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We consider an isothermal self-gravitating system surrounding a central body. This model can represent a galaxy or a globular cluster harboring a central black hole. It can also represent a gaseous atmosphere surrounding a protoplanet. In three dimensions, the Boltzmann-Poisson equation must be solved numerically to obtain the density profile of the gas [Chavanis et al., Phys. Rev. E 109, 014118 (2024)10.1103/PhysRevE.109.014118]. In one and two dimensions, we show that the Boltzmann-Poisson equation can be solved analytically. We obtain explicit analytical expressions of the density profile around a central body which generalize the analytical solutions found by Camm (1950) and Ostriker (1964) in the absence of a central body. Our results also have applications for self-gravitating Brownian particles (Smoluchowski-Poisson system), for the chemotaxis of bacterial populations in biology (Keller-Segel model), and for two-dimensional point vortices in hydrodynamics (Onsager's model). In the case of bacterial populations, the central body could represent a supply of "food" that attracts the bacteria (chemoattractant). In the case of two-dimensional vortices, the central body could be a central vortex.

After reviewing the peculiar thermodynamics and statistical mechanics of self-gravitating systems, we consider the case of a "binary star" consisting of two particles of size a in gravitational interaction in a box of radius R . The caloric curve of this system displays a region of negative specific heat in the microcanonical ensemble, which is replaced by a first-order phase transition in the canonical ensemble. The free energy viewed as a thermodynamic potential exhibits two local minima that correspond to two metastable states separated by an unstable maximum forming a barrier of potential. By introducing a Langevin equation to model the interaction of the particles with the thermal bath, we study the random transitions of the system between a "dilute" state, where the particles are well separated, and a "condensed" state, where the particles are bound together. We show that the evolution of the system is given by a Fokker-Planck equation in energy space and that the lifetime of a metastable state is given by the Kramers formula involving the barrier of free energy. This is a particular case of the theory developed in a previous paper (Chavanis, 2005) for N Brownian particles in gravitational interaction associated with the canonical ensemble. In the case of a binary star (N=2), all the quantities can be calculated exactly analytically. We compare these results with those obtained in the mean field limit N→+∞.

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" that attracts the bacteria (chemoattractant). We consider both microcanonical (fixed energy) and canonical (fixed temperature) descriptions and study the inequivalence of statistical ensembles. At high energies (respectively, high temperatures), the system is in a "gaseous" phase and at low energies (respectively, 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 ρ_{*} 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_{*}^{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)1063-651X10.1103/PhysRevE.65.056123].

We investigate the long-term relaxation of one-dimensional (1D) self-gravitating systems, using both kinetic theory and N-body simulations. We consider thermal and Plummer equilibria, with and without collective effects. All combinations are found to be in clear agreement with respect to the Balescu-Lenard and Landau predictions for the diffusion coefficients. Interestingly, collective effects reduce the diffusion by a factor ∼10. The predicted flux for Plummer equilibrium matches the measured one, which is a remarkable validation of kinetic theory. We also report on a situation of quasikinetic blocking for the same equilibrium.

We calculate density profiles for self-gravitating clusters of an ideal Fermi-Dirac gas with nonrelativistic energy-momentum relation and macroscopic mass at thermal equilibrium. Our study includes clusters with planar symmetry in dimensions D=1,2,3, clusters with cylindrical symmetry in D=2,3, and clusters with spherical symmetry in D=3. Wall confinement is imposed where needed for stability against escape. The length scale and energy scale in use render all results independent of total mass and prove adequate at all temperatures. We present exact analytic expressions for (fully degenerate) T=0 density profiles in four of the six combinations of symmetry and dimensionality. Our numerical results for T>0 describe the emergence, upon quasistatic cooling, of a core with incipient degeneracy surrounded by a more dilute halo. The equilibrium macrostates are found to depend more strongly on the cluster symmetry than on the space dimensionality. We demonstrate the mechanical and thermal stability of spherical clusters with coexisting phases.

We calculate density profiles for self-gravitating clusters of an ideal Bose-Einstein gas with nonrelativistic energy-momentum relation and macroscopic mass at thermal equilibrium. Our study includes clusters with planar symmetry in dimensions D=1,2,3, clusters with cylindrical symmetry in D=2,3, and clusters with spherical symmetry in D=3. Wall confinement is imposed where needed to prevent escape. The length scale and energy scale in use for the gaseous phase render density profiles for gaseous macrostates independent of total mass. Density profiles for mixed-phase macrostates have a condensed core surrounded by a gaseous halo. The spatial extension of the core is negligibly small on the length scale tailored for the halo. The mechanical stability conditions as evident in caloric curves permit multiple macrostates to coexist. Their status regarding thermal equilibrium is examined by a comparison of free energies. The onset of condensation takes place at a nonzero temperature in all cases. The critical singularities and the nature of the phase transition vary with the symmetry of the cluster and the dimensionality of the space.

Finite-N effects unavoidably drive the long-term evolution of long-range interacting N-body systems. The Balescu-Lenard kinetic equation generically describes this process sourced by 1/N effects but this kinetic operator exactly vanishes by symmetry for one-dimensional homogeneous systems: such systems undergo a kinetic blocking and cannot relax as a whole at this order in 1/N. It is therefore only through the much weaker 1/N^{2} effects, sourced by three-body correlations, that these systems can relax, leading to a much slower evolution. In the limit where collective effects can be neglected, but for an arbitrary pairwise interaction potential, we derive a closed and explicit kinetic equation describing this very long-term evolution. We show how this kinetic equation satisfies an H-theorem while conserving particle number and energy, ensuring the unavoidable relaxation of the system toward the Boltzmann equilibrium distribution. Provided that the interaction is long-range, we also show how this equation cannot suffer from further kinetic blocking, i.e., the 1/N^{2} dynamics is always effective. Finally, we illustrate how this equation quantitatively matches measurements from direct N-body simulations.

We determine the caloric curves of classical self-gravitating systems at statistical equilibrium in general relativity. In the classical limit, the caloric curves of a self-gravitating gas depend on a unique parameter ν=GNm/Rc^{2}, called the compactness parameter, where N is the particle number and R the system's size. Typically, the caloric curves have the form of a double spiral. The "cold spiral," corresponding to weakly relativistic configurations, is a generalization of the caloric curve of nonrelativistic classical self-gravitating systems. The "hot spiral," corresponding to strongly relativistic configurations, is similar (but not identical) to the caloric curve of the ultrarelativistic self-gravitating black-body radiation. We introduce two types of normalization of energy and temperature to obtain asymptotic caloric curves describing, respectively, the cold and the hot spirals in the limit ν→0. As the number of particles increases, the cold and the hot spirals approach each other, merge at ν_{S}^{'}=0.128, form a loop above ν_{S}=0.1415, reduce to a point at ν_{max}=0.1764, and finally disappear. Therefore, the double spiral shrinks when the compactness parameter ν increases, implying that general relativistic effects render the system more unstable. We discuss the nature of the gravitational collapse at low and high energies with respect to a dynamical (fast) or a thermodynamical (slow) instability. We also provide an historical account of the developments of the statistical mechanics of classical self-gravitating systems in Newtonian gravity and general relativity.

The long-term dynamics of long-range interacting N-body systems can generically be described by the Balescu-Lenard kinetic equation. However, for one-dimensional homogeneous systems, this collision operator exactly vanishes by symmetry. These systems undergo a kinetic blocking, and cannot relax as a whole under 1/N resonant effects. As a result, these systems can only relax under 1/N^{2} effects, and their relaxation is drastically slowed down. In the context of the homogeneous Hamiltonian mean field model, we present a closed and explicit kinetic equation describing self-consistently the very long-term evolution of such systems, in the limit where collective effects can be neglected, i.e., for dynamically hot initial conditions. We show in particular how that kinetic equation satisfies an H theorem that guarantees the unavoidable relaxation to the Boltzmann equilibrium distribution. Finally, we illustrate how that kinetic equation quantitatively matches with the measurements from direct N-body simulations.

In hierarchical models of structure formation, the first galaxies form in low-mass dark matter potential wells, probing the behavior of dark matter on kiloparsec scales. Even though these objects are below the detection threshold of current telescopes, future missions will open an observational window into this emergent world. In this Letter, we investigate how the first galaxies are assembled in a "fuzzy" dark matter (FDM) cosmology where dark matter is an ultralight ∼10^{-22}  eV boson and the primordial stars are expected to form along dense dark matter filaments. Using a first-of-its-kind cosmological hydrodynamical simulation, we explore the interplay between baryonic physics and unique wavelike features inherent to FDM. In our simulation, the dark matter filaments show coherent interference patterns on the boson de Broglie scale and develop cylindrical solitonlike cores, which are unstable under gravity and collapse into kiloparsec-scale spherical solitons. Features of the dark matter distribution are largely unaffected by the baryonic feedback. On the contrary, the distributions of gas and stars, which do form along the entire filament, exhibit central cores imprinted by dark matter-a smoking gun signature of FDM.

We investigate the secular dynamics of long-range interacting particles moving on a sphere, in the limit of an axisymmetric mean-field potential. We show that this system can be described by the general kinetic equation, the inhomogeneous Balescu-Lenard equation. We use this approach to compute long-term diffusion coefficients, that are compared with direct simulations. Finally, we show how the scaling of the system's relaxation rate with the number of particles fundamentally depends on the underlying frequency profile. This clarifies why systems with a monotonic profile undergo a kinetic blocking and cannot relax as a whole under 1/N resonant effects. Because of its general form, this framework can describe the dynamics of globally coupled classical Heisenberg spins, long-range couplings in liquid crystals, or the orbital inclination evolution of stars in nearly Keplerian systems.

Because the collapse of massive stars occurs in a few seconds, while the stars evolve on billions of years, the supernovae are typical complex phenomena in fluid mechanics with multiple time scales. We describe them in the light of catastrophe theory, assuming that successive equilibria between pressure and gravity present a saddle-center bifurcation. In the early stage we show that the loss of equilibrium may be described by a generic equation of the Painlevé I form. This is confirmed by two approaches, first by the full numerical solutions of the Euler-Poisson equations for a particular pressure-density relation, secondly by a derivation of the normal form of the solutions close to the saddle-center. In the final stage of the collapse, just before the divergence of the central density, we show that the existence of a self-similar collapsing solution compatible with the numerical observations imposes that the gravity forces are stronger than the pressure ones. This situation differs drastically in its principle from the one generally admitted where pressure and gravity forces are assumed to be of the same order. Moreover it leads to different scaling laws for the density and the velocity of the collapsing material. The new self-similar solution (based on the hypothesis of dominant gravity forces) which matches the smooth solution of the outer core solution, agrees globally well with our numerical results, except a delay in the very central part of the star, as discussed. Whereas some differences with the earlier self-similar solutions are minor, others are very important. For example, we find that the velocity field becomes singular at the collapse time, diverging at the center, and decreasing slowly outside the core, whereas previous works described a finite velocity field in the core which tends to a supersonic constant value at large distances. This discrepancy should be important for explaining the emission of remnants in the post-collapse regime. Finally we describe the post-collapse dynamics, when mass begins to accumulate in the center, also within the hypothesis that gravity forces are dominant.

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 [P. H. Chavanis and L. Delfini, Phys. Rev. E 81, 051103 (2010)]. 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 explicitly 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 ΔF divided by the energy of thermal excitation kBT. 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 eN and is considerable for N≫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, 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.

The large-scale circulation of planetary atmospheres such as that of the Earth is traditionally thought of in a dynamical framework. Here we apply the statistical mechanics theory of turbulent flows to a simplified model of the global atmosphere, the quasigeostrophic model, leading to nontrivial equilibria. Depending on a few global parameters, the structure of the flow may be either a solid-body rotation (zonal flow) or a dipole. A second-order phase transition occurs between these two phases, with associated spontaneous symmetry breaking in the dipole phase. This model allows us to go beyond the general theory of marginal ensemble equivalence through the notion of Goldstone modes.

We present a kinetic theory of two-dimensional decaying turbulence in the context of two-body and three-body vortex merging processes. By introducing the equations of motion for two or three vortices in the effective noise due to all the other vortices, we demonstrate analytically that a two-body mechanism becomes inefficient at low vortex density n<<1. When the more efficient three-body vortex mergings are considered (involving vortices of different signs), we show that n~t(-ξ), with ξ=1. We generalize this argument to three-dimensional geostrophic turbulence, finding ξ=5/4, in excellent agreement with direct Navier-Stokes simulations [McWilliams et al., J. Fluid Mech. 401, 1 (1999)].

We study the growth of perturbations in a uniformly collapsing cloud of self-gravitating Brownian particles. This problem shares analogies with the formation of large-scale structures in a universe experiencing a "big-crunch" or with the formation of stars in a molecular cloud experiencing gravitational collapse. Starting from the barotropic Smoluchowski-Poisson system, we derive a new equation describing the evolution of the density contrast in the comoving (collapsing) frame. This equation can serve as a prototype to study the process of self-organization in complex media with structureless initial conditions. We solve this equation analytically in the linear regime and compare the results with those obtained by using the "Jeans swindle" in a static medium. The stability criteria, as well as the laws for the time evolution of the perturbations, differ. The Jeans criterion is expressed in terms of a critical wavelength λ(J) while our criterion is expressed in terms of a critical polytropic index γ(4/3). In a static background, the system is stable for λ<λ(J) and unstable for λ>λ(J). In a collapsing cloud, the system is stable for γ>γ(4/3) and unstable for γ<γ(4/3). If γ=γ(4/3), it is stable for λ<λ(J) and unstable for λ>λ(J). We also study the fragmentation process in the nonlinear regime. We determine the growth of the skewness, the long-wavelength tail of the power spectrum and find a self-similar solution to the nonlinear equations valid for large times. Finally, we consider dissipative self-gravitating Bose-Einstein condensates with short-range interactions and show that, in a strong friction limit, the dissipative Gross-Pitaevskii-Poisson system is equivalent to the quantum barotropic Smoluchowski-Poisson system. This yields new types of nonlinear mean-field Fokker-Planck equations, including quantum effects.

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-level initial conditions, the caloric curve β(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 the phase transitions between magnetized and nonmagnetized states changes: (i) For f(0)>0.10965, the system displays a second-order phase transition; (ii) for 0.109497

In the thermodynamic limit, the time evolution of isolated long-range interacting systems is properly described by the Vlasov equation. This equation admits nonequilibrium dynamically stable stationary solutions characterized by a zero order parameter. We show that the presence of external noise sources, such as a heat bath, can reduce their lifetime and induce at a specific time a dynamical phase transition marked by a nonzero order parameter. This transition may be used as a distinctive experimental signature of the temporary existence of nonequilibrium Vlasov-stable states. In particular, we present evidence of a regime characterized by an order parameter pulse. Our analytical results are corroborated by numerical simulations of a paradigmatic long-range model., (©2011 American Physical Society)

We provide an exact analytical solution of the collapse dynamics of self-gravitating Brownian particles and bacterial populations at zero temperature. These systems are described by the Smoluchowski-Poisson system or Keller-Segel model in which the diffusion term is neglected. As a result, the dynamics is purely deterministic. A cold system undergoes a gravitational collapse, leading to a finite-time singularity: The central density increases and becomes infinite in a finite time t{coll}. The evolution continues in the postcollapse regime. A Dirac peak emerges, grows, and finally captures all the mass in a finite time t{end}, while the central density excluding the Dirac peak progressively decreases. Close to the collapse time, the pre- and postcollapse evolutions are self-similar. Interestingly, if one starts from a parabolic density profile, one obtains an exact analytical solution that describes the whole collapse dynamics, from the initial time to the end, and accounts for non-self-similar corrections that were neglected in previous works. Our results have possible application in different areas including astrophysics, chemotaxis, colloids, and nanoscience.

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.

A simplified thermodynamic approach of the incompressible axisymmetric Euler equations is considered based on the conservation of helicity, angular momentum, and microscopic energy. Statistical equilibrium states are obtained by maximizing the Boltzmann entropy under these sole constraints. We assume that these constraints are selected by the properties of forcing and dissipation. The fluctuations are found to be Gaussian, while the mean flow is in a Beltrami state. Furthermore, we show that the maximization of entropy at fixed helicity, angular momentum, and microscopic energy is equivalent to the minimization of macroscopic energy at fixed helicity and angular momentum. This provides a justification of this selective decay principle from statistical mechanics. These theoretical predictions are in good agreement with experiments of a von Kármán turbulent flow and provide a way to measure the temperature of turbulence and check fluctuation-dissipation relations. Relaxation equations are derived that could provide an effective description of the dynamics toward the Beltrami state and the progressive emergence of a Gaussian distribution. They can also provide a numerical algorithm to determine maximum entropy states or minimum energy states.

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. For small screenings, the interaction is long range. 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 (without screening) 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.

Systems with long-range interactions display a short-time relaxation toward quasistationary 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 (nonzero 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 unmagnetized states in the theoretically magnetized region as well as persisting magnetized states in the theoretically unmagnetized region. We also report the existence of a region with negative specific heats for QSSs both in the numerical and analytical caloric curves.

On the basis of analytical results and molecular dynamics simulations, we clarify the nonequilibrium dynamics of a long-range interacting system in contact with a heat bath. For small couplings with the bath, we show that the system can first be trapped in a Vlasov quasistationary state, then a microcanonical one follows, and finally canonical equilibrium is reached at the bath temperature. We demonstrate that, even out of equilibrium, Hamiltonian reservoirs microscopically coupled with the system and Langevin thermostats provide equivalent descriptions. Our identification of the key parameters determining the quasistationary lifetimes could be exploited to control experimental systems such as the free-electron laser, in the presence of external noise or inherent imperfections.

We study the critical dynamics of the generalized Smoluchowski-Poisson system (for self-gravitating Langevin particles) or generalized Keller-Segel model (for the chemotaxis of bacterial populations). These models [P. H. Chavanis and C. Sire, Phys. Rev. E 69, 016116 (2004)] are based on generalized stochastic processes leading to the Tsallis statistics. The equilibrium states correspond to polytropic configurations with index n similar to polytropic stars in astrophysics. At the critical index n&#95;{3}=d(d-2) (where d>or=2 is the dimension of space), there exists a critical temperature Theta&#95;{c} (for a given mass) or a critical mass M&#95;{c} (for a given temperature). For Theta>Theta&#95;{c} or MM&#95;{c} the system collapses and forms, in a finite time, a Dirac peak containing a finite fraction M&#95;{c} of the total mass surrounded by a halo. We study these regimes numerically and, when possible, analytically by looking for self-similar or pseudo-self-similar solutions. This study extends the critical dynamics of the ordinary Smoluchowski-Poisson system and Keller-Segel model in d=2 corresponding to isothermal configurations with n&#95;{3}-->+infinity . We also stress the analogy between the limiting mass of white dwarf stars (Chandrasekhar's limit) and the critical mass of bacterial populations in the generalized Keller-Segel model of chemotaxis.

We experimentally characterize the fluctuations of the nonhomogeneous nonisotropic turbulence in an axisymmetric von Kármán flow. We show that these fluctuations satisfy relations, issued from the Euler equation, which are analogous to classical fluctuation-dissipation relations in statistical mechanics. We use these relations to estimate statistical temperatures of turbulence.

We study dynamical phase transitions in systems with long-range interactions, using the Hamiltonian mean field model as a simple example. These systems generically undergo a violent relaxation to a quasistationary state (QSS) before relaxing towards Boltzmann equilibrium. In the collisional regime, the out-of-equilibrium one-particle distribution function (DF) is a quasistationary solution of the Vlasov equation, slowly evolving in time due to finite- N effects. For subcritical energy densities, we exhibit cases where the DF is well fitted by a Tsallis q distribution with an index q(t) slowly decreasing in time from q approximately = 3 (semiellipse) to q=1 (Boltzmann). When the index q(t) reaches an energy-dependent critical value q&#95;(crit) , the nonmagnetized (homogeneous) phase becomes Vlasov unstable and a dynamical phase transition is triggered, leading to a magnetized (inhomogeneous) state. While Tsallis distributions play an important role in our study, we explain this dynamical phase transition by using only conventional statistical mechanics. For supercritical energy densities, we report the existence of a magnetized QSS with a very long lifetime.

A generic feature of systems with long-range interactions is the presence of quasistationary states with non-Gaussian single particle velocity distributions. For the case of the Hamiltonian mean-field model, we demonstrate that a maximum entropy principle applied to the associated Vlasov equation explains known features of such states for a wide range of initial conditions. We are able to reproduce velocity distribution functions with an analytic expression which is derived from the theory with no adjustable parameters. A normal diffusion of angles is detected, which is consistent with Gaussian tails of velocity distributions. A dynamical effect, two oscillating clusters surrounded by a halo, is also found and theoretically justified.

We consider the dynamics of a gas of free bosons within a semiclassical Fokker-Planck equation for which we give a physical justification. In this context, we find a striking similarity between the Bose-Einstein condensation in the canonical ensemble, and the gravitational collapse of a gas of classical self-gravitating Brownian particles. The paper is mainly devoted to the complete study of the Bose-Einstein "collapse" within this model. We find that at the Bose-Einstein condensation temperature Tc, the chemical potential mu(t) vanishes exponentially with a universal rate that we compute exactly. Below Tc, we show analytically that square root mu(t) vanishes linearly in a finite time t coll. After t coll, the mass of the condensate grows linearly with time and saturates exponentially to its equilibrium value for large time. We also give analytical results for the density scaling functions, for the corrections to scaling, and for the exponential relaxation time. Finally, we find that the equilibration time (above Tc) and the collapse time T coll(below Tc) both behave like -T -3 c ln|T-Tc|, near Tc.

We propose a general kinetic and hydrodynamic description of self-gravitating Brownian particles in d dimensions. We go beyond the usual approximations by considering inertial effects and finite-N effects while previous works use a mean-field approximation valid in a proper thermodynamic limit (N --> +infinity) and consider an overdamped regime (xi --> +infinity). We recover known models in some particular cases of our general description. We derive the expression of the virial theorem for self-gravitating Brownian particles and study the linear dynamical stability of isolated clusters of particles and uniform systems by using techniques introduced in astrophysics. We investigate the influence of the equation of state, of the dimension of space, and of the friction coefficient on the dynamical stability of the system. We obtain the exact expression of the critical temperature Tc for a multicomponents self-gravitating Brownian gas in d = 2. We also consider the limit of weak frictions, xi --> 0, and derive the orbit-averaged Kramers equation.

We derive the virial theorem appropriate to the generalized Smoluchowski-Poisson (GSP) system describing self-gravitating Brownian particles in an overdamped limit. We extend previous works by considering the case of an unbounded domain and an arbitrary equation of state. We use the virial theorem to study the diffusion (evaporation) of an isothermal Brownian gas above the critical temperature Tc in dimension d = 2 and show how the effective diffusion coefficient and the Einstein relation are modified by self-gravity. We also study the collapse at T = Tc and show that the central density increases logarithmically with time instead of exponentially in a bounded domain. Finally, for d > 2, we show that the evaporation of the system is essentially a pure diffusion slightly slowed down by self-gravity. We also study the linear dynamical stability of stationary solutions of the GSP system representing isolated clusters of particles and investigate the influence of the equation of state and of the dimension of space on the dynamical stability of the system.

We develop variational principles to study the structure and the stability of equilibrium states of axisymmetric flows. We show that the axisymmetric Euler equations for inviscid flows admit an infinite number of steady state solutions. We find their general form and provide analytical solutions in some special cases. The system can be trapped in one of these steady states as a result of an inviscid violent relaxation. We show that the stable steady states maximize a (nonuniversal) function while conserving energy, helicity, circulation, and angular momentum (robust constraints). This can be viewed as a form of generalized selective decay principle. We derive relaxation equations which can be used as numerical algorithm to construct nonlinearly dynamically stable stationary solutions of axisymmetric flows. We also develop a thermodynamical approach to predict the equilibrium state at some fixed coarse-grained scale. We show that the resulting distribution can be divided in two parts: one universal coming from the conservation of robust invariants and one non-universal determined by the initial conditions through the fragile invariants (for freely evolving systems) or by a prior distribution encoding nonideal effects such as viscosity, small-scale forcing, and dissipation (for forced systems). Finally, we derive a parametrization of inviscid mixing to describe the dynamics of the system at the coarse-grained scale. A conceptual interest of this axisymmetric model is to be intermediate between two-dimensional (2D) and 3D turbulence.

We study the formation and the evolution of velocity distribution tails for systems with weak long-range interactions. In the thermal bath approximation, the evolution of the distribution function of a test particle is governed by a Fokker-Planck equation where the diffusion coefficient depends on the velocity. We extend the theory of Potapenko et al [Phys. Rev. E 56, 7159 (1997)] developed for power-law diffusion coefficients to the case of an arbitrary form of diffusion coefficient and friction force. We study how the structure and the progression of the front depend on the behavior of the diffusion coefficient and friction force for large velocities. Particular emphasis is given to the case where the velocity dependence of the diffusion coefficient is Gaussian. This situation arises in Fokker-Planck equations associated with one dimensional systems with long-range interactions such as the Hamiltonian mean field (HMF) model and in the kinetic theory of two-dimensional point vortices in hydrodynamics. We show that the progression of the front is extremely slow (logarithmic) in that case so that the convergence towards the equilibrium state is peculiar. Our general formalism can have applications for other physical systems such as optical lattices.

We study the thermodynamical properties of a self-gravitating gas with two or more types of particles. Using the method of linear series of equilibria, we determine the structure and stability of statistical equilibrium states in both microcanonical and canonical ensembles. We show how the critical temperature (Jeans instability) and the critical energy (Antonov instability) depend on the relative mass of the particles and on the dimension of space. We then study the dynamical evolution of a multicomponent gas of self-gravitating Brownian particles in the canonical ensemble. Self-similar solutions describing the collapse below the critical temperature are obtained analytically. We find particle segregation, with the scaling profile of the slowest collapsing particles decaying with a nonuniversal exponent that we compute perturbatively in different limits. These results are compared with numerical simulations of the two-species Smoluchowski-Poisson system. Our model of self-attracting Brownian particles also describes the chemotactic aggregation of a multi-species system of bacteria in biology.

We present strategies based upon optimization principles in the case of the axisymmetric equations of magnetohydrodynamics (MHD). We derive the equilibrium state by using a minimum energy principle under the constraints of the MHD axisymmetric equations. We also propose a numerical algorithm based on a maximum energy dissipation principle to compute in a consistent way the nonlinearly dynamically stable equilibrium states. Then, we develop the statistical mechanics of such flows and recover the same equilibrium states giving a justification of the minimum energy principle. We find that fluctuations obey a Gaussian shape and we make the link between the conservation of the Casimirs on the coarse-grained scale and the process of energy dissipation. We contrast these results with those of two-dimensional hydrodynamical turbulence where the equilibrium state maximizes a H function at fixed energy and circulation and where the fluctuations are nonuniversal.

We determine an exact asymptotic expression of the blow-up time t(coll) for self-gravitating Brownian particles or bacterial populations (chemotaxis) close to the critical point in d=3. We show that t(coll) = t(*) (eta- eta(c) )(-1/2) with t(*) =0.917 677 02..., where eta represents the inverse temperature (for Brownian particles) or the mass (for bacterial colonies), and eta(c) is the critical value of eta above which the system blows up. This result is in perfect agreement with the numerical solution of the Smoluchowski-Poisson system. We also determine the exact asymptotic expression of the relaxation time close to but above the critical temperature and derive a large time asymptotic expansion for the density profile exactly at the critical point.

We address the postcollapse dynamics of a self-gravitating gas of Brownian particles in D dimensions in both canonical and microcanonical ensembles. In the canonical ensemble, the postcollapse evolution is marked by the formation of a Dirac peak with increasing mass. The density profile outside the peak evolves self-similarly with decreasing central density and increasing core radius. In the microcanonical ensemble, the postcollapse regime is marked by the formation of a "binarylike" structure surrounded by an almost uniform halo with high temperature. These results are consistent with thermodynamical predictions in astrophysics. We also show that the Smoluchowski-Poisson system describing the collapse of self-gravitating Brownian particles in a strong-friction limit is isomorphic to a simplified version of the Keller-Segel equations describing the chemotactic aggregation of bacterial populations. Therefore, our study has direct applications in this biological context.

We discuss the statistical mechanics of a system of self-gravitating fermions in a space of dimension D. We plot the caloric curves of the self-gravitating Fermi gas giving the temperature as a function of energy and investigate the nature of phase transitions as a function of the dimension of space. We consider stable states (global entropy maxima) as well as metastable states (local entropy maxima). We show that for D> or =4, there exists a critical temperature (for sufficiently large systems) and a critical energy below which the system cannot be found in statistical equilibrium. Therefore, for D> or =4, quantum mechanics cannot stabilize matter against gravitational collapse. This is similar to a result found by Ehrenfest (1917) at the atomic level for Coulomb forces. This makes the dimension D=3 of our Universe very particular with possible implications regarding the anthropic principle. Our study joins a long tradition of scientific and philosophical papers that examined how the dimension of space affects the laws of physics.

We address the generalized thermodynamics and the collapse of a system of self-gravitating Langevin particles exhibiting anomalous diffusion in a space of dimension D. This is a basic model of stochastic particles in interaction. The equilibrium states correspond to polytropic configurations similar to stellar polytropes and polytropic stars. The index n of the polytrope is related to the exponent of anomalous diffusion. We consider a high-friction limit and reduce the problem to the study of the nonlinear Smoluchowacute;ski-Poisson system. We show that the associated Lyapunov functional is the Tsallis free energy. We discuss in detail the equilibrium phase diagram of self-gravitating polytropes as a function of D and n, and determine their stability by using turning point arguments and analytical methods. When no equilibrium state exists, we investigate self-similar solutions of the nonlinear Smoluchowski-Poisson system describing the collapse. Our stability analysis of polytropic spheres can be used to settle the generalized thermodynamical stability of self-gravitating Langevin particles as well as the nonlinear dynamical stability of stellar polytropes, polytropic stars and polytropic vortices. Our study also has applications concerning the chemotactic aggregation of bacterial populations.

We introduce a class of generalized Fokker-Planck equations that conserve energy and mass and increase a generalized entropy functional until a maximum entropy state is reached. Nonlinear Fokker-Planck equations associated with Tsallis entropies are a special case of these equations. Applications of these results to stellar dynamics and vortex dynamics are proposed. Our prime result is a relaxation equation that should offer an easily implementable parametrization of two-dimensional turbulence. Usual parametrizations (including a single turbulent viscosity) correspond to the infinite temperature limit of our model. They forget a fundamental systematic drift that acts against diffusion as in Brownian theory. Our generalized Fokker-Planck equations can have applications in other fields of physics such as chemotaxis for bacterial populations. We propose the idea of a classification of generalized entropies in "classes of equivalence" and provide an aesthetic connection between topics (vortices, stars, bacteria, em leader ) which were previously disconnected.

We address the thermodynamics and the collapse of a self-gravitating gas of Brownian particles in D dimensions, in both canonical and microcanonical ensembles. We study the equilibrium density profile and phase diagram of isothermal spheres and, for 2

A 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 an attractive soft -(r(2)+r(2)(0))(-1/2) Coulomb potential. When the range of regularization is sufficiently short, the self-attracting systems exhibit gravitational or collapselike 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.

We study the thermodynamics and the collapse of a self-gravitating gas of Brownian particles. We consider a high-friction limit in order to simplify the problem. This results in the Smoluchowski-Poisson system. Below a critical energy or below a critical temperature, there is no equilibrium state and the system develops a self-similar collapse leading to a finite time singularity. In the microcanonical ensemble, this corresponds to a "gravothermal catastrophe" and in the canonical ensemble to an "isothermal collapse." Self-similar solutions are investigated analytically and numerically.

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-sphere models. Depending on the values of the parameters, three kinds of phase transitions (of zero, 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 dependence on the degeneracy parameter (for fermions) or on the size of the particles (for a hard-sphere gas). Scaling laws are obtained analytically in the asymptotic limit of a small short distance cutoff. Our analytical model captures the essential physics of the problem and compares remarkably well with the full numerical solutions. We also stress some analogies with the liquid-gas transition and with the Blume-Emery-Griffiths model with infinite range interactions. In particular, our system presents two tricritical points at which the transition passes from first order to second order.

We complete previous investigations on the statistics of velocity fluctuations arising from a random distribution of point vortices in two-dimensional hydrodynamics. We show that, in a statistical sense, the velocity created by a point vortex is shielded by cooperative effects on a distance Lambda approximately n(-1/2), the intervortex separation. For R>>Lambda, the "effective" velocity decays as R-2 instead of the ordinary law R(-1) recovered for R<

We extend the formalism of statistical mechanics developed for the two-dimensional (2D) Euler equation to the case of shallow water system. Relaxation equations towards the maximum entropy state are proposed, which provide a parametrization of subgrid-scale eddies in 2D compressible turbulence.

We develop a kinetic theory for point vortices in two-dimensional hydrodynamics. Using standard projection operator techniques, 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.

We use high resolution numerical simulations over several hundred of turnover times to study the influence of small scale dissipation onto vortex statistics in 2D decaying turbulence. A scaling regime is detected when the scaling laws are expressed in units of mean vorticity and integral scale, like predicted in Carnevale et al., Phys. Rev. Lett. 66, 2735 (1991), and it is observed that viscous effects spoil this scaling regime. The exponent controlling the decay of the number of vortices shows some trends toward xi=1, in agreement with a recent theory based on the Kirchhoff model [C. Sire and P. H. Chavanis, Phys. Rev. E 61, 6644 (2000)]. In terms of scaled variables, the vortices have a similar profile with a functional form related to the Fermi-Dirac distribution.

This paper is devoted to a statistical analysis of the fluctuations of velocity and acceleration produced by a random distribution of point vortices in two-dimensional turbulence. We show that the velocity probability density function PDF behaves in a manner which is intermediate between Gaussian and Levy laws, while the distribution of accelerations is governed by a Cauchy law. Our study accounts properly for a spectrum of circulations among the vortices. In the case of real vortices (with a finite core), we show analytically that the distribution of accelerations makes a smooth transition from Cauchy (for small fluctuations) to Gaussian (for large fluctuations), probably passing through an exponential tail. We introduce a function T(V) which gives the typical duration of a velocity fluctuation V; we show that T(V) behaves like V and V-1 for weak and large velocities, respectively. These results have a simple physical interpretation in the nearest neighbor approximation, and in Smoluchowski theory concerning the persistence of fluctuations. We discuss the analogies with respect to the fluctuations of the gravitational field in stellar systems. As an application of these results, we determine an approximate expression for the diffusion coefficient of point vortices. When applied to the context of freely decaying two-dimensional turbulence, the diffusion becomes anomalous and we establish a relationship nu=1+(xi/2) between the exponent of anomalous diffusion nu and the exponent xi which characterizes the decay of the vortex density.