Pierre-Henri Chavanis

Mass-radius relation of Newtonian self-gravitating Bose-Einstein condensates with short-range interactions: I. Analytical results

Several recent astrophysical observations of distant type Ia supernovae have revealed that the content of the universe is made of about 70% of dark energy, 25% of dark matter and 5% of baryonic (visible) matter [1]. Thus, the overwhelming preponderance of matter and energy in the universe is believed to be dark i.e. unobservable by telescopes. The dark energy is responsible for the accelerated expansion of the universe. Its origin is mysterious and presumably related to the cosmological constant. Dark energy is usually interpreted as a vacuum energy and it behaves like a fluid with negative pressure. Dark matter also is mysterious. The suggestion that dark matter may constitute a large part of the universe was raised by Zwicky [2] in 1933. Using the virial theorem to infer the average mass of galaxies within the Coma cluster, he obtained a value much larger than the mass of luminous material. He realized therefore that some mass was “missing” in order to account for observations. This missing mass problem was confirmed later by accurate measurements of rotation curves of disc galaxies [3, 4]. The rotation curves of neutral hydrogen clouds in spiral galaxies measured from the Doppler effect are found to be roughly flat (instead of Keplerian) with a typical rotational velocity v∞ ∼ 200km/s up to the maximum observed radius of about 50 kpc. This mass profile is much more extended than the distribution of starlight which typically converges within ∼ 10 kpc. This implies that galaxies are surrounded by an extended halo of dark matter whose mass M(r) = rv 2/G increases linearly with radius [56]. This can be conveniently modeled by an isothermal self-gravitating gas the density of which scales asymptotically as r −2 [6].

PHASE TRANSITIONS IN SELF-GRAVITATING SYSTEMS

We discuss the nature of phase transitions in self-gravitating systems. We show the connection between the binary star model of Padmanabhan, the thermodynamics of stellar systems and the thermodynamics of self-gravitating fermions. We stress the inequivalence of statistical ensembles for systems with long-range interactions, like gravity. In particular, we contrast the microcanonical evolution of stellar systems from the canonical evolution of self-gravitating Brownian particles. At low energies, self-gravitating Hamiltonian systems experience a gravothermal catastrophe in the microcanonical ensemble. At low temperatures, self-gravitating Brownian systems experience an isothermal collapse in the canonical ensemble. For classical particles, the gravothermal catastrophe leads to a binary star surrounded by a hot halo while the isothermal collapse leads to a Dirac peak containing all the mass. For self-gravitating fermions, the collapse stops when quantum degeneracy comes into play through the Pauli exclusio...

Thermodynamics and collapse of self-gravitating Brownian particles in D dimensions.

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<D<10, determine the onset of instability in the series of equilibria. We also study the dynamics of self-gravitating Brownian particles in a high friction limit leading to the Smoluchowski-Poisson system. Self-similar solutions describing the collapse are investigated analytically and numerically. In the canonical ensemble (fixed temperature), we derive the analytic form of the density scaling profile which decays as f(x) approximately x(-alpha), with alpha=2. In the microcanonical ensemble (fixed energy), we show that f decays as f(x) approximately x(-alpha(max)), where alpha(max) is a nontrivial 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 with, in particular, the formation of a Dirac peak in the density profile.

Nonlinear mean field Fokker-Planck equations. Application to the chemotaxis of biological populations

Abstract.We study a general class of nonlinear mean field Fokker-Planck equations in relation with an effective generalized thermodynamical (E.G.T.) formalism. We show that these equations describe several physical systems such as: chemotaxis of bacterial populations, Bose-Einstein condensation in the canonical ensemble, porous media, generalized Cahn-Hilliard equations, Kuramoto model, BMF model, Burgers equation, Smoluchowski-Poisson system for self-gravitating Brownian particles, Debye-Hückel theory of electrolytes, two-dimensional turbulence... In particular, we show that nonlinear mean field Fokker-Planck equations can provide generalized Keller-Segel models for the chemotaxis of biological populations. As an example, we introduce a new model of chemotaxis incorporating both effects of anomalous diffusion and exclusion principle (volume filling). Therefore, the notion of generalized thermodynamics can have applications for concrete physical systems. We also consider nonlinear mean field Fokker-Planck equations in phase space and show the passage from the generalized Kramers equation to the generalized Smoluchowski equation in a strong friction limit. Our formalism is simple and illustrated by several explicit examples corresponding to Boltzmann, Tsallis, Fermi-Dirac and Bose-Einstein entropies among others.

Gravitational instability of isothermal and polytropic spheres

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 connexion with a generalized thermodynamical approach proposed by Tsallis. We determine in each case the onset of gravitational instability by analytical methods and graphical constructions in the Milne plane. We also discuss the relation between dynamical and thermodynamical stability of stellar systems and gaseous spheres. Our study provides an aesthetic and simple approach to this otherwise complicated subject.

Anomalous diffusion and collapse of self-gravitating Langevin particles in D dimensions.

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.

Growth of perturbations in an expanding universe with Bose-Einstein condensate dark matter

We study the growth of perturbations in an expanding Newtonian universe with Bose-Einstein condensate (BEC) dark matter. We first ignore special relativistic effects and derive a differential equation that governs the evolution of the density contrast in the linear regime. This equation, which takes quantum pressure and self-interaction into account, can be solved analytically in several cases. We argue that an attractive self-interaction can enhance the Jeans instability and fasten the formation of structures. Then, we take pressure effects (coming from special relativity) into account in the evolution of the cosmic fluid and add the contribution of radiation, baryons, and dark energy (cosmological constant). For BEC dark matter with repulsive self-interaction (positive pressure) the scale factor increases more rapidly than in the standard ΛCDM model where dark matter is pressureless, while it increases less rapidly for BEC dark matter with attractive self-interaction (negative pressure). We study the linear development of the perturbations in these two cases and show that the perturbations grow faster in BEC dark matter than in pressureless dark matter. Finally, we consider a “dark fluid” with a generalized equation of state p = (αρ + kρ 2 )c 2 having a component p = kρ 2 c 2 similar to BEC dark matter and a component p = αρc 2 mimicking the effect of the cosmological constant (dark energy). We find optimal parameters that give good

Dynamics and thermodynamics of a simple model similar to self-gravitating systems: the HMF model

Abstract.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.

Coarse-grained distributions and superstatistics

We show an interesting connection between non-standard (non-Boltzmannian) distribution functions arising in the theory of violent relaxation for collisionless stellar systems [D. Lynden-Bell, Mon. Not. R. Astron. Soc. 136 (1967) 101.] and the notion of superstatistics recently introduced by [Beck and Cohen Physica A 322 (2003) 267]. The common link between these two theories is the emergence of coarse-grained distributions arising out of fine-grained distributions. The coarse-grained distribution functions are written as a superposition of Boltzmann factors weighted by a non-universal function. Even more general distributions can arise in case of incomplete violent relaxation (non-ergodicity). They are stable stationary solutions of the Vlasov equation. We also discuss analogies and differences between the statistical equilibrium state of a multi-components self-gravitating system and the metaequilibrium (or quasi-equilibrium) states of a collisionless stellar system. Finally, we stress the important distinction between entropies, generalized entropies, relative entropies and H-functions. We discuss applications of these ideas in two-dimensional turbulence and for other systems with long-range interactions.

Hamiltonian and Brownian systems with long-range interactions: II. Kinetic equations and stability analysis

We discuss the kinetic theory of systems with long-range interactions. We contrast the microcanonical description of an isolated Hamiltonian system described by the Liouville equation from the canonical description of a stochastically forced Brownian system described by the Fokker–Planck equation. We show that the mean-field approximation is exact in a proper thermodynamic limit. For N→+∞, a Hamiltonian system is described by the Vlasov equation. In this collisionless regime, coherent structures can emerge from a process of violent relaxation. These metaequilibrium states, or quasi-stationary states (QSS), are stable stationary solutions of the Vlasov equation. To order 1/N, the collision term of a homogeneous system has the form of the Lenard–Balescu operator. It reduces to the Landau operator when collective effects are neglected. The statistical equilibrium state (Boltzmann) is obtained on a collisional timescale of order N or larger (when the Lenard–Balescu operator cancels out). We also consider the stochastic motion of a test particle in a bath of field particles and derive the general form of the Fokker–Planck equation describing the evolution of the velocity distribution of the test particle. The diffusion coefficient is anisotropic and depends on the velocity of the test particle. For Brownian systems, in the N→+∞ limit, the kinetic equation is a non-local Kramers equation. In the strong friction limit ξ→+∞, or for large times t⪢ξ-1, it reduces to a non-local Smoluchowski equation. We give explicit results for self-gravitating systems, 2D vortices and for the HMF model. We also introduce a generalized class of stochastic processes and derive the corresponding generalized Fokker–Planck equations. We discuss how a notion of generalized thermodynamics can emerge in complex systems displaying anomalous diffusion.