First Star-Forming Structures in Fuzzy Cosmic Filaments.

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.

Supernovae: an example of complexity in the physics of compressible fluids.

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.

Phase transitions in self-gravitating systems and bacterial populations surrounding a central body.

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].

Self-gravitating clusters of Bose-Einstein gas with planar, cylindrical, or spherical symmetry: Gaseous density profiles and onset of condensation.

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.

Self-gravitating clusters of Fermi-Dirac gas with planar, cylindrical, or spherical symmetry: Evolution of density profiles with temperature.

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.

Long-term relaxation of one-dimensional self-gravitating systems.

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.

Caloric curves of classical self-gravitating systems in general relativity.

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.

Kinetic theory of one-dimensional homogeneous long-range interacting systems with an arbitrary potential of interaction.

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.

Microcanonical quasistationarity of long-range interacting systems in contact with a heat bath.

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.

Secular dynamics of long-range interacting particles on a sphere in the axisymmetric limit.

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.

Kinetic theory of one-dimensional homogeneous long-range interacting systems sourced by 1/N

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.

Critical dynamics of self-gravitating Langevin particles and bacterial populations.

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_{3}=d(d-2) (where d>or=2 is the dimension of space), there exists a critical temperature Theta_{c} (for a given mass) or a critical mass M_{c} (for a given temperature). For Theta>Theta_{c} or MM_{c} the system collapses and forms, in a finite time, a Dirac peak containing a finite fraction M_{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_{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.

Dynamical phase transitions in long-range Hamiltonian systems and Tsallis distributions with a time-dependent index.

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_(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.

Maximum entropy principle explains quasistationary states in systems with long-range interactions: the example of the Hamiltonian mean-field model.

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.

Virial theorem and dynamical evolution of self-gravitating Brownian particles in an unbounded domain. I. Overdamped models.

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.

Virial theorem and dynamical evolution of self-gravitating Brownian particles in an unbounded domain. II. Inertial models.

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.

Dynamics of the Bose-Einstein condensation: analogy with the collapse dynamics of a classical self-gravitating Brownian gas.

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.

Relaxation of the distribution function tails for systems described by Fokker-Planck equations.

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.

Self-gravitating Brownian systems and bacterial populations with two or more types of particles.

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.

Generalized thermodynamics and Fokker-Planck equations: applications to stellar dynamics and two-dimensional turbulence.

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.