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- 3 octobre 2006

**Two-dimensional flows with high Reynolds numbers have the striking
property of organizing spontaneously into large scale coherent vortices.
These vortices are common features of geophysical and astrophysical flows
with the well-known example of Jupiter’s Great Red Spot, a huge vortex
persisting for more than three centuries in a turbulent shear between two
zonal jets. Similarly, it is striking to observe that galaxies themselves
follow a kind of organization revealed in the Hubble classification or
in de Vaucouleurs R^ 1/4 law for the surface brightness of Ellipticals
(NGC3379). **

**The global organization of these systems can be understood in terms
of relatively similar statistical mechanics. This was first investigated
by Lynden-Bell [1] in his theory of ``violent relaxation’’ applying to
collisionless stellar systems and rediscovered more recently by Kuz’min
[2], Miller [3] and Robert & Sommeria [4] in the context of two-dimensional
turbulence. A firm mathematical justification of these ideas has been provided
by Robert [5] in terms of Young measures. The formal analogy between galaxies
and two-dimensional vortices resides in the similar morphology of the Vlasov
and the Euler equations (Chavanis [6][7] ; Chavanis et al. [8]). These equations
develop a kind of ``mixing process’’ associated with the damped oscillations
of a protogalaxy initially far from mechanical equilibrium or with the
nonlinear development of the Kelvin-Helmholtz instability in hydrodynamics.
The effectiveness of mixing is measured by an entropy functional which
tends to be maximum at equilibrium. However, complete mixing, which would
give rise to structureless objects, is forbidden by the existence of global
constraints like the conservation of energy. Therefore, the star density
or the vorticity profiles are obtained by maximizing the entropy while
accounting for all the constraints of the Vlasov or the Euler equation.
In this point of view, galaxies and two-dimensional vortices can be considered
as sort of equilibrium states (in a thermodynamical sense) following a
law of chaos : there is a total lack of information at small scales, but
the exciting phenomenon is that microscopic disorder leads to macroscopic
order. This description, of course, requires a hypothesis of ergodicity
which may not be completely fulfilled in practice. The statistical approach
provides, however, a systematic and valuable framework to tackle the problem
of self-organization in two-dimensional turbulence and for stellar systems.
**

**Because of the long range nature and the singularity at short distances
of the gravitational potential, the statistical mechanics of self-gravitating
systems makes problem and the notion of equilibrium is not always well-defined
[9]. First of all, it is easy to show that there is no entropy maximum
if the system is not bounded : its mass would be infinite ! In addition,
even if truncated models are introduced or if the system is artificially
confined to a box, an equilibrium state does not always exist. Below a
minimum energy (or above a maximum radius), there is no critical point of entropy [10] : the system takes a ``core-halo’’ structure and can always increase entropy
by making the core denser and denser and hotter and hotter. This ``gravothermal
catastrophe’’ is probably related to the negative heat capacities of self-gravitating
systems [11]. A priori, the collapse continues up to the formation of a
central singularity. Different dynamical models show that the central density
becomes infinite in a finite time [12][13][14]. This singularity has been
known as ``core collapse’’ and many globular clusters have probably experienced
core collapse. In practice, the formation of hard binaries can release
sufficient energy to stop the collapse and drive a reexpansion of the system
[15]. It has been suggested that a series of oscillations (contraction
and reexpansion) should follow [16]. Since collisionless stellar system
can reach an isothermal distribution during the process of violent relaxation
[1], it is possible that they also undergo the gravothermal catastrophe.
However, the core will cease to shrink when it becomes degenerate in Lynden-Bell’s
sense. The system will then have a core-halo structure which is a global
entropy maximum. This prediction has been fully confirmed by the calculation
of Fermi-Dirac spheres by Chavanis \& Sommeria [17]. **

**While stellar systems exhibit a ``core-halo’’ structure with high
central densities, two-dimensional vortices have a smoother vorticity profile.
However, unlike the stellar case, vorticity can take positive and negative
values leading to a wider variety of structures such as monopoles, rotating
or translating dipoles, tripoles... In a limit of strong mixing, the maximization
of entropy becomes equivalent to a kind of enstrophy minimization and it
is possible to obtain a nice classification of the whole ``zoology’’ of
coherent vortices usually met in two-dimensional flows [18][19]. Isolated
vortices are justified as restricted equilibrium states, introducing the
concept of a ``maximum entropy bubble’’ as a heuristic attempt to take
into account ``incomplete relaxation’’. Incomplete relaxation is also necessary
in the case of stellar systems to avoid the ``infinite mass’’ problem.
**

**To understand more precisely why relaxation does not proceed to completion,
we must ressort to kinetic theory and non equilibrium thermodynamics. The
route to equilibrium can be described simply in terms of convection-diffusion
equations obtained from a general Maximum Entropy Production Principle
[20][8][21][22]. It appears that the optimal diffusion current is the sum
of two terms : a pure diffusion compensated by an appropriate friction (for
stellar systems) or a drift (for two-dimensional vortices) necessary to
conserve energy. The diffusion coefficient is related to small-scale fluctuations
and can freeze the system in a ``bubble’’ when these fluctuations die away
(providing a firmer basis for incomplete relaxation) [23][8]. Finally,
a kind of Einstein relation connects the diffusion coefficient to the friction
or to the drift, indicating some link with the theory of Brownian motion
and Fokker-Planck equations [24][25]. This link is particularly apparent
when we consider the more conventional statistical mechanics of Hamiltonian
systems made of point vortices [26][27] or point mass stars [28] instead
of continuous distributions. **

**The process of diffusion in two-dimensional turbulence can be analyzed
in a completely stochastic framework by studying the statistics of the
velocity field arising from a random distribution of point vortices [29].
This problem is intimately related to the statistics of the gravitational
field produced by a random distribution of stars [24] and this is another
manifestation of the analogy between these two systems. It is possible
to derive exact results concerning the probability density function (p.d.f)
of the velocity fluctuations, their typical duration and their spatial
correlations. The velocity distribution presents a behavior which is intermediate
between Gaussian and Levy laws. For that reason, the results are polluted
by logarithmic corrections. As an application of these ideas, we can obtain
an estimate for the diffusion coefficient of point vortices. When applied
to the context of freely decaying turbulence, the diffusion becomes anomalous
and we can establish a relation nu=1+xi/2 between the coefficient of anomalous
diffusion nu and the exponent xi which characterizes the decay of the vortex
density. We have proposed that xi=1, yielding nu=3/2, in the scaling regime
of the decay. These results are in agreement with numerical simulations
of ``punctuated Hamiltonain dynamics of point vortices’’ performed with
a renormalization group procedure [30].**Proceedings

**References :**

**[1] D. Lynden-Bell : Mon. Not. R. astr. Soc. 136 (1967) 101. **

**[2] G.A Kuz’min in Structural Turbulence, Acad. Naouk CCCP Novosibirsk,
Institute of Thermophysics, Ed. Goldshtik M.A (1982) 103. **

**[3] J. Miller : Phys. Rev. Lett. 65 (1990) 2137.**

**[4] R. Robert & J. Sommeria : J. Fluid. Mech. 229 (1991) 291.
**

**[5] R. Robert : J. Stat. Phys. 65 (1991) 531. **

**[6] P.H. Chavanis : Contribution a la mecanique statistique des tourbillons
bidimensionnels. Analogie avec la relaxation violente des systemes stellaires.
PhD Thesis (1996). Ecole Normale Superieure de Lyon. **

**[7] P.H. Chavanis in Nonlinear Dynamics and Chaos in Astrophysics :
A Festschrift in Honor of George Contopoulos. Annals of the New York Academy
of Sciences 867 (1998) 120. **

**[8] P.H. Chavanis, J. Sommeria & R. Robert : Astrophys. J. 471
(1996) 385. **

**[9] T. Padmanabhan : Phys. Rep. 188 (1990) 285. **

**[10] V.A. Antonov : Vest. Leningr. Gos. Univ. 7 (1962) 135. **

**[11] D. Lynden-Bell & R. Wood : Mon. Not. R. astr. Soc. 138 (1968)
495. **

**[12] R.B. Larson : Mon. Not. R. astr. Soc. 147 (1970) 323. **

**[13] H. Cohn : Astrophys. J. 242 (1980) 765. **

**[14] D. Lynden-Bell & P.P. Eggleton : Mon. Not. R. astr. Soc.
191 (1980) 483. **

**[15] S. Inagaki & D. Lynden-Bell : Mon. Not. R. astr. Soc. 205
(1983) 913. **

**[16] E. Bettwieser & D. Sugimoto : Mon. Not. R. astr. Soc. 208
(1984) 493. **

**[17] P.H. Chavanis & J. Sommeria : Mon. Not. R. astr. Soc. 296
(1998) 569. **

**[18] P.H. Chavanis & J. Sommeria : J. Fluid. Mech. 314 (1996)
267. **

**[19] P.H. Chavanis & J. Sommeria : J. Fluid. Mech. 356 (1998)
259. **

**[20] R. Robert & J. Sommeria : Phys. Rev. Letters 69 (1992) 2776.
**

**[21] P.H. Chavanis & J. Sommeria : Phys. Rev. Letters 78 (1997)
3302. **

**[22] P.H. Chavanis : Mon. Not. R. astr. Soc. 300 (1998) 981. **

**[23] R. Robert & C. Rosier : J. Stat. Phys. 86 (1997) 481-515.
**

**[24] S. Chandrasekhar : Rev. Mod. Phys. 15 (1943) 1. **

**[25] P.H. Chavanis : Phys. Rev. E 58 (1998) R1199. **

**[26] L. Onsager : Nuovo Cimento Supp. 6 (1949) 279. **

**[27] G. Joyce & D. Montgomery : J. Plasma Phys. 10 (1973) 107.
**

**[28] S. Chandrasekhar, Principles of stellar dynamics (Dover, 1943).
**

**[29] P.H. Chavanis & C. Sire : Phys. Rev. E 62 (2000) 490 **

**[29] C. Sire & P.H. Chavanis : Phys. Rev. E 61 (2000) 6644 **