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We study the nature of phase transitions in a self-gravitating classical gas in the presence of a central body. The central body can mimic a black hole at the center of a galaxy or a rocky core (protoplanet) in the context of planetary formation. In the chemotaxis of bacterial populations, sharing formal analogies with self-gravitating systems, the central body can be a supply of ``food'' (chemoattractant). We consider both microcanonical (fixed energy) and canonical (fixed temperature) descriptions and study the inequivalence of statistical ensembles. At high energies (resp. high temperatures), the system is in a ``gaseous'' phase and at low energies (resp. low temperatures) it is in a condensed phase with a ``cusp-halo'' structure, where the cusp corresponds to the rapid increase of the density of the gas at the contact with the central body. For a fixed density $\rho_{*}$ of the central body, we show the existence of two critical points in the phase diagram, one in each ensemble, depending on the core radius $R_{*}$: for small radii $R_{*}R_{*}^{\rm CCP}$, there is no phase transition at all. We study how the nature of these phase transitions changes as a function of the dimension of space. We also discuss the analogies and the differences with phase transitions in the self-gravitating Fermi gas [P.H. Chavanis, Phys. Rev. E 65, 056123 (2002)].

Galaxies show different halo scaling relations such as the Radial Acceleration Relation, the Mass Discrepancy Acceleration Relation (MDAR) or the dark matter Surface Density Relation (SDR). At difference with traditional studies using phenomenological $\Lambda$CDM halos, we analyze the above relations assuming that dark matter (DM) halos are formed through a Maximum Entropy Principle (MEP) in which the fermionic (quantum) nature of the DM particles is dully accounted for. For the first time a competitive DM model based on first physical principles, such as (quantum) statistical-mechanics and thermodynamics, is tested against a large data-set of galactic observables. In particular, we compare the fermionic DM model with empirical DM profiles: the NFW model, a generalized NFW model accounting for baryonic feedback, the Einasto model and the Burkert model. For this task, we use a large sample of 120 galaxies taken from the Spitzer Photometry and Accurate Rotation Curves (SPARC) data-set, from which we infer the DM content to compare with the models. We find that the Radial Acceleration Relation and MDAR are well explained by all the models with comparable accuracy, while the fits to the individual rotation curves, in contrast, show that cored DM halos are statistically preferred with respect to the cuspy NFW profile. However, very different physical principles justify the flat inner halo slope in the most favored DM profiles: while generalized NFW or Einasto models rely on complex baryonic feedback processes, the MEP scenario involves a quasi-thermodynamic equilibrium of the DM particles., Comment: 20 pages, 12 figures. Accepted for publication in The Astrophysical Journal

We study random transitions between two metastable states that appear below a critical temperature in a one dimensional self-gravitating Brownian gas with a modified Poisson equation experiencing a second order phase transition from a homogeneous phase to an inhomogeneous phase. We numerically solve the $N$-body Langevin equations and the stochastic Smoluchowski-Poisson system which takes fluctuations (finite $N$ effects) into account. The system switches back and forth between the two metastable states (bistability) and the particles accumulate successively at the center or at the boundary of the domain. We show that these random transitions exhibit the phenomenology of the ordinary Kramers problem for a Brownian particle in a double-well potential. The distribution of the residence time is Poissonian and the average lifetime of a metastable state is given by the Arrhenius law, i.e. it is proportional to the exponential of the barrier of free energy $\Delta F$ divided by $k_B T$. Since the free energy is proportional to the number of particles $N$ for a system with long-range interactions, the lifetime of metastable states scales as $e^N$ and is considerable for $N\gg 1$. As a result, in many applications, metastable states of systems with long-range interactions can be considered as stable states. However, for moderate values of $N$, or close to a critical point $T_c$, the lifetime of the metastable states is reduced since the barrier of free energy decreases. In that case, the fluctuations become important and the mean field approximation is no more valid. This is the situation considered in this paper. By an appropriate change of notations, our results also apply to bacterial populations experiencing chemotaxis in biology. Their dynamics can be described by a stochastic Keller-Segel model that takes fluctuations into account and goes beyond the usual mean field approximation.

We perform direct numerical simulations of the HMF model starting from non-magnetized initial conditions with a velocity distribution that is (i) gaussian, (ii) semi-elliptical, and (iii) waterbag. Below a critical energy E_c, depending on the initial condition, this distribution is Vlasov dynamically unstable. The system undergoes a process of violent relaxation and quickly reaches a quasi-stationary state (QSS). We find that the distribution function of this QSS can be conveniently fitted by a polytrope with index (i) n=2, (ii) n=1, and (iii) n=1/2. Using the values of these indices, we are able to determine the physical caloric curve T_{kin}(E) and explain the negative kinetic specific heat region C_{kin}=dE/dT_{kin}<0 observed in the numerical simulations. At low energies, we find that the system takes a "core-halo" structure. The core corresponds to the pure polytrope discussed above but it is now surrounded by a halo of particles. We also consider unsteady initial conditions with magnetization M_0=1 and isotropic waterbag distribution and report the complex dynamics of the system creating phase space holes and dense filaments. We show that the kinetic caloric curve is approximately constant, corresponding to a polytrope with index n_0= 3.56. Finally, we consider the collisional evolution of an initially Vlasov stable distribution, and show that the time-evolving distribution function f(v,t) can be fitted by a sequence of polytropic distributions with a time-dependent index n(t) both in the non-magnetized and magnetized regimes. These numerical results show that polytropic distributions (also called Tsallis distributions) provide in many cases a good fit of the QSSs. However, in order to moderate our message, we also report a case where the Lynden-Bell theory provides an excellent prediction of an inhomogeneous QSS.

We analyze the possibility that due to their superfluid properties some compact astrophysical objects may contain a significant part of their matter in the form of a Bose-Einstein condensate. To study the condensate we use the Gross-Pitaevskii equation, with arbitrary non-linearity. By introducing the Madelung representation of the wave function, we formulate the dynamics of the system in terms of the continuity equation and of the hydrodynamic Euler equations. The non-relativistic and Newtonian Bose-Einstein gravitational condensate can be described as a gas, whose density and pressure are related by a barotropic equation of state. In the case of a condensate with quartic non-linearity, the equation of state is polytropic with index one. In the framework of the Thomas-Fermi approximation the structure of the Newtonian gravitational condensate is described by the Lane-Emden equation, which can be exactly solved. The case of the rotating condensate is also discussed. General relativistic configurations with quartic non-linearity are studied numerically with both non-relativistic and relativistic equations of state, and the maximum mass of the stable configuration is determined. Condensates with particle masses of the order of two neutron masses (Cooper pair) and scattering length of the order of 10-20 fm have maximum masses of the order of 2 M_sun, maximum central density of the order of 0.1-0.3 10^16 g/cm^3 and minimum radii in the range of 10-20 km. In this way we obtain a large class of stable astrophysical objects, whose basic astrophysical parameters (mass and radius) sensitively depend on the mass of the condensed particle, and on the scattering length. We also propose that the recently observed neutron stars with masses in the range of 2-2.4 M_sun are Bose-Einstein Condensate stars.

We develop the suggestion that dark matter could be a Bose-Einstein condensate. We determine the mass-radius relation of a Newtonian self-gravitating Bose-Einstein condensate with short-range interactions described by the Gross-Pitaevskii-Poisson system. We numerically solve the equation of hydrostatic equilibrium describing the balance between the gravitational attraction and the pressure due to quantum effects (Heisenberg's uncertainty principle) and short-range interactions (scattering). We connect the non-interacting limit to the Thomas-Fermi limit. We also consider the case of attractive self-interaction. We compare the exact mass-radius relation obtained numerically with the approximate analytical relation obtained with a Gaussian ansatz. An overall good agreement is found.

We provide a detailed discussion of out-of-equilibrium phase transitions in the Hamiltonian Mean Field (HMF) model in the framework of Lynden-Bell's statistical theory of the Vlasov equation. For two-levels initial conditions, the caloric curve $\beta(E)$ only depends on the initial value $f_0$ of the distribution function. We evidence different regions in the parameter space where the nature of phase transitions between magnetized and non-magnetized states changes: (i) for $f_0>0.10965$, the system displays a second order phase transition; (ii) for $0.109497

We study the minimization of potential enstrophy at fixed circulation and energy in an oceanic basin with arbitrary topography. For illustration, we consider a rectangular basin and a linear topography h=by which represents either a real bottom topography or the beta-effect appropriate to oceanic situations. Our minimum enstrophy principle is motivated by different arguments of statistical mechanics reviewed in the article. It leads to steady states of the quasigeostrophic (QG) equations characterized by a linear relationship between potential vorticity q and stream function psi. For low values of the energy, we recover Fofonoff flows [J. Mar. Res. 13, 254 (1954)] that display a strong westward jet. For large values of the energy, we obtain geometry induced phase transitions between monopoles and dipoles similar to those found by Chavanis and Sommeria [J. Fluid Mech. 314, 267 (1996)] in the absence of topography. In the presence of topography, we recover and confirm the results obtained by Venaille and Bouchet [Phys. Rev. Lett. 102, 104501 (2009)] using a different formalism. In addition, we introduce relaxation equations towards minimum potential enstrophy states and perform numerical simulations to illustrate the phase transitions in a rectangular oceanic basin with linear topography (or beta-effect)., Comment: 26 pages, 28 figures

According to thermodynamics, the specific heat of Boltzmannian short-range interacting systems is a positive quantity. Less intuitive properties are instead displayed by systems characterized by long-range interactions. In that case, the sign of specific heat depends on the considered statistical ensemble: negative specific heat can be found in isolated systems, which are studied in the framework of the microcanonical ensemble; on the other hand, it is generally recognized that a positive specific heat should always be measured in systems in contact with a thermal bath, for which the canonical ensemble is the appropriate one. We demonstrate that the latter assumption is not generally true: one can in principle measure negative specific heat also in the canonical ensemble if the system under scrutiny is non-Boltzmannian and/or out-of-equilibrium., Comment: 5 pages, 3 figures

We study the maximization of the Tsallis functional at fixed mass and energy in the HMF model. We give a thermodynamical and a dynamical interpretation of this variational principle. This leads to q-distributions known as stellar polytropes in astrophysics. We study phase transitions between spatially homogeneous and spatially inhomogeneous equilibrium states. We show that there exists a particular index q_c=3 playing the role of a canonical tricritical point separating first and second order phase transitions in the canonical ensemble and marking the occurence of a negative specific heat region in the microcanonical ensemble. We apply our results to the situation considered by Antoni & Ruffo [Phys. Rev. E 52, 2361 (1995)] and show that the anomaly displayed on their caloric curve can be explained naturally by assuming that, in this region, the QSSs are polytropes with critical index q_c=3. We qualitatively justify the occurrence of polytropic (Tsallis) distributions with compact support in terms of incomplete relaxation and inefficient mixing (non-ergodicity). Our paper provides an exhaustive study of polytropic distributions in the HMF model and the first plausible explanation of the surprising result observed numerically by Antoni & Ruffo (1995). In the course of our analysis, we also report an interesting situation where the caloric curve presents both microcanonical first and second order phase transitions.

We consider a system of particles interacting via a screened Newtonian potential and study phase transitions between homogeneous and inhomogeneous states in the microcanonical and canonical ensembles. Like for other systems with long-range interactions, we obtain a great diversity of microcanonical and canonical phase transitions depending on the dimension of space and on the importance of the screening length. We also consider a system of particles in Newtonian interaction in the presence of a ``neutralizing background''. By a proper interpretation of the parameters, our study describes (i) self-gravitating systems in a cosmological setting, and (ii) chemotaxis of bacterial populations in the original Keller-Segel model.

A simplified thermodynamic approach of the incompressible 2D Euler equation is considered based on the conservation of energy, circulation and microscopic enstrophy. Statistical equilibrium states are obtained by maximizing the Miller-Robert-Sommeria (MRS) entropy under these sole constraints. The vorticity fluctuations are Gaussian while the mean flow is characterized by a linear $\bar{\omega}-\psi$ relationship. Furthermore, the maximization of entropy at fixed energy, circulation and microscopic enstrophy is equivalent to the minimization of macroscopic enstrophy at fixed energy and circulation. This provides a justification of the minimum enstrophy principle from statistical mechanics when only the microscopic enstrophy is conserved among the infinite class of Casimir constraints. A new class of relaxation equations towards the statistical equilibrium state is derived. These equations can provide an effective description of the dynamics towards equilibrium or serve as numerical algorithms to determine maximum entropy or minimum enstrophy states. We use these relaxation equations to study geometry induced phase transitions in rectangular domains. In particular, we illustrate with the relaxation equations the transition between monopoles and dipoles predicted by Chavanis and Sommeria [J. Fluid. Mech. 314, 267 (1996)]. We take into account stable as well as metastable states and show that metastable states are robust and have negative specific heats. This is the first evidence of negative specific heats in that context. We also argue that saddle points of entropy can be long-lived and play a role in the dynamics because the system may not spontaneously generate the perturbations that destabilize them., Comment: 26 pages, 10 figures

Using a Maximum Entropy Production Principle (MEPP), we derive a new type of relaxation equations for two-dimensional turbulent flows in the case where a prior vorticity distribution is prescribed instead of the Casimir constraints [Ellis, Haven, Turkington, Nonlin., 15, 239 (2002)]. The particular case of a Gaussian prior is specifically treated in connection to minimum enstrophy states and Fofonoff flows. These relaxation equations are compared with other relaxation equations proposed by Robert and Sommeria [Phys. Rev. Lett. 69, 2776 (1992)] and Chavanis [Physica D, 237, 1998 (2008)]. They can provide a small-scale parametrization of 2D turbulence or serve as numerical algorithms to compute maximum entropy states with appropriate constraints. We perform numerical simulations of these relaxation equations in order to illustrate geometry induced phase transitions in geophysical flows., Comment: 21 pages, 9 figures

We study the motion of N=2 overdamped Brownian particles in gravitational interaction in a space of dimension d=2. This is equivalent to the simplified motion of two biological entities interacting via chemotaxis when time delay and degradation of the chemical are ignored. This problem also bears some similarities with the stochastic motion of two point vortices in viscous hydrodynamics [Agullo & Verga, Phys. Rev. E, 63, 056304 (2001)]. We analytically obtain the density probability of finding the particles at a distance r from each other at time t. We also determine the probability that the particles have coalesced and formed a Dirac peak at time t (i.e. the probability that the reduced particle has reached r=0 at time t). Finally, we investigate the variance of the distribution and discuss the proper form of the virial theorem for this system. The reduced particle has a normal diffusion behaviour for small times with a gravity-modified diffusion coefficient =r_0^2+(4k_B/\xi\mu)(T-T_*)t, where k_BT_{*}=Gm_1m_2/2 is a critical temperature, and an anomalous diffusion for large times ~t^(1-T_*/T). As a by-product, our solution also describes the growth of the Dirac peak (condensate) that forms in the post-collapse regime of the Smoluchowski-Poisson system (or Keller-Segel model) for T

Systems with long-range interactions display a short-time relaxation towards Quasi Stationary States (QSSs) whose lifetime increases with system size. The application of Lynden-Bell's theory of "violent relaxation" to the Hamiltonian Mean Field model leads to the prediction of out-of-equilibrium first and second order phase transitions between homogeneous (zero magnetization) and inhomogeneous (non-zero magnetization) QSSs, as well as an interesting phenomenon of phase re-entrances. We compare these theoretical predictions with direct $N$-body numerical simulations. We confirm the existence of phase re-entrance in the typical parameter range predicted from Lynden-Bell's theory, but also show that the picture is more complicated than initially thought. In particular, we exhibit the existence of secondary re-entrant phases: we find un-magnetized states in the theoretically magnetized region as well as persisting magnetized states in the theoretically unmagnetized region., Comment: Submitted to Phys. Rev E

We apply the Nyquist method to the Hamiltonian Mean Field (HMF) model in order to settle the linear dynamical stability of a spatially homogeneous distribution function with respect to the Vlasov equation. We consider the case of Maxwell (isothermal) and Tsallis (polytropic) distributions and show that the system is stable above a critical kinetic temperature T_c and unstable below it. Then, we consider a symmetric double-humped distribution, made of the superposition of two decentered Maxwellians, and show the existence of a re-entrant phase in the stability diagram. When we consider an asymmetric double-humped distribution, the re-entrant phase disappears above a critical value of the asymmetry factor Delta>1.09. We also consider the HMF model with a repulsive interaction. In that case, single-humped distributions are always stable. For asymmetric double-humped distributions, there is a re-entrant phase for 143.9. Finally, we extend our results to arbitrary potentials of interaction and mention the connexion between the HMF model, Coulombian plasmas and gravitational systems. We discuss the relation between linear dynamical stability and formal nonlinear dynamical stability and show their equivalence for spatially homogeneous distributions. We also provide a criterion of dynamical stability for inhomogeneous systems.

We introduce a stochastic model of two-dimensional Brownian vortices associated with the canonical ensemble. The point vortices evolve through their usual mutual advection but they experience in addition a random velocity and a systematic drift generated by the system as a whole. The statistical equilibrium state of this stochastic model is the Gibbs canonical distribution. We consider a single species system and a system made of two types of vortices with positive and negative circulations. At positive temperatures, like-sign vortices repel each other (plasma case) and at negative temperatures, like-sign vortices attract each other (gravity case). We derive the stochastic equation satisfied by the exact vorticity field and the Fokker-Planck equation satisfied by the N-body distribution function. We present the BBGKY-like hierarchy of equations satisfied by the reduced distribution functions and close the hierarchy by considering an expansion of the solutions in powers of 1/N, where N is the number of vortices, in a proper thermodynamic limit. For spatially inhomogeneous systems, we derive the kinetic equations satisfied by the smooth vorticity field in a mean field approximation valid for $N\to +\infty$. For spatially homogeneous systems, we study the two-body correlation function, in a Debye-H\"uckel approximation valid at the order O(1/N). The results of this paper can also apply to other systems of random walkers with long-range interactions such as self-gravitating Brownian particles and bacterial populations experiencing chemotaxis. Furthermore, for positive temperatures, our study provides a kinetic derivation, from microscopic stochastic processes, of the Debye-H\"uckel model of electrolytes.

Starting from the Liouville equation, we derive the exact hierarchy of equations satisfied by the reduced distribution functions of the single species point vortex gas in two dimensions. Considering an expansion of the solution in powers of 1/N in a proper thermodynamic limit $N\to +\infty$, and neglecting some collective effects, we derive a kinetic equation satisfied by the smooth vorticity field which is valid at order $O(1/N)$. This equation was obtained previously [P.H. Chavanis, Phys. Rev. E, 64, 026309 (2001)] from a more abstract projection operator formalism. If we consider axisymmetric flows and make a markovian approximation, we obtain a simpler kinetic equation which can be studied in great detail. We discuss the properties of these kinetic equations in regard to the $H$-theorem and the convergence (or not) towards the statistical equilibrium state. We also study the growth of correlations by explicitly calculating the time evolution of the two-body correlation function in the linear regime. In a second part of the paper, we consider the relaxation of a test vortex in a bath of field vortices and obtain the Fokker-Planck equation by directly calculating the second (diffusion) and first (drift) moments of the increment of position of the test vortex. A specificity of our approach is to obtain general equations, with a clear physical meaning, that are valid for flows that are not necessarily axisymmetric and that take into account non-Markovian effects. A limitations of our approach, however, is that it ignores collective effects.

We develop the kinetic theory of point vortices in two-dimensional hydrodynamics and illustrate the main results of the theory with numerical simulations. We first consider the evolution of the system "as a whole" and show that the evolution of the vorticity profile is due to resonances between different orbits of the point vortices. The evolution stops when the profile of angular velocity becomes monotonic even if the system has not reached the statistical equilibrium state (Boltzmann distribution). In that case, the system remains blocked in a sort of metastable state with a non standard distribution. We also study the relaxation of a test vortex in a steady bath of field vortices. The relaxation of the test vortex is described by a Fokker-Planck equation involving a diffusion term and a drift term. The diffusion coefficient, which is proportional to the density of field vortices and inversely proportional to the shear, usually decreases rapidly with the distance. The drift is proportional to the gradient of the density profile of the field vortices and is connected to the diffusion coefficient by a generalized Einstein relation. We study the evolution of the tail of the distribution function of the test vortex and show that it has a front structure. We also study how the temporal auto-correlation function of the position of the test vortex decreases with time and find that it usually exhibits an algebraic behavior with an exponent that we compute analytically. We mention analogies with other systems with long-range interactions.

We derive the exact expression of the diffusion coefficient of a self-gravitating Brownian gas in two dimensions. Our formula generalizes the usual Einstein relation for a free Brownian motion to the context of two-dimensional gravity. We show the existence of a critical temperature T_{c} at which the diffusion coefficient vanishes. For T

We adapt the statistical mechanics of the shallow-water equations to the case where the flow is forced at small scales. We assume that the statistics of forcing is encoded in a prior potential vorticity distribution which replaces the specification of the Casimir constraints in the case of freely evolving flows. This determines a generalized entropy functional which is maximized by the coarse-grained PV field at statistical equilibrium. Relaxation equations towards equilibrium are derived which conserve the robust constraints (energy, mass and circulation) and increase the generalized entropy.

We discuss the structure of the equilibrium states of a regularized Keller-Segel model describing the chemotaxis of bacterial populations. We consider the limit of high degradation of the secreted chemical where analytical results can be obtained. Below a critical effective temperature, the system experiences a second order phase transition from a homogeneous phase to an inhomogeneous phase formed by two domains with uniform concentration separated by a thin interface (domain wall). We study the properties of the interface and determine the bifurcation between a circular shape (spot) and a stripe as a function of the control parameters. We show the analogy with the structure of Jupiter's Great red spot which also consists of two phases with uniform potential vorticity separated by a thin annular jet.

Systems with long-range interactions can reach a Quasi Stationary State (QSS) as a result of a violent collisionless relaxation. If the system mixes well (ergodicity), the QSS can be predicted by the statistical theory of Lynden-Bell (1967) based on the Vlasov equation. When the initial distribution takes only two values, the Lynden-Bell distribution is similar to the Fermi-Dirac statistics. Such distributions have recently been observed in direct numerical simulations of the HMF model (Antoniazzi et al. 2006). In this paper, we determine the caloric curve corresponding to the Lynden-Bell statistics in relation with the HMF model and analyze the dynamical and thermodynamical stability of spatially homogeneous solutions by using two general criteria previously introduced in the literature. We express the critical energy and the critical temperature as a function of a degeneracy parameter fixed by the initial condition. Below these critical values, the homogeneous Lynden-Bell distribution is not a maximum entropy state but an unstable saddle point. We apply these results to the situation considered by Antoniazzi et al. For a given energy, we find a critical initial magnetization above which the homogeneous Lynden-Bell distribution ceases to be a maximum entropy state, contrary to the claim of these authors. For an energy U=0.69, this transition occurs above an initial magnetization M_{x}=0.897. In that case, the system should reach an inhomogeneous Lynden-Bell distribution (most mixed) or an incompletely mixed state (possibly fitted by a Tsallis distribution). Thus, our theoretical study proves that the dynamics is different for small and large initial magnetizations, in agreement with numerical results of Pluchino et al. (2004). This new dynamical phase transition may reconcile the two communities.

We determine the mass-radius relation of relativistic white dwarf stars (a self-gravitating degenerate Fermi gas at T=0) in a D-dimensional universe and study the influence of the dimension of space on the laws of physics when we combine quantum mechanics, special relativity and gravity. We exhibit characteristic dimensions D=1, D=2, D=3, D={1\over 2}(3+\sqrt{17}), D=4, D=2(1+\sqrt{2}) and show that quantum mechanics cannot balance gravitational collapse for D\ge 4, even in the non-relativistic regime. This is similar to a result found by Ehrenfest (1917) at the atomic level for Coulomb forces (in Bohr's model) and for the Kepler problem. This makes the dimension of our universe D=3 very particular. We discuss some historic aspects concerning the discovery of the Chandrasekhar (1931b) limiting mass in relation with previous investigations by Anderson (1929) and Stoner (1930). We also show the analogy between the Chandrasekhar limiting mass of white dwarf stars and the critical mass of biological populations in the two-dimensional chemotactic problem.

We complete previous investigations on the dynamical stability of barotropic stars and collisionless stellar systems. A barotropic star that minimizes the energy functional at fixed mass is a nonlinearly dynamically stable stationary solution of the Euler-Poisson system. Formally, this minimization problem is similar to a condition of ``canonical stability'' in thermodynamics. A stellar system that maximizes an H-function at fixed mass and energy is a nonlinearly dynamically stable stationary solution of the Vlasov-Poisson system. Formally, this maximization problem is similar to a condition of ``microcanonical stability'' in thermodynamics. Using a thermodynamical analogy, we provide a derivation and an interpretation of the nonlinear Antonov first law in terms of ``ensembles inequivalence'': a spherical stellar system with f=f(epsilon) and f'(epsilon)<0 is nonlinearly dynamically stable with respect to the Vlasov-Poisson system if the corresponding barotropic star with the same equilibrium density distribution is nonlinearly dynamically stable with respect to the Euler-Poisson system. This is similar to the fact that ``canonical stability implies microcanonical stability'' in thermodynamics. The converse is wrong in case of ``ensembles inequivalence'' which is generic for systems with long-range interactions like gravity. We show that criteria of nonlinear dynamical stability can be obtained very simply from purely graphical constructions by using the method of series of equilibria and the turning point argument of Poincare, as in thermodynamics.

We adapt the formalism of the statistical theory of 2D turbulence in the case where the Casimir constraints are replaced by the specification of a prior vorticity distribution. A new relaxation equation is obtained for the evolution of the coarse-grained vorticity. It can be used as a thermodynamical parametrization of forced 2D turbulence (determined by the prior), or as a numerical algorithm to construct arbitrary nonlinearly dynamically stable stationary solutions of the 2D Euler equation., Comment: In Proceedings of the Workshop on ``Interdisciplinary Aspects of Turbulence'' at Ringberg Castle, Tegernsee, Germany (Max-Planck Institut fur Astrophysik, 2005)

We study the relaxation of a test particle immersed in a bath of field particles interacting via weak long-range forces. To order 1/N in the $N\to +\infty$ limit, the velocity distribution of the test particle satisfies a Fokker-Planck equation whose form is related to the Landau and Lenard-Balescu equations in plasma physics. We provide explict expressions for the diffusion coefficient and friction force in the case where the velocity distribution of the field particles is isotropic. We consider (i) various dimensions of space $d=3,2$ and 1 (ii) a discret spectrum of masses among the particles (iii) different distributions of the bath including the Maxwell distribution of statistical equilibrium (thermal bath) and the step function (water bag). Specific applications are given for self-gravitating systems in three dimensions, Coulombian systems in two dimensions and for the HMF model in one dimension.

We present first elements of kinetic theory appropriate to the inhomogeneous phase of the HMF model. In particular, we investigate the case of strongly inhomogeneous distributions for $T\to 0$ and exhibit curious behaviour of the force auto-correlation function and friction coefficient. The temporal correlation function of the force has an oscillatory behaviour which averages to zero over a period. By contrast, the effects of friction accumulate with time and the friction coefficient does not satisfy the Einstein relation. On the contrary, it presents the peculiarity to increase linearly with time. Motivated by this result, we provide analytical solutions of a simplified kinetic equation with a time dependent friction. Analogies with self-gravitating systems and other systems with long-range interactions are also mentioned.

We discuss the nature of quasi-stationary states (QSS) with non-Boltzmannian distribution in systems with long-range interactions in relation with a process of incomplete violent relaxation based on the Vlasov equation. We discuss several attempts to characterize these QSS and explain why their prediction is difficult in general., Comment: Talk given at the 3rd NEXT-Sigma-Phi Conference, Crete, Aug.2005, 10 pages

We discuss the meaning of Tsallis functional in astrophysics. The energy functional of a polytropic star is similar to Tsallis free energy and the H-function associated with a stellar polytrope is similar to Tsallis entropy. More generally, the energy functional of a barotropic star is similar to a generalized free energy and the H-function associated with a spherical stellar system is similar to a generalized entropy. Their optimization under appropriate constraints determines a condition of nonlinear dynamical stability, not a condition of thermodynamical stability. The criterion of dynamical stability for barotropic stars described by the Euler-Poisson system is similar to a condition of canonical stability and the criterion of dynamical stability for collisionless stellar systems described by the Vlasov-Poisson system is similar to a condition of microcanonical stability. Hence, stellar systems are stable whenever corresponding barotropic stars are stable but the converse is wrong in general. This provides a new interpretation of Antonov first law in terms of ensemble inequivalence. In this context, Tsallis functional is not a true entropy or a true free energy. This is a useful H-function making the bridge between polytropic and isothermal distributions. A thermodynamical analogy can be developed to investigate the nonlinear dynamical stability problem but the notions of entropy, free energy and temperature are essentially effective. Similar ideas apply to other systems with long-range interactions such as 2D vortices and the HMF model. We also discuss two other interpretations of Tsallis functional in relation with generalized kinetic equations and quasi-stationary states of collisional stellar systems.

We discuss the dynamics and thermodynamics of the Hamiltonian Mean Field model (HMF) which is a prototypical system with long-range interactions. The HMF model can be seen as the one Fourier component of a one-dimensional self-gravitating system. Interestingly, it exhibits many features of real self-gravitating systems (violent relaxation, persistence of metaequilibrium states, slow collisional dynamics, phase transitions,...) while avoiding complicated problems posed by the singularity of the gravitational potential at short distances and by the absence of a large-scale confinement. We stress the deep analogy between the HMF model and self-gravitating systems by developing a complete parallel between these two systems. This allows us to apply many technics introduced in plasma physics and astrophysics to a new problem and to see how the results depend on the dimension of space and on the form of the potential of interaction. This comparative study brings new light in the statistical mechanics of self-gravitating systems. We also mention simple astrophysical applications of the HMF model in relation with the formation of bars in spiral galaxies.

We develop the analogy between self-gravitating Brownian particles and bacterial populations. In the high friction limit, the self-gravitating Brownian gas is described by the Smoluchowski-Poisson system. These equations can develop a self-similar collapse leading to a finite time singularity. Coincidentally, the Smoluchowski-Poisson system corresponds to a simplified version of the Keller-Segel model of bacterial populations. In this biological context, it describes the chemotactic aggregation of the bacterial colonies. We extend these classical models by introducing a small-scale regularization. In the gravitational context, we consider a gas of self-gravitating Brownian fermions and in the biological context we consider finite size effects. In that case, the collapse stops when the system feels the influence of the small-scale regularization. A phenomenon of ``explosion'', reverse to the collapse, is also possible.

We investigate a model describing the dynamics of a gas of self-gravitating Brownian particles. This model can also have applications for the chemotaxis of bacterial populations. We focus here on the collapse phase obtained at sufficiently low temperature/energy and on the post-collapse regime following the singular time where the central density diverges. Several analytical results are illustrated by numerical simulations.

We compute statistical equilibrium states of rotating self-gravitating systems enclosed within a box by maximizing the Fermi-Dirac entropy at fixed mass, energy and angular momentum. We increase the rotation up to the Keplerian limit and describe the flattening of the configuration until mass shedding occurs. At the maximum rotation, the system develops a cusp at the equator. We draw the equilibrium phase diagram of the rotating self-gravitating Fermi gas and discuss the structure of the caloric curve as a function of degeneracy parameter and angular velocity. We argue that systems described by the Fermi-Dirac distribution in phase space do not bifurcate to non-axisymmetric structures, in continuity with the case of polytropes with index n>0.808 (the Fermi gas at T=0 corresponds to n=3/2). This contrasts with the study of Votyakov et al. (2002) who consider a Fermi-Dirac distribution in configuration space and find ``double star'' structures (their model at T=0 corresponds to n=0). We also discuss the influence of rotation on the onset of the gravothermal catastrophe for globular clusters. On general grounds, we complete previous investigations concerning the nature of phase transitions in self-gravitating systems. We emphasize the inequivalence of statistical ensembles regarding the formation of binaries (or low-mass condensates) in the microcanonical ensemble and Dirac peaks (or massive condensates) in the canonical ensemble. We also describe an hysteretic cycle between the gaseous phase and the condensed phase that are connected by a ``collapse'' or an ``explosion''. This notion of hysteresis in self-gravitating systems is new., Comment: submitted to A&A

The formation of large-scale vortices is an intriguing phenomenon in two-dimensional turbulence. Such organization is observed in large-scale oceanic or atmospheric flows, and can be reproduced in laboratory experiments and numerical simulations. A general explanation of this organization was first proposed by Onsager (1949) by considering the statistical mechanics for a set of point vortices in two-dimensional hydrodynamics. Similarly, the structure and the organization of stellar systems (globular clusters, elliptical galaxies,...) in astrophysics can be understood by developing a statistical mechanics for a system of particles in gravitational interaction as initiated by Chandrasekhar (1942). These statistical mechanics turn out to be relatively similar and present the same difficulties due to the unshielded long-range nature of the interaction. This analogy concerns not only the equilibrium states, i.e. the formation of large-scale structures, but also the relaxation towards equilibrium and the statistics of fluctuations. We will discuss these analogies in detail and also point out the specificities of each system., Comment: Chapter of the forthcoming "Lecture Notes in Physics" volume: ``Dynamics and Thermodynamics of Systems with Long Range Interactions'', T. Dauxois, S. Ruffo, E. Arimondo, M. Wilkens Eds., Lecture Notes in Physics Vol. 602, Springer (2002)

We discuss the statistical mechanics of violent relaxation in stellar systems following the pioneering work of Lynden-Bell (1967). The solutions of the gravitational Vlasov-Poisson system develop finer and finer filaments so that a statistical description is appropriate to smooth out the small-scales and describe the ``coarse-grained'' dynamics. In a coarse-grained sense, the system is expected to reach an equilibrium state of a Fermi-Dirac type within a few dynamical times. We describe in detail the equilibrium phase diagram and the nature of phase transitions which occur in self-gravitating systems. Then, we introduce a small-scale parametrization of the Vlasov equation and propose a set of relaxation equations for the coarse-grained dynamics. These relaxation equations, of a generalized Fokker-Planck type, are derived from a Maximum Entropy Production Principle (MEPP). We make a link with the quasilinear theory of the Vlasov-Poisson system and derive a truncated model appropriate to collisionless systems subject to tidal forces. With the aid of this kinetic theory, we qualitatively discuss the concept of ``incomplete relaxation'' and the limitations of Lynden-Bell's theory., Comment: In Proceedings of the Conference on Multiscale Problems in Science and Technology, edited by N. Antonic, C.J. van Duijn, W. Jager, and A. Mikelic (Springer, Berlin, 2002)

We complete previous investigations on the thermodynamics of self-gravitating systems by studying the grand canonical, grand microcanonical and isobaric ensembles. We also discuss the stability of polytropic spheres in the light of a generalized thermodynamics 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., Comment: Submitted to A&A

We discuss the nature of phase transitions in the self-gravitating Fermi gas at non-zero temperature. This study can be relevant for massive neutrinos in Dark Matter models and for collisionless self-gravitating systems experiencing a ``violent relaxation''. Below a critical energy (in the microcanonical ensemble) or below a critical temperature (in the canonical ensemble), the system undergoes a gravitational collapse leading to a compact object with a massive degenerate nucleus (fermion ball)., Comment: Invited talk at DARK 2002, 4th Int. Conf. on Dark Matter in Astro and Particle Physics, Cape Town, Feb. 2002

Phase diagram of microcanonical ensembles of self-attracting particles is studied for two types of short-range potential regularizations: self-gravitating fermions and classical particles interacting via attractive soft $-(r^2+r_0^2)^{-1/2}$ Coulomb potential. When the range of regularization is sufficiently short, the self-attracting systems exhibit gravitational or collapse-like transition. As the fermionic degeneracy or the softness radius increases, the gravitational phase transition crosses over to a normal first-order phase transition, becomes second-order at a critical point, and finally disappears. Applicability of a commonly used saddle-point or mean-field approximation and importance of metastable states is discussed., Comment: 9 pages, 11 figures

We address the thermodynamics (equilibrium density profiles, phase diagram, instability analysis...) and the collapse of a self-gravitating gas of Brownian particles in D dimensions, in both canonical and microcanonical ensembles. In the canonical ensemble, we derive the analytic form of the density scaling profile which decays as f(x)=x^{-\alpha}, with alpha=2. In the microcanonical ensemble, we show that f decays as f(x)=x^{-\alpha_{max}}, where \alpha_{max} is a non-trivial exponent. We derive exact expansions for alpha_{max} and f in the limit of large D. Finally, we solve the problem in D=2, which displays rather rich and peculiar features.

We discuss the statistical mechanics of rotating self-gravitating systems by allowing properly for the conservation of angular momentum. We study analytically the case of slowly rotating isothermal spheres by expanding the solutions of the Boltzmann-Poisson equation in a series of Legendre polynomials, adapting the procedure introduced by Chandrasekhar (1933) for distorted polytropes. We show how the classical spiral of Lynden-Bell & Wood (1967) in the temperature-energy plane is deformed by rotation. We find that gravitational instability occurs sooner in the microcanonical ensemble and later in the canonical ensemble. According to standard turning point arguments, the onset of the collapse coincides with the minimum energy or minimum temperature state in the series of equilibria. Interestingly, it happens to be close to the point of maximum flattening. We determine analytically the generalization of the singular isothermal solution to the case of a slowly rotating configuration. We also consider slowly rotating configurations of the self-gravitating Fermi gas at non zero temperature., Comment: Submitted to A&A

We discuss the nature of phase transitions in self-gravitating systems both in the microcanonical and in the canonical ensemble. We avoid the divergence of the gravitational potential at short distances by considering the case of self-gravitating fermions and hard spheres models. Three kinds of phase transitions (of zeroth, first and second order) are evidenced. They separate a ``gaseous'' phase with a smoothly varying distribution of matter from a ``condensed'' phase with a core-halo structure. We propose a simple analytical model to describe these phase transitions. We determine the value of energy (in the microcanonical ensemble) and temperature (in the canonical ensemble) at the transition point and we study their dependance with the degeneracy parameter (for fermions) or with the size of the particles (for a hard spheres gas). Scaling laws are obtained analytically in the asymptotic limit of a small short distance cut-off. Our analytical model captures the essential physics of the problem and compares remarkably well with the full numerical solutions., Comment: Submitted to Phys. Rev. E. New material added

We complete the existing literature on the structure and stability of polytropic gas spheres reported in the classical monograph of Chandrasekhar (1942). For isolated polytropes with index $1

Self-gravitating systems are expected to reach a statistical equilibrium state either through collisional relaxation or violent collisionless relaxation. However, a maximum entropy state does not always exist and the system may undergo a ``gravothermal catastrophe'': it can achieve ever increasing values of entropy by developing a dense and hot ``core'' surrounded by a low density ``halo''. In this paper, we study the phase transition between ``equilibrium'' states and ``collapsed'' states with the aid of a simple relaxation equation [Chavanis, Sommeria and Robert, Astrophys. J. 471, 385 (1996)] constructed so as to increase entropy with an optimal rate while conserving mass and energy. With this numerical algorithm, we can cover the whole bifurcation diagram in parameter space and check, by an independent method, the stability limits of Katz [Mon. Not. R. astr. Soc. 183, 765 (1978)] and Padmanabhan [Astrophys. J. Supp. 71, 651 (1989)]. When no equilibrium state exists, our relaxation equation develops a self-similar collapse leading to a finite time singularity., Comment: 54 pages. 25 figures. Submitted to Phys. Rev. E

We develop a kinetic theory for point vortices in two-dimensional hydrodynamics. Using standard projection operator technics, we derive a Fokker-Planck equation describing the relaxation of a ``test'' vortex in a bath of ``field'' vortices at statistical equilibrium. The relaxation is due to the combined effect of a diffusion and a drift. The drift is shown to be responsible for the organization of point vortices at negative temperatures. A description that goes beyond the thermal bath approximation is attempted. A new kinetic equation is obtained which respects all conservation laws of the point vortex system and satisfies a H-theorem. Close to equilibrium this equation reduces to the ordinary Fokker-Planck equation., Comment: 50 pages. To appear in Phys. Rev. E

We investigate the stability of bounded self-gravitating systems in the canonical ensemble by using a thermodynamical approach. Our study extends the earlier work of Padmanabhan [Astrophys. J. Supp. 71, 651 (1989)] in the microcanonical ensemble. By studying the second variations of the free energy, we find that instability sets in precisely at the point of minimum temperature in agreement with the theorem of Katz [ Mon. Not. R. astr. Soc. 183, 765 (1978)]. The perturbation that induces instability at this point is calculated explicitly; it has not a ``core-halo'' structure contrary to what happens in the microcanonical ensemble. We also study Jeans type gravitational instability of isothermal gaseous spheres described by Navier-Stokes equations. We show analytically the equivalence between dynamical stability and thermodynamical stability and the fact that the stability of isothermal gas spheres does not depend on the viscosity. This confirms the findings of Semelin et al. [astro-ph/9908073] who used numerical methods. We also give a simpler derivation of the geometric hierarchy of scales inducing instability discovered by these authors. The density profiles that trigger these instabilities are calculated analytically; they present more and more oscillations that also follow a geometric progression. This suggests that the system will fragmentate in a series of `clumps' and that these `clumps' will themselves fragmentate in substructures. The fact that both the domain sizes leading to instability and the `clumps' sizes within a box follow a geometric progression with the same ratio suggests a fractal-like behaviour. This gives further support to the interpretation of de Vega et al. [Nature, 383, 56 (1996)]., Comment: Submitted to Astron. Astrophys

We extend the formalism of the statistical theory developed for the 2D Euler equation to the case of shallow water system. Relaxation equations towards the maximum entropy state are proposed, which provide a parametrization of sub-grid scale eddies in 2D compressible turbulence., Comment: submitted to Phys. Fluids

In the problem of planetary formation one seeks a mechanism to gather small solid particles together into larger accumulations of solid matter. Here we describe a scenario in which turbulence mediates this process by aggregating particles into anticyclonic regions. If, as our simulations suggest, anticyclonic vortices form as long-lived coherent structures, the process becomes more powerful because such vortices trap particles effectively. Even if the turbulence is decaying, following the upheaval that formed the disk, there is enough time to make the dust distribution quite lumpy., Comment: 16 pages, 9 figures