite region of space. This incomplete relaxation is qualitatively discussed by Lynden-Bell (1967) in his statistical description of the "violent relaxation'' of collisionless stellar systems (e.g., elliptical galaxies). A similar limitation occurs in the context of two-dimensional turbulence to understand the confinement of vortices that form after a rapid merging (see, e.g., Chavanis & Sommeria 1998 and in particular Brands et al. 1999 for a discussion of Tsallis entropy in fluid dynamics). This incomplete relaxation can be described by kinetic equations of the form proposed by Chavanis et al. (1996) with a space dependant diffusion coefficient. Convincing numerical evidence of this kinetic confinement has been given by Robert & Rosier (1997) in two-dimensional turbulence. In addition, in astrophysics, specific processes (e.g., the evaporation of stars, tidal effects, etc.) must be taken into account and can restrict the domain of applicability of statistical mechanics. These limitations are usually accounted for by introducing truncated models like the Michie-King model for globular clusters or the models proposed by Stiavelli & Bertin (1987) or Hjorth & Madsen (1993) for elliptical galaxies. These models lead to composite configurations with an isothermal core and a polytropic envelope. With these modifications (justified by precise physical arguments), ordinary statistical mechanics provides in general a good explanation of astrophysical phenomena and is consistent with observations. Note that the above-mentioned truncated models differ from pure polytropes, so that they cannot be justified by generalized thermodynamics. On the other hand, the "gravothermal catastrophe'' (Lynden-Bell & Wood 1968) should not throw doubt on the validity of statistical mechanical arguments applied to self-gravitating systems. On the contrary, it is gratifying that standard thermodynamics is consistent with the natural tendency of self-gravitating systems to undergo a gravitational collapse (Jeans instability). The gravothermal catastrophe has been confirmed by sophisticated numerical simulations (see, e.g., Larson 1970; Cohn 1980) based on standard kinetic theory (Landau-Fokker-Planck equations) and it is expected to be at work in globular clusters. The existence of negative specific heats in the microcanonical ensemble (and the resulting inequivalence of statistical ensembles) is a direct consequence of the Virial theorem for self-gravitating systems and is not in contradiction with ordinary thermodynamics (Lynden-Bell & Lynden-Bell 1977; Padmanabhan 1990). Finally, self-gravitating systems have a well-defined (albeit unusual) thermodynamical limit in which the number of particles Nand the volume R³ go to infinity keeping N/R fixed (see, e.g., de Vega & Sanchez 2001).

Therefore, we are tempted to believe that ordinary statistical mechanics is still relevant to describe self-gravitating systems (and two-dimensional vortices). An isothermal distribution corresponds to the most probable distribution reached by a system after a complex evolution during which microscopic information is lost. Accordingly, it maximizes the Boltzmann entropy, which is a measure of the number of microstates associated with a given macrostate (see, e.g., Ogorodnikov 1965; Lynden-Bell 1967). This definition of the entropy does not depend whether the particles are in interaction or not. The effects of non-locality and non-extensivity intervene only when the entropy is maximized at fixed energy. It is implicitly assumed, however, that all accessible microstates are equiprobable, which is the fundamental postulate of statistical mechanics, and that the system converges at equilibrium towards the most probable (macroscopic) distribution. Generalized thermodynamics is interesting to extend standard results of thermodynamics to a larger class of functionals which possess nice mathematical properties, but it should not be presented (in our point of view) as an improvement or a replacement of classical thermodynamics.

We agree with Tsallis and co-workers that the kinetic theory of self-gravitating systems (see, e.g., Kandrup 1981) and two-dimensional vortices (see, e.g., Chavanis 2001) encounters some problems due to the occurence of memory effects, spatial delocalization and logarithmic divergences in the diffusion coefficient. For these reasons, and also for the problems of incomplete relaxation (lack of ergodicity) mentioned previously, the system may not necessarily reach the maximum entropy state described by the Boltzmann distribution, even in the mixing region. The deviation from the Boltzmann distribution is never too severe (see, e.g., the isothermal core of globular clusters and of elliptical galaxies) so that ordinary statistical mechanics provides a fairly good prediction from any initial condition. It may happen that the effective equilibrium distribution can be fitted by the q-distribution proposed by Tsallis and co-workers better than by the Boltzmann distribution. This is the case for some examples of statistical equilibrium in two-dimensional turbulence (see the discussion of Brands et al. 1999). However, this agreement may be fortuitous rather than dictated by a general physical principle since there is a parameter q in the theory which, in practice, must be adjusted in each case. The fit can only improve since this family of distribution includes the Boltzmann distribution as a particular limit (q=1). In fact, it is advocated by Tsallis (private communication) that the q-parameter is not free but uniquely determined by the microscopic dynamics of the system. It can be relatively easy to determine if the system is simple enough but it can also be very hard to determine in the case of complex systems like those involved in fluid mechanics and stellar dynamics. In that case, it must be regarded as a fitting parameter. Unfortunately, the power of prediction of the Tsallis theory is limited as long as a precise prescription for determining q is not given. Tsallis entropy could be considered, however, as a heuristic attempt to take into account non-ergodic effects in complex systems. In view of these remarks, it would be of interest to see if the non Markovian and non-local generalized kinetic equations proposed by Kandrup (1981) in stellar dynamics and by Chavanis (2001) in vortex dynamics allow for a stationary distribution of the form conjectured by Tsallis, instead of the Boltzmann distribution. From our point of view, we believe that they do not select any "universal'' distribution except in the approximation where (i) memory effects are ignored (ii) a local approximation is made (in the stellar context) (iii) ergodicity is assumed, in which case the classical Boltzmann distribution is obtained. It is hard to believe that extremely complicated effects of non ergodicity, spatial delocalization and memory can be encapsulated in a simple functional with a prescribed value of q. However, if it can be proved that these generalized kinetic equations rigorously converge towards a Tsallis distribution with $q\neq 1$, this would of course be a strong argument in favour of generalized thermodynamics. Unfortunately, the study of these generalized kinetic equations seems of considerable difficulty and will demand an extended effort.