A&A 401, 1542 (2003)
DOI: 10.1051/00046361:20021779
Gravitational instability of isothermal and polytropic
spheres
P. H. Chavanis
Laboratoire de Physique Quantique, Université Paul Sabatier, 118
route de Narbonne, 31062 Toulouse, France
Received 3 July 2002 / Accepted 19 November 2002
Abstract
We complete previous investigations on the thermodynamics
of selfgravitating systems by studying the grand canonical, grand
microcanonical and isobaric ensembles. We also discuss the
stability of polytropic spheres in connexion with a generalized
thermodynamical approach proposed by Tsallis. We determine in each
case the onset of gravitational instability by analytical methods
and graphical constructions in the Milne plane. We also discuss the
relation between dynamical and thermodynamical stability of stellar
systems and gaseous spheres. Our study provides an aesthetic and
simple approach to this otherwise complicated subject.
Key words: stellar dynamics  hydrodynamics  instabilities  gravitation
1 Introduction
The statistical mechanics of systems interacting via longrange forces
exhibit peculiar features such as negative specific heats,
inequivalence of statistical ensembles and phase transitions. These
curious behaviours have been first discussed in the astrophysical
literature during the elaboration of a thermodynamics for stars
(Eddington 1926), globular clusters (LyndenBell & Wood 1968), black
holes (Hawking 1974) and galaxies (Padmanabhan 1990). They have been
recently rediscovered in different fields of physics such as nuclear
physics, plasma physics, BoseEinstein condensates, atomic clusters,
twodimensional turbulence... The main challenge is represented by the
construction of a thermodynamic treatment of systems with longrange
forces and by the analogies and differences among the numerous domains
of application (see Chavanis 2002e and other contributions in that book).
Gravity provides a fundamental example of unshielded longrange
interaction for which ideas of statistical mechanics and
thermodynamics can be developed and tested. For systems with
longrange interactions, the meanfield approximation is known to be
exact in a suitable thermodynamic limit. Therefore, the
structure and stability of selfgravitating systems at statistical
equilibrium can be analyzed in terms of the maximization of a
thermodynamical potential. This thermodynamical approach leads to
isothermal configurations which have been studied for a long time in
the context of stellar structure (Chandrasekhar 1942) and galactic
structure (Binney
& Tremaine 1987). As is wellknown, isothermal spheres have infinite
mass so that the system must be confined within a box (Antonov
problem) in order to prevent evaporation and make the thermodynamical
approach rigorous. It is also wellknown that isothermal
configurations only correspond to metastable equilibrium states
(i.e., local maxima of the thermodynamical potential), not true
equilibrium states. These metastable equilibrium states are expected
to be relevant, however, for the timescales contemplated in
astrophysics. In particular, globular clusters described by
MichieKing models are probably in such metastable states.
The series of equilibria of finite isothermal spheres can be
parametrized by the density contrast between the center and the
boundary of the system. For sufficiently low density contrasts, the
system is thermodynamically stable. However, instability
occurs at sufficiently large concentrations: at some point in
the series of equilibria, the solutions cease to be local maxima of
the thermodynamical potential and become unstable saddle points. The
crucial point to realize is that the onset of instability depends on
the statistical ensemble considered: microcanonical (MCE), canonical
(CE) or grand canonical (GCE). This contrasts with ordinary systems,
with short range interactions, for which the statistical ensembles are
equivalent at the thermodynamic limit, except near a phase
transition. This suggests that gravitating systems in virial
equilibrium are similar to normal (extensive) systems at the verge of
a phase transition (Padmanabhan 1990).
The thermodynamical stability of selfgravitating systems can be
studied by different technics. Horwitz &
Katz (1978) use a field theory and write the density of states,
the partition function and the grand partition function in MCE, CE and
GCE respectively as a path integral for a formal field .
In the
meanfield approximation, the integral is dominated by the distribution
which maximizes a specific action
.
In the grand canonical ensemble,
is the Liouville action. Horwitz & Katz (1978) solve the problem
numerically and find that the series of equilibria become unstable for
a density contrast 1.58, 32.1 and 709 in GCE, CE and MCE
respectively. It is normal that the critical density contrast (hence
the stability of the system) increases when more and more constraints
are added on the system (conservation of mass in CE, conservation of
mass and energy in MCE). This field theory has been rediscussed
recently by de Vega & Sanchez (2002) who confirmed previous results
and proposed interesting developements.
The thermodynamical stability of selfgravitating systems can also be
settled by studying whether an isothermal sphere is a maximum or a
saddle point of an appropriate thermodynamical potential: the entropy
in MCE, the free energy in CE and the grand potential in GCE. The
change of stability can be determined very easily from the topology of
the equilibrium phase diagram by using the turning point criterion of
Katz (1978) who has extended Poincaré's theory on linear series of
equilibria (see also LyndenBell & Wood 1968). This method is very
powerful but it does not provide the form of the perturbation profile
that triggers the instability. This perturbation profile can be
obtained by computing explicitly the second order variations of the
thermodynamical potential and reducing the problem of stability to the
study of an eigenvalue equation. This study was first performed by
Antonov (1962) in MCE and revisited by Padmanabhan (1989) with a
simpler mathematical treatment. This analysis was extended in CE by
Chavanis (2002a) who showed in addition the equivalence
between thermodynamical stability and dynamical stability with respect
to NavierStokes equations (Jeans problem). Remarkably, this stability
analysis can be performed analytically by using simple graphical
constructions in the Milne plane.
In the present paper, we propose to extend these analytical methods to
more general situations in order to provide a complete description of
the thermodynamics of spherical selfgravitating systems. In
Sect. 2, we review the stability limits of isothermal
spheres in different ensembles by using the turning point
criterion. In Sect. 3, we consider specifically the grand
canonical and grand microcanonical ensembles and evaluate the second
order variations of the associated thermodynamical potential. In
Sect. 4, we briefly discuss the connexion between
thermodynamics and statistical mechanics (and field theory) for
selfgravitating systems. In Sect. 5, we consider the
case of generalized thermodynamics proposed by Tsallis (1988) and
leading to stellar polytropes (Plastino & Plastino 1997; Taruya &
Sakagami 2002a,b; Chavanis 2002b). In Sect. 6, we
discuss the stability of isothermal and polytropic spheres under an
external pressure (Bonnor problem). We provide a new and entirely
analytical solution of this old problem. Finally, in Sect. 7
we discuss the relation between dynamical and thermodynamical
stability of stellar systems and gaseous spheres.
2 Thermodynamics of selfgravitating systems
2.1 Thermodynamical ensembles
Consider a system of N particles, each of mass m, interacting via Newtonian gravity. Let
denote the distribution function of the system and
d
the spatial density. The total mass and total energy can be expressed as



(1) 



(2) 
where K is the kinetic energy and W the potential energy. The
gravitational potential
is related to the spatial density by
the Poisson equation

(3) 
The Boltzmann entropy is given by the standard formula (within an
additional constant term)

(4) 
which can be obtained by a combinatorial analysis (Ogorodnikov 1965).
For the moment, we shall assume that the system is confined within a
spherical box of radius R so that its volume is fixed.
In the microcanonical ensemble, the system is isolated and
conserves energy E and particle number N (or mass
M=Nm). Statistical equilibrium corresponds to the state that
maximizes the entropy S at fixed E and N. Introducing Lagrange
multipliers 1/T and
for each constraint, the first order
variations of entropy satisfy the relation

(5) 
where T is the temperature and
the chemical potential. Equation (5)
can be regarded as the first principle of thermodynamics.
In the canonical ensemble, the temperature and the particle
number are fixed allowing the energy to fluctuate. In that case, the
relevant thermodynamical parameter is the free energy (more precisely
the Massieu function)
which is related to S by a
Legendre transformation. According to Eq. (5), the first
variations of J satisfy
.
Therefore, at statistical equilibrium, the system is in the
state that maximizes the free energy J at fixed particle number N and temperature T.
In the grand canonical ensemble, both temperature and chemical
potential are fixed, allowing the energy and the particle number to
fluctuate. The relevant thermodynamical potential is now the grand
potential
.
Its first variations
satisfy
.
At
statistical equilibrium, the system is in the state that maximizes
G at fixed T and .
The microcanonical, canonical and grand canonical ensembles are the
most popular. However, we are free to define other thermodynamical
ensembles. For example, Lecar & Katz (1981) have introduced a grand
microcanonical ensemble (GMCE) in which the fugacity and the
energy are fixed. This corresponds to a thermodynamical potential
whose first variations satisfy
.
At statistical equilibrium, the
system is in the state that maximizes
at fixed E and .
2.2 The isothermal distribution
For extensive systems, it is wellknown that the statistical ensembles
described previously are all equivalent at the thermodynamic
limit. Therefore, the choice of a particular ensemble (in general the
grand canonical one) is only dictated by reasons of convenience. This
is not the case for systems with longrange interactions such as
gravitational systems. Indeed, the statistical ensembles are not
interchangeable and the stability limits differ from one ensemble to
the other. Therefore, the choice of the ensemble is imposed by the
physical properties of the system under consideration. If we want to
describe in terms of statistical mechanics globular clusters
(LyndenBell & Wood 1968) and elliptical galaxies (LyndenBell 1967;
Hjorth & Madsen 1993; Chavanis & Sommeria 1998), which can be
treated as isolated systems in a first approximation, the good choice
is the microcanonical ensemble. If now the system is in contact with a
radiation background imposing its temperature, the relevant ensemble
is the canonical one. This ensemble may be appropriate to star or
galaxy formation (Penston 1969; Chavanis 2002a,b), white dwarf and
neutron stars (Hertel & Thirring 1971; Chavanis 2002c,d) and dark
matter made of massive neutrinos (Bilic & Viollier 1997; Chavanis
2002g). This canonical description is also exact in a model
of selfgravitating Brownian particles (Chavanis et al. 2002; Sire &
Chavanis 2002). Finally, the grand canonical ensemble may be
appropriate to the interstellar medium (or possibly the largescale
structures of the universe), assuming that a statistical description
is relevant (de Vega et al. 1998). Possible applications of the grand
microcanonical ensemble are discussed by Lecar & Katz (1981).
If we just cancel the first order variations of the thermodynamical
potential (under constraints appropriate to the ensemble considered),
it is straightforward to check that each ensemble yields the
MaxwellBoltzmann distribution

(6) 
Therefore, the statistical equilibrium state of a selfgravitating system
is isothermal whatever the ensemble
considered. The differences will come from the second order variations
of the thermodynamical potential. The choice of the
statistical ensemble will affect the stability of the system.
The density field associated with the distribution function (6) is given by

(7) 
where we have defined the fugacity z and the inverse temperature
by
the relations

(8) 
The distribution function (6) can then be written in
terms of as

(9) 
This distribution function is a global maximum of the
thermodynamical potential under the usual constraints and for a
given density field
(see Padmanabhan 1989; Chavanis
2002a). Substituting this optimal distribution in Eqs. (2)
and (4), we get



(10) 



(11) 
It will be more convenient in the sequel to study the stability problem directly from these equations, without loss of generality.
2.3 The Emden equation and the Milne variables
Inserting the relation (7) in Eq. (3), we find
that the gravitational potential
is a solution of the BoltzmannPoisson
equation

(12) 
If we introduce the function
where
is the gravitational potential at r=0, the density
field can be written

(13) 
where
is the central density. Introducing the normalized
distance
,
the BoltzmannPoisson
equation can be written in the standard Emden form (Chandrasekhar
1942)

(14) 
with
at .
This equation has to be solved between
and
corresponding to the normalized box radius

(15) 
We also recall the definition of the Milne variables that will be used in the sequel

(16) 
These variables satisfy the identities



(17) 



(18) 
which can be directly derived from Eq. (14).
The phase portrait of classical isothermal spheres is represented in Fig. 1 and forms a spiral. The spiral starts at
(u,v)=(3,0) for
and tends to the limit point
as
.
This limit point corresponds to the singular solution e
.
For finite isothermal spheres, we must consider only the part of the curve between
and
.
2.4 The equilibrium phase diagram
The thermodynamical parameters can be expressed in terms of the values
of the Milne variables at the normalized box radius .
Writing
and
,
the energy and the
temperature can be written as (Padmanabhan 1989; Chavanis 2002a):



(19) 



(20) 
We need also to express
in terms of the fugacity z. Using the expression of the gravitational potential at the box radius
and comparing Eqs. (7) and (13), we get

(21) 
Expressing
in terms of
by Eq. (15) and
introducing the Milne variables (16), we obtain

(22) 
We can also normalize the fugacity by the energy instead of the
temperature. Combining Eqs. (22), (20) and
(19), we get

(23) 
It is easy to show that statistical equilibrium states only exist for
sufficiently small values of the control parameters ,
,
and
defined previously. Above a critical value, there is
no possible equilibrium state and the system is expected to
collapse. For example, for
(which
corresponds to sufficiently negative energies), we have the wellknow
gravothermal catastrophe (Antonov 1962; LyndenBell & Wood
1968). Similarly, for
(which corresponds to
small temperatures or large masses), we have an isothermal collapse
(Chavanis 2002a; Chavanis et al. 2002). We can show the existence of
such bounds by a simple graphical construction, without numerical
work. Indeed, the equilibrium structure of the system (characterized
by its degree of central concentration )
is determined by the
intersection between the spiral in the (u,v) plane and the curves
defined by Eqs. (19), (20) and (22). If there is
no intersection, the system cannot be in statistical equilibrium. This
graphical construction is made explicitly in Fig. 1 in
MCE, CE and GCE.

Figure 1:
This graphical construction, first introduced
by Padmanabhan (1989) in MCE, shows very simply the existence of an
upper bound for ,
and
above which there is no
hydrostatic equilibrium for an isothermal gas (dotted line:
;
solid line:
;
dashed line:
with similar convention for
and ). 
Open with DEXTER 
In the microcanonical ensemble, the control parameters are the energy E and the particle number N. The parameter conjugate to the energy
(with respect to the entropy S) is the inverse temperature .
The curve giving the inverse temperature as a function of
energy for a fixed particle number N is drawn in
Fig. 2. The parameter conjugate to the particle number N is
the ratio ,
related to the fugacity z. In order to represent
the fugacity as a function of N, for a fixed energy, we use
Eqs. (19) and (23). The
vs. N curve can then
be deduced from Figs. 3 and 4. According to
standard turning point arguments (Katz 1978), the series of equilibria
becomes unstable at the point of minimum energy (for a given mass) or
at the point of minimum mass (for a given negative energy). The condition
implies (Padmanabhan 1989)
4u_{0}^{2}+2u_{0}v_{0}11u_{0}+3=0.

(24) 
The values of
at which the parameter
is extremum are determined by the intersections between the spiral in the (u,v) plane and the parabole defined by Eq. (24), see Fig. 5. The change of stability occurs for the smallest value of .
Its numerical value is
corresponding to a density contrast of 709. Gravitational instability occurs in MCE when the specific heats
becomes zero, passing from negative to positive values.

Figure 2:
Temperature vs. energy plot at fixed particle number. The series of equilibria becomes unstable at MCE in the microcanonical ensemble and at CE in the canonical ensemble. 
Open with DEXTER 

Figure 3:
Fugacity vs. particle number at fixed energy. The series of equilibria becomes unstable at GMCE1 in the grand microcanonical ensemble. 
Open with DEXTER 

Figure 4:
Enlargement of Fig. 3 near the spiral. In the grand microcanonical ensemble, a change of stability occurs at GMCE2 and GMCE3. Between these points the series of equilibria is stable again. After GMCE3, the series becomes and remains unstable. 
Open with DEXTER 

Figure 5:
Graphical construction determining the turning points of the control parameter at which a change of stability occurs in MCE, CE and GCE. 
Open with DEXTER 

Figure 6:
Fugacity vs. mass at fixed temperature. The series of equilibria becomes unstable at GCE in the grand canonical ensemble. 
Open with DEXTER 

Figure 9:
Evolution of
as a function of the central concentration .
In dashed line we have represented the asymptotic value
e^{4} corresponding to the singular solution. The
and
curves have a similar behaviour. 
Open with DEXTER 

Figure 10:
Evolution of
as a function of the central concentration .
In dashed line we have represented the asymptotic value
e^{8} corresponding to the singular solution. Contrary to Fig. 9, the second extremum
is above the asymptote
.
This appears to be the reason for the regain of stability between
and
. 
Open with DEXTER 
In the canonical ensemble, the control parameters are the inverse
temperature
and the particle number N. The parameter
conjugate to the inverse temperature (with respect to the free energy J) is E. The curve giving E as a function of
for a
fixed particle number can be deduced from Fig. 2. The
parameter conjugate to the particle number is ,
related to the
fugacity z. We can represent the fugacity as a function of mass, for
a given temperature, by using the relations (22) and (20). The corresponding curve can be deduced from the
vs.
plot represented in Fig. 6. In the canonical
ensemble, the series of equilibria becomes unstable at the point of
minimum temperature (for a given mass) or at the point of maximum mass
(for a given temperature). The condition dd
is
equivalent to (Chavanis 2002a)
The values of
at which the parameter
is extremum are
determined by the intersections between the spiral in the (u,v) plane
and the straight line defined by Eq. (25), see
Fig. 5. The change of stability occurs for the smallest
value of .
Its numerical value is
corresponding
to a density contrast of 32.1. Gravitational instability in CE
occurs when the specific heats becomes negative passing by
.
In the grand canonical ensemble, the control parameters are the
inverse temperature
and the ratio
related to the fugacity z. The parameter conjugate to the inverse temperature (with respect to the grand potential G) is E. The
curve giving E as a function of
for a fixed fugacity z is
determined by Eqs. (22) and (23) and can be deduced from
Figs. 7 and 8. The parameter conjugate to the ratio
is the particle number N. The curve giving N as a function
of z for a given temperature can be deduced from
Fig. 6. In the grand canonical ensemble,
the series of equilibria becomes unstable at the point of maximum
temperature (for a given fugacity) or at the point of maximum fugacity
(for a given temperature). The condition dd
implies
The values of
at which the parameter
is extremum (see
Fig. 9) are determined by the intersections between the
spiral in the (u,v) plane and the hyperbole defined by
Eq. (26), see Fig. 5. The change of stability
occurs for the smallest value of .
Its numerical value is
corresponding to a density contrast of 1.58.
In the grand microcanonical ensemble, the control parameters are the
energy E and the ratio
related to the fugacity z. The
parameter conjugate to the energy (with respect to the potential )
is .
The curve giving
as a function of E for a fixed
fugacity z can be deduced from Figs. 7 and 8. The parameter conjugate to the ratio
is the
particle number N. The N vs.
plot for a given energy can be
deduced from Figs. 3 and 4. In the grand
microcanonical ensemble, the series of equilibria becomes unstable at
the point of maximum energy (for a given fugacity) or at the point of
maximum fugacity (for a given positive energy). The condition ddimplies
8u_{0}^{2}v_{0}10u_{0}v_{0}17u_{0}+21=0.

(27) 
The values of
at which the parameter
is extremum (see
Fig. 10) are determined by the intersections between
the spiral in the (u,v) plane and the curve defined by
Eq. (27), see Fig. 11. The series of equilibria
becomes unstable for
(density contrast 1.66) which
corresponds to the first intersection. At that point, the
vs.
curve rotates clockwise (see Fig. 7). However, at the
second turning point
(density contrast 11.6) the
vs.
curve rotates anticlockwise (see Fig. 8)
so that stability is regained (this is proved in Katz 1978). Stability
is lost again at
(density contrast 92.7) and never
recovered afterwards. This curious behaviour in GMCE was first noted
by Lecar & Katz (1981). A similar lostgainlost stability behaviour
also occurs for selfgravitating fermions and hard sphere models in
MCE and CE (Chavanis & Sommeria 1998; Chavanis 2002c). In that case,
it is related to the occurence of phase transitions between
"gaseous'' and "condensed'' states.

Figure 11:
Graphical construction determining the turning points of the control parameter
in GMCE. The series of equilibria is stable for
,
unstable for
,
stable again for
and unstable for
. 
Open with DEXTER 
3 Thermodynamical stability of selfgravitating systems
We now determine the condition of thermodynamical stability by
computing the second order variations of the thermodynamical potential
explicitly. This study has already been done in MCE (Padmanabhan 1989)
and CE (Chavanis 2002a). In the following, we perform the
analysis in GCE and GMCE.
3.1 The grand canonical ensemble
The second order variations of the grand potential G are

(28) 
An isothermal sphere is stable in the grand canonical ensemble if
for all variations
of the density
profile. We shall restrict ourselves to spherically symmetric
perturbations since only such perturbations can trigger gravitational
instability for non rotating systems (Horwitz & Katz 1978). In the
grand canonical ensemble, the mass of the system is not
conserved. Therefore, the potential at the wall changes according to

(29) 
On the other hand, according to the Gauss theorem, we have

(30) 
These boundary conditions suggest the introduction of the new function

(31) 
which satisfies

(32) 
where we have used Eqs. (29) and (30) to obtain the
third equality (the second equality is obvious for a spherically
symmetric system).
The variations
and
are related to each other by the Poisson equation

(33) 
In terms of the new variable (31), Eq. (33) becomes

(34) 
Expressing the second order variations of G in terms of X, we get

(35) 
Integrating by parts, we can put Eq. (35) in the form

(36) 
The boundary terms arising from the integration by parts are seen to vanish by virtue of Eq. (32). We can therefore reduce the stability analysis of isothermal spheres in the grand canonical ensemble to the eigenvalue problem

(37) 
If all the eigenvalues
are negative, then the isothermal
sphere is a maximum of the thermodynamical potential G (and is
therefore stable in GCE). If at least one eigenvalue is positive, it
is an unstable saddle point. The condition of marginal stability
determines the transition between stability and
instability. We thus have to solve the equation

(38) 
Integrating once and using the boundary conditions (32) at r=0, the constant of integration is seen to vanish and we get

(39) 
Introducing the dimensionless variables defined in Sect. 2.3, we are led to solve the problem



(40) 



(41) 
Let us denote by

(42) 
the differential operator which occurs in Eq. (40). Using the
Emden Eq. (14), we readily check that

(43) 
Therefore, the general solution of Eq. (40) which satisfies the condition X=0 at the origin is

(44) 
where c_{1} is an arbitrary constant. The boundary condition
implies

(45) 
This relation determines the value of
at which the series of equilibria becomes unstable in the grand canonical ensemble. In terms of the Milne variables, Eq. (45) is equivalent to
which is precisely the condition (26) obtained with the turning point criterion. We are now in a position to determine the structure of the perturbation profile that triggers the instability in GCE. Using Eq. (34), we have

(47) 
with Eq. (44) for X. This yields

(48) 
or, introducing the Milne variables,

(49) 
We note that Eq. (49) coincides with the expression
found in the canonical ensemble (Chavanis 2002a). The number of nodes in
the profile
is determined by the number of intersections
between the spiral in the (u,v) plane and the line v=2. We see in
Fig. 5 that there is no intersection for
,
where 1.78 is the stability limit in
GCE. Therefore, collecting the results obtained by different authors,
the perturbation that triggers gravitational instability has two nodes
in MCE ("corehalo'' structure), one node in CE and no node in GCE
(see Fig. 12).
3.2 The grand microcanonical ensemble
The grand microcanonical ensemble is interesting in its own right but
also because it exhibits a lostgainlost stability behaviour similar
to that found for selfgravitating fermions and hard sphere models
(Chavanis 2002c). Now, the stability analysis can be conducted
analytically in GMCE for classical isothermal spheres while only
numerical results are available for fermions and hard sphere
systems. On a technical point of view, GMCE is the most complicated
and richest of all thermodynamical ensembles. We can argue, however,
that GMCE may not be realized in practice. It is not clear, indeed,
what mechanism can be devised so that two systems can share particles
without also sharing energy. One example might be the case of
relativistic particles created and destroyed at equilibrium in a cosmological
settling.
In GMCE, we have to maximize the thermodynamical potential



(50) 
at fixed energy

(51) 
We vary the density profile and use Eq. (51) to determine the corresponding variation of temperature. After some algebra, we find that the second order variations of the thermodynamical potential can be written



(52) 
Introducing the variable X defined previously and integrating by parts, we can put the problem in the form



(53) 
with



(54) 
The stability analysis reduces therefore to the study of the eigenvalue equation



(55) 
The condition of marginal stability ()
reads



(56) 
Integrating once, we obtain



(57) 
The gravitational potential
is related to the Emden function
by the relation
.
Using
and Eq. (20), we get

(58) 
Inserting this relation in Eq. (57) and introducing the dimensionless variables defined in Sect. 2.3, we find that



(59) 
where we have integrated by parts and used the identity
e
equivalent to the Emden Eq. (14). Writing
V 
= 

(60) 
W 
= 

(61) 
we are led to solve the problem



(62) 



(63) 
Seeking a solution of Eq. (62) in the form

(64) 
and using the identities (43), we find that

(65) 
The constant b is determined by the boundary condition
.
We thus find that the general solution of Eqs. (62) and (63) is
where we have used Eq. (61). The values of
at which
a change of stability occurs ()
are obtained by inserting
the solution (66) for X in Eq. (60). We need the
identities
which are derived in Appendix A. After simplification, we
obtain the condition
8u_{0}^{2}v_{0}10u_{0}v_{0}17u_{0}+21=0,

(70) 
which coincides with the condition (27) deduced from the
turning point criterion. The perturbation profile at the critical
points at which
is given by Eq. (47) with
Eq. (66) for X. Using Eqs. (13), (62)
and (66), we get

(71) 
where c_{1} is a constant. The nodes of the perturbation profile are determined by the condition

(72) 
with
.
The number of nodes can be found by a graphical
construction. In Fig. 13, we plot the lines defined by
Eq. (72) in the (u,v) plane for the four first critical
points in the series of equilibria. The corresponding perturbation
density profiles are plotted in Fig. 14. The first mode of
instability
has no node. The profile of the second mode
increases (in absolute value) with distance. The
absence of node at
is a signature of the fact that
stability is regained at that point. The mode
at which
stability is lost again has one node, the mode
two nodes
etc.

Figure 13:
Graphical construction determining the number of nodes for the different modes of instability in GMCE. The horizontal lines correspond to the condition (72) for the four first critical points. 
Open with DEXTER 

Figure 14:
Density perturbation profiles for the four first critical points in the series of equilibria in GMCE. We have fixed the constant c_{1} such that
at r=0. 
Open with DEXTER 
4 Connexion with statistical mechanics
In this section, we briefly discuss the relation between the
thermodynamics and the statistical mechanics (and field theory) of
selfgravitating systems. We shall just discuss the physical ideas
without entering into technical details.
In the microcanonical ensemble, the object of fundamental interest is
the density of states



(73) 
where

(74) 
is the Hamiltonian of the selfgravitating gas. By definition, g(E)dE is proportional to the number of microstates (specified by the position and the velocity of all N particles) with energy between E and
.
The entropy
is defined by

(75) 
As is customary in statistical mechanics, we decompose phase space
into macrocells and divide these macrocells into a large number of
microcells. A macrostate is characterized by the number of
particles in each macrocell (irrespective of their precise position
and velocity in the cell), or equivalently by the smooth distribution
function
.
If we call W_{i} the number of
microstates corresponding to the macrostate i, we can rewrite
Eq. (73) formally in the form



(76) 
where the sum runs over all macrostates with energy E (and mass M). In addition,
is the Boltzmann entropy (4) of the macrostate i (i.e.,
)
and
s_{i}=S_{i}/N the entropy per particle. The thermodynamic limit for
selfgravitating systems corresponds to
in such
a way that all the dimensionless control parameters defined in
Sect. 2.4 remain finite. This can be written for example
with N/R finite (and ,
). This is a very unusual thermodynamic limit due to the
longrange nature of the interactions.
Now, at the thermodynamic limit, the sum in Eq. (76)
is dominated by the macrostate f_{*} which maximizes the Boltzmann
entropy (4) at fixed mass and energy:



(77) 
It has to be noted that the sum in Eq. (76) is sharply
peaked around f_{*}, which corresponds to what is called a large
deviation property in mathematics. Therefore, the meanfield approximation
is exact for selfgravitating systems at the thermodynamic limit.
In fact, if we look at the problem more carefully, the argument
leading to Eq. (77) is not so obvious because the entropy
S[f] has no global maximum, except if we introduce smallscale and
largescale cutoffs. The introduction of these cutoffs is necessary
to define correctly the density of state (73), otherwise the
integral would diverge (Padmanabhan 1990). The divergence at large
scales is associated to the fact that a stellar system has the
tendency to evaporate under the effect of encounters. In
reality, stellar systems are not allowed to extend to infinity because
they interact with the surrounding. For example, globular clusters
are subject to the tides of a nearby galaxy so that the Boltzmann
distribution has to be truncated at high energies. The MichieKing
model,



(78) 
which is a truncated isothermal, can take into account the evaporation
of high energy stars and provides a good description of about 80%
of globular clusters (Binney & Tremaine 1987). It can be derived from
the FokkerPlanck equation (taking into account the encounters between
stars) by imposing that the distribution function vanishes at the
escape energy
.
In that case, the system is not
truly static since it gradually loses stars but we can consider that a
globular cluster passes by a succession of quasiequilibrium
configurations. Instead of working with truncated models, it is more
convenient to confine the system within a box of radius R, the box
radius playing the role of a tidal radius. The divergence at small
scales of the density of states reflects the natural tendency of a
stellar system to form binaries. In this extreme case, the size a of
the stars provides a smallscale cutoff. The influence of a
smallscale cutoff on the form of the equilibrium phase diagram is
discussed by Chavanis (2002c) and Chavanis & Ispolatov (2002). When
the smallscale cutoff
,
which is the limit relevant
for globular clusters since the size of the stars is in general much
smaller than the interstellar distance, the entropy S[f] possesses
two maxima (for sufficiently high energies). The local entropy maximum
describes a "gaseous'' phase insensitive to the small scale
cutoff. It corresponds to the isothermal spheres studied
previously. The global entropy maximum is dominated by the small scale
cutoff and corresponds to a "condensed'' phase made of a few stars
close together (or simply a single binary) surrounded by a diffuse
halo. This "corehalo'' structure is the most probable configuration
of a stellar system so it is expected to be reached at equilibrium.
Since the observed globular clusters, described by the MichieKing
model, do not possess this corehalo structure, we conclude that they
are not in true equilibrium but that they are slowly evolving in that
direction. However, the formation of binaries can take extremely long
times, much larger than the age of the universe. Indeed, equilibrium
statistical mechanics tells nothing about the relaxation time. For
the timescales contemplated in astrophysics, globular clusters can be
considered as metastable equilibrium states corresponding to
local entropy maxima. Indeed, the probability of transition from a
gaseous state to a condensed state (corresponding to the formation of
a binary) is extremely small (Katz & Okamoto 2000; Chavanis
& Ispolatov 2002). Therefore, in the calculation of the density of
states (73), we must discard by hands the states which
correspond to collapsed configurations (binaries) since they cannot be
reached in the timescales of interest. Accordingly, Eq. (77)
is correct with f_{*} being the local maximum of S[f]. This is
precisely the situation studied in Sects. 2 and 3. In fact, this picture is correct only for globular
clusters with high energy E, well above the Antonov limit .
Because of evaporation, the energy of a globular cluster
slowly decreases until it reaches the point at which the system cannot
be in equilibrium anymore (e.g., Katz 1980). At that point, an isothermal
sphere becomes unstable and the system undergoes a gravitational
collapse (gravothermal catastrophe). The evolution proceeds
selfsimilarly with the formation of a high density core with a
shrinking radius and a small mass (Hénon 1961; Cohn 1980; LyndenBell &
Eggleton 1980). This core collapse concerns typically 20% of
globular clusters. In theory, the central density becomes infinite in
a finite time. In practice, the formation of binaries can release
sufficient energy to stop the collapse (Hénon 1961) and even drive a
reexpansion of the system (Inagaki & LyndenBell 1983). Then, a
series of gravothermal oscillations should follow (Bettwieser &
Sugimoto 1984).
In the canonical ensemble, the object of interest is the partition function



(79) 
The free energy is defined by



(80) 
The partition function is related to the density of states by the Laplace transform



(81) 
Using Eq. (76), it can be rewritten



(82) 
where
is the Massieu function and the sum runs over all macrostates with mass M. In the thermodynamic limit, the partition function is dominated by the contribution of the macrostate which maximizes the Massieu function
at fixed mass. In this limit, the mean field approximation is exact and we have



(83) 
Again, the Massieu function J[f] has no global maximum in the
absence of cutoff. If we introduce a small scale cutoff a and a
large scale cutoff R and consider the limit ,
it is found
that the Massieu function J[f] has two maxima (for sufficiently high
temperatures). The local maximum corresponds to isothermal "gaseous''
configurations and the global maximum to a "condensed'' structure in
which all the particles are packed together (Aronson & Hansen 1972;
Chavanis 2002c; Chavanis & Ispolatov 2002). As
,
the
density profile of the condensed state tends to a Dirac peak
(Kiessling 1989). As in the microcanonical ensemble, the "gaseous''
configurations are metastable but they can be long lived and relevant
for astrophysical purposes. This justifies why we only select the
contribution of the local maximum of J[f] in
Eq. (83). This treatment is, however, valid only for
sufficiently high temperatures. Below ,
a phase transition
similar to the gravothermal catastrophe occurs in the canonical
ensemble and leads to a condensed structure containing almost all the
mass. The inequivalence of statistical ensembles regarding the
formation of binaries (in MCE) or Dirac peak (in CE) is further
discussed in Appendices A and B of Sire & Chavanis (2002).
Finally, the grand canonical partition function is defined by



(84) 
where z is the fugacity. Applying the standard HubbardStratanovich
transformation, it can be rewritten in the form of a path integral
(Horwitz & Katz 1978; Padmanabhan 1991; de Vega & Sanchez 2002)



(85) 



(86) 
for the Liouville action



(87) 
It should be emphasized that
is a formal field which is not the gravitational field. In the meanfield approximation, the path
integral is dominated by the contribution of the field which maximizes the Liouville action (87). The cancellation
of the first order variations gives



(88) 
which is similar to the BoltzmannPoisson Eq. (12). This allows one to identify
with the
equilibrium gravitational potential
(up to a negative proportionality
factor). The second order variations of A at the critical point must
be negative for
to be a maximum. This yields the
condition



(89) 
which is equivalent to the maximization of the grand potential G(see Sect. 3.1). Therefore, the thermodynamical approach is
equivalent to the statistical mechanics or field theory approach. We
note, however, that the Liouville action
is not the same
functional as the grand potential
although they give the
same critical points and the same conditions of stability.
5 Generalized thermodynamics and Tsallis entropy
5.1 General considerations
It has been recently argued that the classical Boltzmann entropy may not be relevant for nonextensive systems and that Tsallis entropies, also called qentropies, should be used instead. These entropies can be written

(90) 
where q is a real number. For
,
this expression returns the classical Boltzmann entropy (4)
as a particular case.
The first application of Tsallis generalized thermodynamics in stellar
systems was made by Plastino & Plastino (1997) who showed that the
extremization of Tsallis entropy leads to stellar polytropes with an index

(91) 
For
,
or
,
we recover isothermal
spheres. For q>9/7 (i.e. n<5), the density drops to zero at a finite distance so
that the mass is finite. Then, Taruya & Sakagami (2002a) considered
the stability problem in the microcanonical ensemble by extending
Padmanabhan's classical analysis of the Antonov instability to the
case of polytropic distribution functions. A configuration is stable
in the sense of generalized thermodynamics if it corresponds to a
maximum of Tsallis entropy at fixed mass and energy. They considered
polytropic spheres confined within a box and showed that polytropes
with index
become unstable above a certain density contrast
while polytropes with
are always stable. On the other hand,
if we consider the canonical ensemble, we have to select maxima of
Tsallis free energy
at fixed mass and (the generalized inverse temperature). It is found (Taruya
& Sakagami 2002b; see also Sect. 5.5) that polytropes with index
become unstable above a certain density contrast while
polytropes with
are always stable. In our sense, the main
interest of the qentropies (and their attractive nature) is to
offer a simple generalization of classical thermodynamics which leads
to models that are still analytically tractable since exponentials
are replaced by power laws. This approach is interesting to develop, at
least formally, as it offers a nice connexion between polytropic and
isothermal spheres, which are the most popular mathematical models of
selfgravitating systems. We shall therefore devote a section to this
generalization. However, in Sect. 7, we shall argue that
Tsallis entropies are just a particular case of Hfunctions S[f]whose maximization at fixed mass and energy determines nonlinearly
stable stationary solutions of the Vlasov equation (Tremaine et al.
1986). We shall also argue that the maximization of the Jfunction
at fixed mass determines nonlinearly stable
stationary solutions of the EulerJeans equations. This discussion
should demystify the concept of generalized thermodynamics introduced
by Tsallis (1988).
5.2 Stellar polytropes
The extremization of Tsallis entropy at fixed mass and energy yields (Plastino & Plastino 1997; Taruya & Sakagami 2002a)
f 
= 

(92) 
A 
= 

(93) 
where
and
(related to )
are Lagrange
multipliers which can be called inverse temperature and chemical
potential in the generalized sense. The distribution function (92) corresponds to stellar polytropes that were first
introduced by Plummer (1911). Their derivation from a variational
principle was discussed by Ipser (1974) among others. Note that
Eq. (92) is valid for
.
For
,
we set f=0. The spatial density
d
and the pressure
d
can be expressed as
with B(a,b) being the
function and use has been made of
Eq. (91). Furthermore, in obtaining Eq. (95), we have
used the identity
B(m+1,n)=mB(m,n)/(m+n). The integrability
condition d
requires that
n>1/2. Eliminating the gravitational potential between the relations (94) and (95), we recover the wellknown fact that
stellar polytropes satisfy the equation of state

(96) 
like gaseous polytropes (note, however, that these systems do not have
the same phase space distribution function, unlike isothermal
spheres). In the present context, the polytropic constant is given by

(97) 
Using Eqs. (92) and (94), the distribution function can be written as a function of the density as



(98) 
with



(99) 
Using the foregoing relations, it is possible to express the total energy and the entropy in terms of
and p according to (Taruya & Sakagami 2002b; see also Sect. 7.8)



(100) 



(101) 
5.3 The LaneEmden equation
The equilibrium structure of a polytrope is determined by substituting the relation (94) between
and the gravitational potential
inside the Poisson Eq. (3). This is equivalent to using the condition of hydrostatic equilibrium
with the equation of state (96). Letting

(102) 
where
is the central density, we can reduce the condition of hydrostatic equilibrium to the LaneEmden equation (Chandrasekhar 1942)

(103) 
with boundary conditions
,
.
For n<5, the density vanishes at some radius identified with the
radius of the polytrope (in terms of the normalized distance, this
corresponds to
at
). Such polytropic
configurations (with infinite density contrasts) will be refered to as
complete polytropes. We shall also consider the case of incomplete polytropes, with arbitrary index, enclosed within a box of radius R. For these configurations, the LaneEmden equation must be solved
between
and
with

(104) 
We also recall the expression of the Milne variables that will be used in the sequel,

(105) 
These variables satisfy the identities
which can be derived from Eq. (103). The phase portrait of
polytropes in the (u,v) plane and the properties that we shall need in
the following are summarized in Chavanis (2002b).
5.4 Turning point criterion
Eliminating A between Eqs. (93) and (97), we find that

(108) 
where we have defined (Taruya & Sakagami 2002b)

(109) 
In a preceding paper (Chavanis 2002b), we have established that the mass of the configuration is related to the central density
(through the parameter )
by the relation

(110) 
We had conjectured that
plays the role of a temperature in the context of Tsallis generalized thermodynamics. This is indeed the case since, using Eq. (108),
can be expressed in terms of as

(111) 
For
,
this parameter becomes equivalent to the parameter
introduced in the study of isothermal spheres (Chavanis 2002a).
In the framework of generalized thermodynamics, the parameter conjugate to the energy with respect to Tsallis entropy S_{q} is the inverse temperature
(microcanonical description) and the parameter conjugate to the inverse temperature with respect to Tsallis free energy
is E (canonical description). The ET curve has been plotted and discussed in Chavanis (2002b) for different values of the polytropic index. According to the turning point criterion, the series of equilibria becomes unstable in the canonical ensemble for dd.
We have shown that this condition is equivalent to

(112) 
For ,
the polytropes are always stable since the
curve is monotonic. The polytropes start to be unstable
(for sufficiently high density contrasts) when n>3. This
generalized thermodynamical stability criterion coincides with the
Jeans dynamical stability criterion based on the NavierStokes
equations for a polytropic gas (Chavanis 2002b). Since K is
considered as a constant when we analyze the stability of polytropic
spheres with respect to the NavierStokes equations, and since K is
related to the temperature via Eq. (108), this corresponds to a
sort of "canonical situation''. This explains why the condition of
dynamical stability for gaseous polytropes is equivalent to the
condition of thermodynamical stability for stellar polytropes in the
canonical ensemble. This equivalence is further discussed in
Sect. 7.

Figure 15:
Fugacity vs. mass (for a given temperature) in the framework of generalized thermodynamics for different polytropic index. For ,
has one turning point but not ;
for ,
both
and
have one turning point; for n>5,
and
have an infinity of turning points. For n<5, complete polytropes correspond to the terminal point of the
curve. For n>5, the curve has a spiral behaviour towards the limit point
corresponding to the singular sphere

Open with DEXTER 
We now turn to the grand canonical situation. Combining
Eqs. (94), (93) and (109), we can write the density
field as

(113) 
This expression suggests to define a generalized fugacity by the relation
so that z is equivalent to the usual fugacity
when
.
Taking r=R in Eq. (113) and comparing with Eq. (102), we get

(114) 
where we have used
.
Expressing the central density in terms of ,
using Eqs. (104) and (108), we find

(115) 
From Eqs. (114) and (115), we get

(116) 
Eliminating M thanks to Eq. (111) and introducing the Milne variables (105), we finally obtain

(117) 
For
,
this parameter becomes equivalent to the
one introduced in Sect. 2.4 for isothermal spheres (note that
for sufficiently small index n, the parameter
can become
negative when v_{0}>1; therefore, the definition of z given
previously is not valid for small n). According to the turning
point criterion, the series of equilibria becomes unstable in the
grand canonical ensemble when dd.
This gives the
condition
2(1v_{0})(n1)u_{0}v_{0}=0.

(118) 
The values of
at which
is extremum can be determined
by a graphical construction like in Chavanis (2002b). They correspond to the
intersections between the hyperbole defined by Eq. (118)
and the solution curve in the (u,v) plane (which depends on the index n). From these graphical
constructions, we deduce that the function
has one maximum for
.
For n=5, the LaneEmden equation can be solved
analytically and we get

(119) 
which is maximum for
.
For n>5, the function
has an infinite number of extrema. The onset of
gravitational instability corresponds to the first maximum. The
fugacity vs. mass plot (for a given temperature) can be deduced from
Fig. 15. Collecting the results obtained by different
authors, incomplete polytropes with high density contrasts are
unstable in MCE for n>5, in CE for n>3 and in GCE for
all n. Complete polytropes are always stable in MCE, stable in CE only
for n<3 and never stable in GCE.
In the following, we analyze the stability of polytropic spheres by
explicitly calculating the second order variations of the
thermodynamical potential. This study has been done by Taruya
& Sakagami (2002a, b) in MCE and CE. We shall consider CE in relation
with our previous study (Chavanis 2002b), GCE and GMCE.
5.5 Generalized thermodynamical stability in the canonical ensemble
In the canonical ensemble, the thermodynamical parameter is the free energy (Massieu function)
which is explicitly given by

(120) 
where we have used Eqs. (100) and (101). Using the
equation of state (96), the second order variations of free
energy can be put in the form

(121) 
As in previous papers, we introduce the function q(r) defined by

(122) 
and satisfying the perturbed Gauss theorem

(123) 
The conservation of mass imposes
q(0)=q(R)=0. After an integration by parts,
we can rewrite Eq. (121) in the form

(124) 
The point of marginal stability is therefore determined by the condition

(125) 
This is the same equation as that determining the dynamical stability
of gaseous polytropes (Chavanis 2002b). We have solved this equation
and showed that the boundary condition q(R)=0 returns the condition
of Eq. (112) brought by the turning point criterion. The
structure of the density perturbation profile that triggers
instability is also discussed in our previous paper.
5.6 Generalized thermodynamical stability in the grand canonical ensemble
The thermodynamical potential in the grand canonical ensemble is
,
i.e.



(126) 
Its second order variations are given by Eq. (121) like for the free energy. Only the boundary conditions are different since the mass is not conserved anymore. Introducing the function X(r) defined by Eq. (31), we obtain after an integration by parts



(127) 
The point of marginal stability is determined by the equation



(128) 
with the boundary conditions (32). Integrating once, we get



(129) 
In terms of the dimensionless variables introduced in Sect. 5.3, we have to solve the problem



(130) 



(131) 
Let us introduce the differential operator



(132) 
Using the LaneEmden Eq. (103), it is easy to check that



(133) 
Therefore, the general solution of Eq. (130) satisfying
X(0)=0 is



(134) 
The critical value of
at which the series of equilibria
becomes unstable in GCE is determined by the boundary condition
.
Using Eq. (134) and introducing the Milne
variables (105), we find that this condition is equivalent to
Eq. (118) in agreement with the turning point criterion. The
density perturbation profile triggering instability can be deduced
from Eqs. (47) and (134). We get



(135) 
with
.
Equation (135) coincides with the
expression found in CE. The number of nodes of the perturbation
profile can be determined by a graphical construction like in Chavanis
(2002b). First, consider the case n=5 which can be solved
analytically. The solution of the equation
is
.
Since
,
we
conclude that the perturbation profile in GCE has no node contrary to
the equivalent situation in CE. This conclusion remains true for all
index .
Indeed, the critical case
would
occur for an index n such that
belongs to the
solution curve (substitute
in Eq. (118)). This
is never the case for
(see Chandrasekhar 1942). Therefore,
the structure of the density profile for n<5 is the same as for
n=5 so it has no node. For n>5, the first mode of instability has no
node, the second mode of instability one node etc.
5.7 Generalized thermodynamical stability in the grand microcanonical ensemble
In the grand microcanonical ensemble, the fixed parameters are the fugacity and the energy. The normalized energy is given by (Taruya & Sakagami 2002a)



(136) 
Eliminating
in Eq. (117) in profit of E, using Eqs. (111) and (136), we find that the relevant control parameter



(137) 
is related to
by



(138) 
In the grand microcanonical ensemble, we have to maximize the
thermodynamical potential
at fixed E and .
The entropy and the energy are expressed in terms of and K by Eqs. (100), (101) and (96). We vary the
density and use the energy constraint to determine the corresponding
variation of K (which is related to the inverse temperature ). After some algebra, we find that the second order variations
of
are given by



(139) 
(To simplify the formulae, we have not written the positive proportionality factor
in front of
.) Introducing the variable X defined
previously and integrating by parts, we can put the problem in the form



(140) 
with



(141) 
The condition of marginal stability reads



(142) 
We now need to relate the gravitational potential
to the
LaneEmden function .
Integrating the equation of hydrostatic
equilibrium for a polytropic equation of state, we
readily find that



(143) 
The constant can be determined from the boundary condition
.
Using the relation



(144) 
which results from Eqs. (110) and (104), we find that Eq. (143) can be written

(145) 
Inserting this relation in Eq. (142) and introducing the dimensionless variables defined in Sect. 5.3, we find that



(146) 
where we have integrated by parts and used the identity
equivalent to the LaneEmden Eq. (103). We are led
therefore to solve a problem of the form



(147) 



(148) 
Using the identities (133), we find that the general solution of Eq. (147) satisfying X(0)=0 is

(149) 
The constant b is determined by the boundary condition
.
Substituting the resulting expression for
in Eq. (146), and using the identities of Appendix B, we obtain after tedious algebra a condition for
which is equivalent to the one obtained from the turning point argument, i.e. by setting
.
6 The isobaric ensemble
We now consider the stability of isothermal spheres in contact with a
medium exerting a constant pressure P on their boundary. In this
isobaric ensemble, the volume of the system can change unlike in
the Antonov problem. This situation was first considered by Ebert
(1955), Bonnor (1956) and McCrea (1957) who found that an isothermal
gas becomes gravitationally unstable above a critical pressure
or above a maximum compression. We shall complete these
results by supplying an entirely analytical derivation of the stability
criterion and by showing the equivalence between thermodynamical and
dynamical stability (Jeans problem). We shall also consider the case
of polytropic gas spheres under external pressure.
6.1 Isothermal gas spheres under external pressure
Let us consider an isothermal gas sphere with mass M at temperature T and under pressure P. The first principle of thermodynamics can
be written
.
In the
isothermalisobaric ensemble (TBE), the relevant thermodynamical
potential is the Gibbs energy
.
Its first variations satisfy
so that, at statistical equilibrium, the system is in
the state that maximizes
at fixed temperature T and
pressure P.
For an isothermal gas,
.
Using Eq. (13), the pressure on the boundary of the sphere is



(150) 
The central density
can be eliminated via the relation (15). Eliminating the radius R via the relation (20) and introducing the Milne variables (16), we obtain after simplification



(151) 
Similarly, the normalized volume of the sphere is given by



(152) 
Using the expansion of the Milne variables for
,



(153) 
we find that the first order correction to Boyle's law due to selfgravity is



(154) 
with
.
This formula can also be derived from the Virial theorem



(155) 
by using the isothermal equation of state and by assuming that the density
is uniform (in which case the
potential energy
W=3GM^{2}/5R). As reported by Bonnor (1956),
Eq. (154) was first suggested by Terletsky (1952) as a simple
attempt to take into account gravitational effects in the perfect gas
law. It appears here as a systematic expansion in terms of the
concentration factor .
For large concentrations,
Eq. (154) is not valid anymore and Eqs. (151)(152) must be used instead. The PV curve is plotted in
Fig. 16. It presents a striking spiral behaviour towards the
limit point
corresponding to the
singular sphere. This is similar to the ET curve and other curves
studied in Sect. 2.4. This is interesting to note, for
historical reasons, because this diagram was discoved before the
fundamental papers of Antonov (1962) and LyndenBell & Wood (1968) on
the gravothermal catastrophe. Like the minimum energy of Antonov,
there exists a maximum pressure
above which no hydrostatic equilibrium for an
isothermal gas is possible. Alternatively, this corresponds to a minimum
temperature
for a given mass
and pressure. From the turning point criterion, the series
of equilibria becomes unstable in TBE at the point
of maximum pressure. The condition dd
leads to
The critical value
is determined by the intersection between
the spiral in the (u,v) plane and the straight line (156). Its numerical value is
corresponding to a
density contrast of 14.0. More compressed isothermal spheres,
ie. those for which
are thermodynamically unstable.

Figure 16:
The PV diagram for isothermal spheres. The series of equilibria becomes unstable at TBE in the isothermalisobaric ensemble. 
Open with DEXTER 
Using Eqs. (151)(152) and the identities (17)(18), we readily obtain



(157) 
and



(158) 
Therefore, the "compressibility''
can be expressed as



(159) 
Using
u_{0}=3PV/NkT and
according to Eqs. (151) and (152), we can check that
Eq. (159) is equivalent to Eq. (2.16) of Bonnor (1956), but it
has been obtained here in a much more direct manner. Gravitational
instability in TBE occurs when the compressibility becomes negative passing by .
We now consider the dynamical stability of isothermal gas spheres
under external pressure and show the equivalence with the
thermodynamical criterion. This problem was studied numerically by
Yabushita (1968) but we provide here an entirely analytical
solution. In a preceding paper, we have found that
the equation of radial pulsations for a gas with an isothermal
equation of state could be written (Chavanis 2002a)



(160) 
where
is the growth rate of the perturbation (such that
)
and q is defined by Eq. (122). In the isobaric ensemble, the boundary conditions are given by Eq. (C.4) of Appendix C. Introducing dimensionless variables and considering the case of marginal stability ,
we are led to solve the problem



(161) 
with F(0)=0 and



(162) 
Now, the function
can be expressed as (Chavanis 2002a)



(163) 
Substituting this relation in the boundary condition (162), using the Emden Eq. (14) and introducing the Milne variables (16), we obtain the condition
which coincides with the criterion (156) of thermodynamical
stability. The expressions of the density perturbation profile and of
the velocity profile at the point of marginal stability are given in
Chavanis (2002a). Only the value of the critical point changes. Due to the different boundary conditions, we now have
at the surface of the gaseous sphere since the sphere
can oscillate. The perturbation profile
has one node
for the first mode of instability, two nodes for the second mode
etc. The velocity profile
has no node for the first mode
(except at the origin), one node for the second mode etc.

Figure 17:
Temperature vs. enthalpy plot at fixed pressure and mass. The series of equilibria becomes unstable at HBE in the isenthalpicisobaric ensemble. 
Open with DEXTER 
We could also consider the situation in which the pressure P and the
enthalpy H=E+PV are fixed (instead of the temperature). This
corresponds to the isenthalpicisobaric ensemble (HBE). Since
,
the entropy must be
a maximum at fixed H and P at statistical equilibrium. Eliminating
the radius R in Eqs. (19) and (20) in profit of the
pressure P, we get after some algebra
The TH curve in HBE is plotted in Fig. 17. It spirals around
the fixed point
corresponding to the singular solution. There is no equilibrium state
for
.
This implies a minimum enthalpy for a fixed
pressure or a maximum pressure for a fixed enthalpy. This is similar
to the Antonov (1962) criterion, the pressure playing the same role
as the box radius. The series of equilibria becomes unstable in HBE at
the point of minimum enthalpy. The condition ddgives
16u_{0}^{2}+8u_{0}v_{0}38u_{0}+3v_{0}=0. 


(167) 
The critical value of
is 25.8 corresponding to a density
contrast of 390.
The specific heat at fixed pressure
can be deduced from Eqs. (165) and (166). After simplification, we find



(168) 
For the ideal gas without gravity
(u_{0},v_{0})=(3,0) we recover the
wellknown result
.
Gravitational instability
occurs in the isenthalpicisobaric ensemble when C_{P}=0 passing from
negative to positive values and in the isothermalisobaric ensemble when C_{P}<0.
6.2 Polytropic gas spheres under external pressure
We now consider a polytropic gas sphere of index n (or
)
and mass M under pressure P. Using Eqs. (96) and (102), the pressure on the boundary of the sphere is given by



(169) 
The central density
can be eliminated via the relation (104). Eliminating the radius R via the relation (110) and introducing the Milne variables (105), we obtain after simplification



(170) 
Similarly, the normalized volume of the sphere is given by



(171) 
Using the expansion of the Milne variables for
,



(172) 
we find that the first order correction to the classical adiabatic law due to selfgravity is



(173) 
Equation (173) can also be obtained from the Virial theorem (155) as before. The PV curve is plotted in
Figs. 18 and 19 for different values of the
polytropic index n. The volume is extremum at points
such
that dd.
This leads to the condition
.
The pressure is extremum at
points
such that dd.
This leads to the
condition
2u_{0}+(3n)v_{0}=0. 


(174) 
The number of extrema can be obtained by a graphical construction in the (u,v) plane like in Chavanis (2002b). For n<3, there is no extremum for
and .
The pressure is a decreasing function of the volume and vanishes for a maximum volume
corresponding to the complete polytrope (such that
). For n=1, the solution of the LaneEmden equation is
for
and we explicitly obtain



(175) 
for
.
For ,
there is one extremum for
and .
This implies the existence of a maximum pressure
and a minimum volume .
In particular, for n=5, the LaneEmden equation can be solved analytically and we get



(176) 
The pressure is maximum for
and the volume is minimum for
.
For n>5 there is an infinity of
extrema for
and .
The PV curve spirals towards
the limit point
corresponding to the
singular sphere. From turning point arguments, the series of
equilibria becomes thermodynamically unstable at the point of maximum
pressure (for n>3). Polytropes with n<3 are always stable in the
isobaric ensemble.

Figure 18:
The PV diagram for polytropes with
(specifically n=1) and
(specifically n=4). 
Open with DEXTER 

Figure 19:
The PV diagram for polytropes with n>5 (specifically n=10). 
Open with DEXTER 
Using Eqs. (170), (171) and the identities (106) and (107), we readily obtain



(177) 
and



(178) 
Therefore, the "compressibility''
can be expressed as



(179) 
We now consider the dynamical stability of polytropic gas spheres
under external pressure. The equation
of radial pulsations for a gas with a polytropic equation of state
can be written (Chavanis 2002b)



(180) 
In the isobaric ensemble, the boundary conditions are given by
Eq. (C.4) of Appendix C. Introducing
dimensionless variables and considering the case of marginal stability
,
we are led to solve the problem



(181) 
with F(0)=0 and



(182) 
Now, the function
can be expressed as (Chavanis 2002b)



(183) 
Substituting this relation in the boundary condition (182), using the LaneEmden Eq. (103) and introducing the Milne variables (105), we obtain the condition
2u_{0}+(3n)v_{0}=0, 


(184) 
which coincides with the criterion (174) of thermodynamical
stability. The expressions of the density perturbation profile and of
the velocity profile at the point of marginal stability are given in
Chavanis (2002b). Only the value of the critical point changes. For ,
the density perturbation profile has one node
and the velocity profile no node (except at the origin). For n>5,
the perturbation profile
has one node for the first
mode of instability, two nodes for the second mode etc. The velocity
profile
has no node for the first mode (except at the
origin), one node for the second mode etc.
7 Stability of galaxies and stars
In this section, we discuss in detail the relation between the
dynamical and the thermodynamical stability of stellar systems and
gaseous stars following our previous investigations (Chavanis
2002a, b). This discussion should clarify the notion of generalized
thermodynamics introduced by Tsallis (1988).
7.1 Collisional vs. collisionless relaxation
Basically, a stellar system is a collection of N stars in
gravitational interaction. Due to the development of encounters
between stars, this system is expected to relax towards a statistical
equilibrium state. The relaxation time can be estimated by
where
is the dynamical time,
i.e. the time it takes a star to cross the system (Binney & Tremaine
1987). For globular clusters,
,
years
so that the relaxation time is of the same order as their age (10^{10} years). Therefore, globular clusters form a discret
Hamiltonian system which can be studied by ordinary methods of
statistical mechanics. In particular, it can be shown rigorously that
the Boltzmann entropy
is the correct form of entropy for
these systems (in a suitable thermodynamic limit) although they are
nonextensive and nonadditive. The absence of entropy maximum in an
unbounded domain simply reflects the natural tendency of a stellar
system to evaporate under the effect of encounters. However,
evaporation is a slow process so that a globular cluster passes by a
succession of quasiequilibrium states corresponding to truncated
isothermals (see Sect. 4). It should be emphasized that
statistical mechanics is based on the postulate that the evolution is
ergodic. Now, the kinetic theory of selfgravitating systems is not
wellunderstood (Kandrup 1981). Indeed, it is possible to prove a
Htheorem for the Boltzmann entropy only if simplifying assumptions
are implemented (Markovian approximation, local approximation,
regularization of logarithmic divergences,...). It would be of
interest to determine more precisely the influence of nonideal
effects on the relaxation of stars. However, these effects are not
expected to induce a strong deviation to Boltzmann's law. In
particular, the diffusion of stars is normal or slightly anomalous,
i.e. polluted by logarithmic corrections (Lee 1968). Hence, Tsallis
entropies, which often arise in the case of anomalous diffusion, do
not seem to be justified in the context of "collisional'' stellar
systems. However, there are many effects that can induce a lack of
ergodicity and a deviation to Boltzmann's law. To a large extent, the
dynamics of selfgravitating systems remains misunderstood and
additional work (using numerical simulations) is necessary to clarify
the process of collisional relaxation.
For elliptical galaxies, on the other hand, the relaxation time due to
close encounters is larger than their age by about ten orders of
magnitude (
,
years, age
years). Therefore, their dynamics is essentially
encounterless and described by the selfconsistent VlasovPoisson
system. Elliptical galaxies form therefore a continuous
Hamiltonian system. Now, due to a complicated mixing process in phase
space, the VlasovPoisson system can achieve a metaequilibrium state
(on a coarsegrained scale) on a very short timescale. A statistical
theory appropriate to this violent relaxation was developed by
LyndenBell in 1967 (see a short review in Chavanis 2002f). For
collisionless stellar systems, the infinite mass problem is solved by
realizing that violent relaxation is incomplete so that the
galaxies are more confined that one would expect on the basis of
statistical considerations.
Collisionless and collisional relaxations are therefore two distinct
processes in the life of a selfgravitating system and they must be
studied by different types of statistical mechanics. Note that a
globular cluster first experiences a collisionless relaxation leading
to a virialized state and then evolves more slowly towards another
equilibrium state due to stellar encounters. We can discuss these two
successive equilibrium states in a more formal manner. Let us consider
a fixed interval of time t and let the number of particles
.
In that case, the system is rigorously
described by the Vlasov equation, and a metaequilibrium state is
achieved on a timescale independant on N as a result of violent
relaxation. Alternatively, if we fix N and let
,
the system is expected to relax to an equilibrium state
resulting from a collisional evolution. These two equilibrium states
are of course physically distinct. This implies that the order of the
limits
and
is not
interchangeable for selfgravitating systems (Chavanis 2002e).
7.2 The VlasovPoisson system
In Sects. 24, we have implicitly
considered the case of collisional stellar systems, such as globular
clusters, relaxing via twobody encounters. We now turn to the case of
collisionless stellar systems, such as elliptical galaxies, described
by the Vlasov equation

(185) 
coupled with the Poisson Eq. (3). The Vlasov Eq. (185) simply states
that, in the absence of encounters, the distribution function f is
conserved by the flow in phase space. This equation conserves the usual invariants (1) and (2) but also
an additional class of invariants called the Casimirs

(186) 
where h is any continuous function of f. This infinite set of
additional constraints is due to the collisionless nature of the
evolution and results from the conservation of f and the
incompressibility of the flow in phase space. As we shall see, they
play a crucial role in the statistical mechanics of violent
relaxation. We note that the conservation of the Casimirs is equivalent to the conservation of the moments

(187) 
It is wellknown that any distribution function of the form
,
where
is the
individual energy of a star, is a stationary solution of the Vlasov
equation. This is a particular case of the Jeans theorem (Binney &
Tremaine 1987). We shall assume furthermore that
so
that high energy stars are less probable than low energy stars. Any
such distribution function can be obtained by extremizing a functional
of the form



(188) 
at fixed mass M and energy E, where C(f) is a convex function,
i.e.
.
Furthermore, maxima of S[f] at fixed mass
and energy are dynamically stable with respect to the VlasovPoisson
system (Ipser 1974; Ipser & Horwitz 1979). This is a sufficient
condition of nonlinear stability. We are tempted to believe that
it is also a necessary condition of nonlinear stability although this
seems to go against the theorem of Doremus et al. (1971) which states
that all distribution functions with
are dynamically
stable (even if they are minima or saddle points of S at fixed Eand M). In fact, this criterion is only a condition of linear
stability (Binney & Tremaine 1987) and it may not be valid for box
confined models (its general applicability has also been
criticized). In the following, we shall consider that
is dynamically stable "in a strong sense'' if and only if it is a
maximum of S[f] at fixed E and M. This is equivalent to
being a minimum of E at fixed S and M (Ipser &
Horwitz 1979). It can be shown that stellar systems satisfying this
requirement are necessarily spherically symmetric.
Introducing appropriate Lagrange multipliers and writing the
variational principle in the form



(189) 
one finds that



(190) 
Since C' is a monotonically increasing function of f, we can inverse this relation to obtain



(191) 
where
F(x)=(C')^{1}(x). From the identity



(192) 
resulting from Eq. (190),
is a monotonically decreasing function of energy if .
We note also that for each stellar systems with
,
there exist a corresponding barotropic gas with the same equilibrium density distribution. Indeed
d
,
d
,
hence .
Writing explicitly the pressure in the form



(193) 
and taking its gradient, we obtain the condition of hydrostatic equilibrium



(194) 
Therefore, the Jeans equations of stellar dynamics reduce in these cases of spherical systems with isotropic pressure to the equation of hydrostatics for a barotropic gas.
7.3 Violent relaxation of collisionless stellar systems
The Vlasov equation does not select a universal form of entropy
S[f], unlike the Boltzmann equation for example. Therefore, the
notion of equilibrium is relatively subtle. When coupled to the
Poisson equation, the Vlasov equation develops very complex filaments
as a result of a mixing process in phase space. In this sense, the
finegrained distribution function
will never
converge towards a stationary solution
.
This is
consistent with the fact that the functionals S[f] are rigorously
conserved by the flow (they are particular Casimirs). However, if we
introduce a coarsegraining procedure, the coarsegrained distribution
function
will relax towards a
metaequilibrium state
.
This
collisionless relaxation is called violent relaxation
(LyndenBell 1967) or chaotic mixing. It can be shown that
calculated with the coarsegrained distribution
function at time t>0 is larger than S[f] calculated with the
finegrained distribution function at t=0 (a monotonic increase of
H is not implied). This is similar to the Htheorem in kinetic
theory except that S[f] is not necessarily the Boltzmann
entropy. For that reason, S[f] is sometimes called a Hfunction
(Tremaine et al. 1986). Boltzmann and Tsallis functionals
and
are particular Hfunctions corresponding to isothermal
stellar systems and stellar polytropes.
It is important to note, furthermore, that among the infinite set of
integrals conserved by the Vlasov equation, some are more robust than
others. For example, the moments
d
d
calculated with the coarsegrained distribution function vary with time for n>1 since
.
By contrast, the mass and the
energy are approximately conserved on the coarsegrained scale. We
shall therefore classify the collisionless invariants in two groups:
the energy E and the mass M will be called robust integrals
since they are conserved by the coarsegrained dynamics while the
moments M_{n} with n> 1 will be called fragile integrals
since they vary under the operation of coarsegraining. Note that, in
practice, the moments M_{n>1} are also altered at the finegrained
scale because, intrinsically, the system is not continuous but made of
stars. Therefore, the graininess of the medium can break the
conservation of some moments, in general those of high order, even if
we are in a regime where the evolution is essentially encounterless.
7.4 LyndenBell's theory of violent relaxation
Now, the question of fundamental interest is the following: given some
initial condition, can we predict the equilibrium distribution
function that the system will eventually achieve? LyndenBell (1967)
tried to answer this question by using statistical mechanics
arguments. He introduced the density probability
of finding the value of the distribution function
in
at equilibrium. Then, using a combinatorial
analysis he showed that the relevant measure is the Boltzmann entropy

(195) 
which is a functional of .
This mixing entropy has been
justified rigorously by Michel & Robert (1994), using the concept of
Young measures. Therefore, although the system is nonextensive, the
Boltzmann entropy is the correct measure. Assuming ergodicity (which
may not be realized in practice, see Sect. 7.5) the most
probable equilibrium state, i.e. most mixed state, is obtained by
maximizing
taking into account all the constraints of the dynamics. The optimal equilibrium state can be written

(196) 
where
accounts for the conservation of the fragile moments
dd
d
and ,
are the usual Lagrange multipliers for M and E (robust integrals). The distinction between these two types of constraints is important (see below) and was not explicitly made by LyndenBell (1967). The "partition function''

(197) 
is determined by the local normalization condition d.
The condition of thermodynamical stability reads



(198) 
for all variation
which does not
change the constraints to first order. The equilibrium coarsegrained
distribution function
d
can
be expressed as

(199) 
Taking the derivative of Eq. (199), it is easy to show that

(200) 
where f_{2} is the centered local variance of the distribution
.
The integrability condition d
requires that the temperature is positive
(). Then, from Eq. (200), we deduce that f is a
monotonically decreasing function of .
Therefore, from
Sect. 7.2, we know that for each function
of the
form (199) there exists a Hfunction
whose
extremization at fixed mass and energy returns Eq. (199).
Furthermore, it can be shown that the condition (198) of
thermodynamical stability (maximum of
at fixed E and M_{n}) implies that
is a nonlinearly stable solution
of the VlasovPoisson system. Therefore, it is a maximum of
at fixed mass and energy. This functional depends
on the initial conditions through the Lagrange multipliers
,
or equivalently through the function
,
which are determined by the fragile moments M_{n>1}. Therefore,
is nonuniversal and can take a wide variety
of forms. In general,
differs from the ordinary
Boltzmann entropy
derived in the context of
collisional stellar systems. This is due to the additional constraints
(Casimir invariants) brought by the Vlasov equation. For a given
initial condition, the Hfunction
maximized by the
system at statistical equilibrium is uniquely determined by the
statistical theory according to Eqs. (190) and (199).
In general,
is not an entropy, unlike ,
as
it cannot be directly obtained from a combinatorial analysis. There
is a case, however, where
is an entropy. If the
initial condition is made of patches with value
and
f_{0}=0 (twolevels approximation), then the entropy
is
equivalent to the FermiDirac entropy
d
d
and LyndenBell's distribution (199) coincides with the FermiDirac distribution (LyndenBell
1967; Chavanis & Sommeria 1998). Moreover, all the moments can be
expressed in terms of the mass alone as
.
In the
nondegenerate (or dilute) limit
,
the
FermiDirac entropy reduces to the Boltzmann entropy
d
d
.
Quite generally, in the dilute
limit, the equilibrium distribution is a sum of isothermal distributions.
We can note some general predictions of the statistical theory, which
are valid for any initial condition
.
(i) The
predicted distribution function is a function
of the stellar energy alone in the
nonrotating case and a function
of the
Jacobi energy
alone in the rotating case. (ii)
so that
high energy stars are less probable than low energy stars. (iii)
is strictly positive for all energies
(it never vanishes). (iv)
is always smaller
than the maximum value
of the initial condition. (v)
According to Eqs. (200) and (192), we have the general
relation
.
(vi) For a boxconfined
system, there always exist a global maximum of
entropy. Therefore, contrary to collisional stellar systems, it is not
necessary to introduce a smallscale cutoff to make the
thermodynamical approach rigorous. This is due to the Liouville
theorem which puts an upper bound on the value of the distribution
function like the Pauli exclusion principle in quantum mechanics (see
Chavanis et al. 1996; Chavanis & Sommeria 1998).
7.5 Incomplete relaxation
The prediction of LyndenBell's statistical theory crucially depends
on the initial conditions. This is because we need to know the value
of the moments M_{n>1} which can only be deduced from the
finegrained distribution function at t=0 (say). Indeed, these
moments are fragile, or microscopic, constraints which cannot be
mesured from the coarsegrained flow at t>0 since they are altered
by the coarsegraining procedure. This is a specificity of continuous
Hamiltonian systems which contrasts with ordinary (discrete)
Hamiltonian systems for which the constraints are robust, or
macroscopic, and can be evaluated at any time (as the mass M and
the energy E). Now, LyndenBell (1967) gives arguments according to
which elliptical galaxies should be nondegenerate (except possibly in
the nucleus). In that case, the dilute limit of his theory applies in
the main body of the galaxy and leads to an isothermal
distribution. The theoretical justification of an isothermal
distribution for collisionless stellar systems was considered as a
triumph in the 1960's. Indeed, LyndenBell's statistical theory of
violent relaxation could explain the observed isothermal core of
elliptical galaxies without recourse to collisions that operate on a
much longer timescale.
However, it is easy to see that the statistical theory of violent
relaxation cannot describe the halo of elliptical galaxies. Indeed, at
large distances, the density decreases as r^{2} resulting in the
infinite mass problem. LyndenBell (1967) has related this
mathematical difficulty to the physical problem of incomplete
relaxation and argued that his distribution function has to be
modified at high energies. Indeed, the relaxation is effective only in
a finite region of space and persists only for a finite period of
time. Therefore, there is no reason to maximize entropy in the whole
available space. The ergodic hypothesis which sustains the statistical
theory applies only in a restricted domain of space, in a sort of
"maximum entropy bubble'', surrounded by an unmixed region which is
only poorly sampled by the system. Accordingly, the physical picture
that emerges is the following: during violent relaxation, the system
has the tendency to reach the most mixed state described by the
distribution (199). However, as it approaches equilibrium, the
mixing becomes less and less efficient and the system settles on a
state which is not the most mixed state.
How can we take into account incomplete relaxation? A first
possibility is to confine the system artificially inside a box, where
the box delimits the typical region in which the statistical theory
applies (Chavanis & Sommeria 1998). Another possibility is to use a
parametrization of the coarsegrained dynamics of collisionless
stellar systems in the form of relaxation equations involving a space
and time dependant diffusion coefficient related to the finegrained
fluctuations of the distribution function (Chavanis et al. 1996;
Chavanis 1998). This space and time dependant diffusion coefficient
can freeze the system in a "maximum entropy bubble'' and lead
to the formation of selfconfined stellar systems. Finally, a third
possibility is to construct a model of incomplete violent relaxation
which provides a depletion of stars in the high energy tail of the
distribution function. If the system is isolated, as most elliptical
galaxies are, Hjorth & Madsen (1993) have proposed a dynamical
scenario attempting to take into account incomplete relaxation in the
halo. They introduce a twostep process: (i) in a first step, they
assume that violent relaxation proceeds to completion in a finite spatial region, of radius ,
which represents roughly
the core of the galaxy (where the fluctuations are important). At this
stage, the escape energy represents no special threshold so that
negative as well as positive energy states are populated in that
region. (ii) After the relaxation process is over, positive energy
particles leave the system and particles with
move in orbits beyond ,
thereby
changing the distribution function to something significantly
"thinner'' than a Boltzmann distribution. The crucial point to realize
is that the differential energy distribution
,
where
d
is the number of stars with energy between
and
,
will be discontinuous at the
escape energy
since there is a finite number of particles
within
after the relaxation process. It can be shown that
this discontinuity implies necessarily that
for
(Jaffe
1987). Therefore, the distribution function consistent with this
scenario is (Hjorth & Madsen 1993)



(201) 
This model corresponds formally to a composite configuration with an
isothermal core in which violent relaxation is efficient and a
polytropic halo (with index n=4) which is only partially relaxed (in
reality, there is a small deviation to the ideal value n=4 because
the potential created by the core is not exactly Keplerian; see Hjorth
& Madsen 1993). For this model, the density decreases as r^{4} like
in elliptical galaxies. Furthermore, this model can reproduce de
Vaucouleur's R^{1/4} law for the surface brightness of ellipticals
(Binney & Tremaine 1987). Alternatively, if the system is subject to
tidal forces (this may be the case for dark matter halos), one
can propose an extended MichieKing model of the form (Chavanis 1998)



(202) 
which takes into account the specificities of LyndenBell's distribution
function (in particular the exclusion principle associated with
Liouville's theorem) and leads to a confinement of the density
at large distances (i.e. the density drops to zero at a finite
distance identified with the tidal radius).
7.6 Tsallis entropies
The statistical theory of LyndenBell encounters some problems because
the distribution function
never vanishes. Therefore, we
have to invoke incomplete relaxation and introduce dynamical
constraints in order to explain the observation of confined galaxies
and solve the infinite mass problem. This is probably the correct
approach to the problem. Another line of thought, defended by Tsallis
and coworkers, is to change the form of entropy in order to obtain a
distribution function that vanishes at some maximum energy. Therefore,
Tsallis generalized thermodynamics is essentially an attempt to take
into account incomplete mixing and lack of ergodicity in complicated
systems. If we assume that the system is classically described by the
Boltzmann entropy
(which is true for collisionless stellar
systems only in a coarsegrained sense and in a dilute limit), then
Tsallis proposes to replace
by a qentropy
S_{q}[f]. The parameter q measures the efficiency of mixing (q=1if the system mixes well) but is not determined by the theory.
This approach is attractive but the justification given by Tsallis and
coworkers is misleading. Fundamentally, ordinary thermodynamics (in
the sense of LyndenBell ) is the correct approach to the problem.
The Boltzmann entropy
has a clear physical meaning as it is
proportional to the logarithm of the disorder, where the disorder is
equal to the number of microstates consistent with a given
macrostate. Of course, if the system does not mix well, statistical
mechanics loses its power of prediction. However, the state resulting
from incomplete violent relaxation is always a nonlinearly stable
stationary solution of the Vlasov equation on a coarsegrained scale (i.e., for
). Therefore, if
,
it
maximizes a Hfunction
d
d
at fixed mass M and energy
E. The function C(f) is influenced
by thermodynamics (mixing) but it cannot be determined by pure
thermodynamical arguments because of incomplete relaxation.
The Hfunction S[f] selected by the system depends both on the
initial condition and on the type of mixing. Tsallis entropy
S_{q}[f] is just a particular Hfunction leading to confined
structures (stellar polytropes). Now, it is wellknown that stellar
polytropes give a poor fit of elliptical galaxies. In particular, they
cannot reproduce de Vaucouleur's law, neither the r^{4} density
decrease of these objects. Indeed, for stellar polytropes, the density
decreases as r^{5} for n=5, the density drops to zero at a finite
distance for n<5 and the mass is infinite for n>5. We conclude,
therefore, that observations of elliptical galaxies do not support the
prediction of Tsallis. The model of Hjorth & Madsen (1993) based on
the statistical theory of LyndenBell improved by a physically
motivated scenario of incomplete relaxation provides a much better
agreement with observations. This equilibrium state cannot be
described by Tsallis distribution which fails to reproduce
simultaneously an isothermal core and an external confinement. In
fact, the model of Hjorth & Madsen (1993) can be obtained by
maximizing a Hfunction with



(203) 
at fixed mass and energy. Therefore, their model is consistent with a
composite Tsallis model with q=1 (complete mixing) in the core
and
(incomplete mixing) in the halo. More
generally, we could argue that q varies in space in order to take
into account a variable mixing efficiency (very similarly, stars are
sometimes approximated by composite polytropic models with a space
dependent index in order to distinguish between convective and
radiative regions). However, this gives to Tsallis distribution a
practical, not a fundamental justification. It can just fit
locally the distribution function of collisionless stellar systems
resulting from incomplete violent relaxation.
In addition, not all stable stationary solutions of the Vlasov equation can be
obtained by maximizing a Hfunction at fixed mass and energy. This
maximization problem leads to
with
.
For spherical stellar systems, the
distribution function can depend on the specific angular momentum
in addition to the stellar energy
and this is what happens in certain models of incomplete
violent relaxation with anisotropic velocity distribution (LyndenBell
1967). On the other hand, for rotating stellar systems, the
maximization of S[f] at fixed mass, energy and angular momentum
predicts that
with f'<0, where
is
the Jacobi energy. However, it is wellknown that such distribution
function cannot account for the triaxial structure of elliptical
galaxies. More general distribution functions must be constructed in
agreement with the Jeans theorem (Binney & Tremaine 1987). However,
it is possible that relevant distribution functions maximize a
Hfunction with additional constraints. These constraints may
correspond to adiabatic invariants that are approximately conserved
during violent relaxation. For example, anisotropic velocity
distributions can be obtained by maximizing a Hfunction while
conserving the individual distributions of angular momentum
d
d
(or
other distribution) in addition to mass and energy. This leads to
with
and
(
is
called the anisotropy radius). It is expected that this maximization
problem is a condition of nonlinear stability. For rotating systems, it
is not known whether we can construct distribution functions yielding
triaxial bodies by maximizing a Hfunction at fixed M, E,
and additional (adiabatic) invariants. These ideas may be a route
to explore.
7.7 Dynamical stability of collisionless stellar systems
On a formal point of view, the maximization problem determining the
nonlinear stability of a collisionless stellar system is similar
to the one which determines the thermodynamical stability of a
collisional stellar system. In thermodynamics, we maximize the
Boltzmann entropy
at fixed mass M and energy E in order
to determine the most probable equilibrium state. In stellar dynamics,
the maximization of a Hfunction S[f] at fixed mass and energy
determines a nonlinearly stable stationary solution of the
VlasovPoisson system. Therefore, we can use a thermodynamical
analogy to analyse the dynamical stability of collisionless stellar
systems. In this analogy, the functional S[f] plays the role of a
generalized entropy, the Lagrange multiplier
plays the role of
a generalized inverse temperature ,
the curve the role of a generalized caloric curve and the maximization of S[f]at fixed E and M corresponds to a microcanonicial description.
This analogy explains why we take C to be convex (and not concave)
in Eq. (188). Then, we must look for maxima of S (not minima)
like in thermodynamics. We can also introduce a Jfunction defined
by
which is similar to a free energy in
thermodynamics (J is the Legendre transform of S). The
maximization of J[f] at fixed M and
corresponds to a
canonical description.
The condition that f is a maximum of
S at fixed mass and energy is equivalent to the
condition that
is negative for all perturbations that
conserve mass and energy to first order. This condition can be written



(204) 
The condition that f is a maximum of
at fixed mass and temperature is equivalent to the condition that
is negative for all perturbations that
conserve mass. This can be written



(205) 
According to Eq. (192),
can be rewritten



(206) 
which is the usual form considered in the literature. We emphasize,
however, that the role of the first order constraints is crucial in
the stability analysis and is often underestimated. If f is a
maximum of J at fixed M and
then it is necessarily a
maximum of S at fixed M and E. Indeed, if the inequality (205) is satisfied for all perturbations that conserve mass, it
is a fortiori satisfied for perturbations that conserve mass and
energy. However, the reciprocal is wrong in general. In
thermodynamics, this means that canonical stability implies
microcanonical stability (but the converse is wrong in
general). Exploiting this thermodynamical analogy, we can use the
turning point arguments of Katz (1978) to settle whether a solution of
the form (190) is a maximum or a minimum (or a saddle point) of
S or J. The canonical and microcanonical descriptions will be
inequivalent if the curve
presents turning points leading
to regions where
dT<0. This corresponds to regions of negative
specific heats in thermodynamics. In these regions, the curve S(E)has a convex dip. This convex dip is the signal of phase transitions
in thermodynamics.
We can also try to reduce the stability problem to the study of an
eigenvalue equation. To that purpose, it is convenient to first
maximize S[f] at fixed mass, energy and density
in order to obtain a functional .
Now, maximizing S[f]at fixed mass, energy and density is equivalent to maximizing S[f]at fixed kinetic energy K and density. Introducing a Lagrange
multiplier
for the kinetic energy and a space dependant
Lagrange multiplier
for the density constraint, we get



(207) 
where F is the same function as in Sect. 7.2. Since



(208) 
we conclude that Eq. (207) is a maximum of S at fixed
K and .
This is also a maximum of
at fixed
and .
With the distribution function (207), we can write the density and the pressure as



(209) 
where



(210) 
and



(211) 
We can now express E and S as functionals of
and .
The energy (2) is simply given by



(212) 
On the other hand, according to Eq. (188), we have



(213) 
Setting
,
we get



(214) 
Integrating by parts and using
,
we find that



(215) 
Integrating by parts one more time and using Eqs. (209)(211), we finally obtain



(216) 
We now consider the maximization of
at fixed mass and
energy. This is similar to a microcanonical situation in
thermodynamics. In that case,
is not fixed and its variation
is related to the change of density
via the
energy constraint (212), using Eqs. (209)(211). This implies in particular that the barotropic relation
,
valid at equilibrium, is not preserved when we
consider variations around equilibrium. Indeed, eliminating
in Eq. (209), we find that
so that
enters explicitly
in the relation between p and .
In the case of isothermal
stellar systems and stellar polytropes described by Boltzmann and
Tsallis entropies, it is possible to reduce the stability problem to
the study of an eigenvalue equation (Padmanabhan 1989; Taruya &
Sakagami 2002a). However, it seem difficult to make this reduction in the
general case.
7.8 Dynamical stability of gaseous spheres
We now consider the maximization of
at
fixed
and M. This is similar to a canonical situation in
thermodynamics. According to Eqs. (212) and (216), we
have



(217) 
Since
is constant, the barotropic relation
remains
valid around equilibrium. Using the relation
h'(x)=g(x) obtained
from Eqs. (210) and (211) by a simple integration
by parts, it is easy to check that Eq. (209) yields



(218) 
Therefore, the functional (217) can be
rewritten as



(219) 
If we cancel the first order variations of J at fixed mass and ,
we get
and we recover Eq. (191). On the other hand, the condition that Eq. (191) is a maximum of J at fixed M and
requires that the second order variations of J be negative. This leads to the inequality



(220) 
for all perturbations
that conserve mass. Using the condition of hydrostatic equilibrium,
can be rewritten



(221) 
We note, in passing, the similarity with the pseudoenergy (or Arnol'd
invariant) in twodimensional hydrodynamics (see Chavanis
2002h). Therefore, Arnol'd theorems can be directly extended to (3D)
selfgravitating bodies. In particular, in a bounded domain, the
system is nonlinearly stable, for any type of perturbation, if
,
where
is the
smallest eigenvalue of
(for isothermal spheres this
corresponds to
and for polytropic spheres to
where
is defined by Eqs. (15)
and (104) respectively). Furthermore, we have already shown
in the case of isothermal stellar systems and stellar polytropes that
the stability condition (220) can be reduced to the study of an
eigenvalue equation. This reduction can be generalized to any stellar
system described by a Hfunction S[f] of the form (188). Considering spherically symmetrical perturbations,
introducing the variable q(r) defined by Eq. (122) and repeating exactly the same steps as for isothermal and polytropic
spheres (see Chavanis 2002a and Sect. 5.5), we end up on
the eigenvalue problem



(222) 
with appropritate boundary conditions (see Appendix C).
If all the 's are negative, then Eq. (191) is a maximum of
J at fixed M and .
We now consider the stability of a gaseous system with an equation of
state .
The energy of this gaseous configuration is
(LyndenBell & Sanitt 1969)



(223) 
We observe that the Jfunction (219) associated with a
stellar system is equal (up to a negative proportionality factor
)
to the energy (223) of the corresponding
barotropic gas. The stability theorem of Chandrasekhar for gaseous
spheres requires that
(see Binney &
Tremaine 1987). This leads to the inequality (220) and to the
eigenvalue Eq. (222). We now consider the dynamical
stability of a selfgravitating gaseous system with respect to the
NavierStokes or Euler equations (Jeans problem). This problem can be
solved explicitly for isothermal and polytropic spheres (Chavanis
2002a,b). Adapting the same procedure for a general barotropic
equation of state and denoting by
the growth rate of the
perturbation (such that
etc.), we
obtain the eigenvalue equation



(224) 
with appropriate boundary conditions (see Appendix C).
This equation is equivalent to the Eddington (1926) equation of
pulsation but it proved to be more convenient in our previous
investigations. We observe that Eqs. (222) and (224) are
similar and that they coincide for marginal stability. This first
demonstates the equivalence between the stability criterion of
Eddington based on the equation of pulsation (224) and the
stability criterion of Chandrasekhar based on the minimization of the
energy (223). This also shows the equivalence between
the maximization of the Jfunction for a stellar system at fixed
and M and the dynamical stability (with respect to the
JeansEuler equations) of the corresponding barotropic star. Note
that we have only studied the dynamical stability of barotropic stars
with respect to small perturbations. However, we conjecture that the
maximization of J at fixed M and
is a condition of nonlinear dynamical stability for the JeansEuler equation. The
proof is expected to be similar to the corresponding one for the
Vlasov equation.
7.9 A new interpretation of Antonov's first law
We can now combine the previous results in order to provide a new
interpretation of Antonov's first law (Binney & Tremaine 1987) based
on a thermodynamical analogy. We have seen in Sect. 7.7 that
a stellar system with
and
is
dynamically stable (nonlinearly) with respect to collisionless
perturbations if f is a maximum of a Hfunction S[f] at fixed
E and M. According to the discussion following Eqs. (204)
and (205), a sufficient condition of dynamical stability
is that f is a maximum of the Jfunction J[f] at fixed and M. Now, according to Sect. 7.8, this is equivalent to
the dynamical stability of the corresponding barotropic
gas. Therefore, we come to the conclusion that "a stellar system is
dynamically stable whenever the corresponding barotropic star is
stable'', which is Antonov's first law. Owing to the thermodynamical
analogy, Antonov's first law is similar to the property that
"canonical stability implies microcanonical stability'' in
thermodynamics. We note that the reciprocal of Antonov's first law is
not true in general. If a stellar system is stable this does not
necessarily imply that the corresponding barotropic gas is
stable. Similarly, in thermodynamics, microcanonical stability does
not necessarily imply canonical stability. This is the case only if
the statistical ensembles are equivalent. We know, however, that this
equivalence is broken for selfgravitating systems.
Let us illustrate these results in the case of polytropes. Taruya &
Sakagami (2002a, b) have investigated the
stability of stellar systems in the framework of Tsallis
generalized thermodynamics (see also Sect. 5.2). Due to the
thermodynamical analogy discussed in Sects. 7.7 and 7.8, the study of Taruya & Sakagami can be used to
determine the dynamical stability of collisionless stellar
polytropes and polytropic gaseous spheres. In the generalized
thermodynamical approach, it is found that the statistical ensembles
are inequivalent. This implies that the dynamical stability of stellar
polytropes and gaseous polytropes will also be inequivalent (Chavanis
2002b). Selfgravitating systems described by Tsallis entropy are
thermodynamically stable in the microcanonical ensemble if n<5 or if
the density contrast is sufficiently low for n>5. In that case, they
are maxima of S_{q}[f] at fixed M and E. According to
Sect. 7.7, this implies that stellar polytropes are
dynamically stable solutions of the Vlasov equation for n<5 (and if
the density contrast is sufficiently low for n>5). On the other
hand, selfgravitating systems described by Tsallis entropy are
thermodynamically stable in the canonical ensemble if n<3 or if the
density contrast is sufficiently low for n>3. In that case, they
are maxima of J_{q}[f] at fixed M and .
According to
Sect. 7.8, this implies that polytropic gaseous spheres are
dynamically stable solutions of the JeansEuler equations if, and
only, if n<3 (or if, and only, if the density contrast is
sufficiently low for n>3). We note, in particular, that gaseous
polytropes with 3<n<5 are unstable for sufficiently high density
contrasts (e.g., complete polytropes) while corresponding stellar
polytropes are stable. This shows that the reciprocal of Antonov's
first law is wrong for polytropes. Due to the thermodynamical analogy,
this can be related to an inequivalence of statistical ensembles in a
region of negative specific heats (Chavanis 2002b). Our present study
shows that the results obtained for isothermal stellar systems and
stellar polytropes can be generalized to any Hfunction.
8 Conclusion
We have performed an exhaustive study of the thermodynamics of
selfgravitating systems in various ensembles. This paper completes
previous investigations on the subject and all ensembles have now been
treated. Contrary to ordinary (extensive) systems, we have to perform
a specific study in each ensemble since the stability limits differ
from one to the other. Remarkably, the thermodynamical stability
problem can be studied analytically or with simple graphical
constructions. The dynamical stability of isothermal gaseous spheres
can also be studied analytically both in Newtonian mechanics for the
EulerJeans equations (Chavanis 2002a) and in general relativity for
the Einstein equations (Chavanis 2002d).
These results can be of relevance both for astrophysicists and
statistical mechanicians and could make a bridge between these two
communities. On an astrophysical point of view, they show that we
must be careful to precisely define the ensemble in which we work
since they are not equivalent. This does not affect the structure of
the equilibrium configuration but it may affect its stability. On a
physical point of view, this study fills an important gap in the
statisical mechanics literature since the case of selfgravitating
systems is not discussed at all in standard textbooks of statistical
mechanics and thermodynamics.
We have also discussed the relevance of Tsallis generalized
thermodynamics for stellar systems. Collisionless stellar systems such
as elliptical galaxies can achieve a metaequilibrium state as a result
of a violent relaxation. The Boltzmann entropy
is the correct entropy for these systems (in a coarsegrained sense
and assuming that the system is nondegenerate). It measures the
disorder where the disorder is equal to the number of microstates
consistent with a given macrostate. However,
is
not maximized by the system because of incomplete relaxation (this is
independant on the fact that
has no
maximum!). In any case, the state resulting from incomplete violent
relaxation is a nonlinearly stable solution of the Vlasov equation (on
a coarsegrained scale). If
,
it maximizes a
Hfunction
d
d
,
where C(f) is
a convex function, at fixed mass and energy. Therefore, we can use a
thermodynamical analogy to study the dynamical stability of
collisionless stellar systems. Tsallis entropies S_{q}[f] are a
particular class of Hfunctions leading to stellar polytropes. Stellar
polytropes do not give a good fit of elliptical galaxies. A better
model of incomplete violent relaxation, motivated by precise physical
arguments, consists of an isothermal core and a polytropic halo with
index
(Hjorth
& Madsen 1993). This can be fitted by a composite
Tsallis (polytropic) model with q=1 in the core (complete mixing)
and
in the envelope (incomplete mixing). More generally,
the state resulting from incomplete violent relaxation does not
necessarily maximize a Hfunction at fixed mass and energy. Due to the
Jeans theorem, the distribution function can depend on other integrals
than the stellar energy or the Jacobi energy. It is possible,
nevertheless, that relevant distribution functions arising from
incomplete violent relaxation maximize a Hfunction at fixed mass,
energy, angular momentum and additional (adiabatic)
invariants.
Acknowledgements
I am grateful to J. Katz and T. Padmanabhan
for stimulating discussions and encouragement.
Appendix A: Some useful identities for isothermal spheres
Using the Emden equation (14), we have



(A.1) 
which establishes Eq. (67). On the other hand, using an integration by
parts, we obtain
Combining with Eq. (A.1) and introducing the Milne variables, we establish Eq. (68). To establish Eq. (69), we start from the identity



(A.3) 
which results from a simple integration by parts. Thus,



(A.4) 
where we have used the Emden Eq. (14). Combining this
relation with Eq. (A.2), we obtain



(A.5) 
Now,
From Eqs. (A.5) and (A.6) results Eq. (69).
Appendix B: Some useful identities for polytropic spheres
Using integrations by parts similar to those performed in Appendix A, we can easily establish the identities
Appendix C: Equation of pulsations and boundary conditions
In this paper and previous ones, we have studied the dynamical stability of barotropic gaseous spheres with the equation of pulsation



(C.1) 
where
is the growth rate of the perturbation (such that
,
etc.) and
dr represent the mass perturbation within the sphere of radius r which is related to the velocity by (Chavanis 2002a)



(C.2) 
Obviously q(0)=0 and more precisely
for
.
If the system is confined within a sphere of fixed radius R, the
conservation of mass imposes q(R)=0. This can also be obtained from
Eq. (C.2) if we set
.
If the system is free and
the density vanishes at its surface, i.e. ,
like for a
complete polytrope, then Eq. (C.2) again implies that q(R)=0(Chavanis 2002b). If, finally, the gaseous sphere supports a fixed external
pressure (isobaric ensemble) the boundary condition requires that the
Lagrangian derivative of the pressure vanishes at the surface of the
configuration, i.e.



(C.3) 
Using dd
and Eqs. (C.2) and (122),
we get for an arbitrary barotropic equation of state



(C.4) 
If we set
d
and introduce the
Lagrangian displacement
,
we can easily check that Eq. (C.1) is equivalent to the
Eddington (1926) equation of pulsations



(C.5) 
where
dd.
The boundary condition at the origin is .
On the other hand, if the system is enclosed within a box .
Finally, if the pressure at the surface of the sphere is constant (possibly zero), the boundary condition (C.4) becomes



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