Abstract:
We develop the idea proposed by Barge & Sommeria (1995) that large-scale vortices present in the solar nebula can concentrate dust particles and facilitate the formation of planetesimals and planets. We introduce an exact vortex solution of the incompressible 2D Euler equation (Kida vortex) and study the motion of dust particles in that vortex. In particular, we derive an analytical expression of the capture time as a function of the friction coefficient and determine the parameters leading to an optimal capture.

Abstract:
We consider a cosmological model based on a quadratic equation of state (where is the Planck density and is the cosmological density) “unifying” vacuum energy, radiation, and dark energy. For , it reduces to leading to a phase of early accelerated expansion (early inflation) with a constant density equal to the Planck density ？g/m3 (vacuum energy). For , we recover the equation of state of radiation . For , we get leading to a phase of late accelerated expansion (late inflation) with a constant density equal to the cosmological density ？g/m3 (dark energy). The temperature is determined by a generalized Stefan-Boltzmann law. We show a nice “symmetry” between the early universe (vacuum energy + radiation) and the late universe (radiation + dark energy). In our model, they are described by two polytropic equations of state with index and respectively. Furthermore, the Planck density in the early universe plays a role similar to that of the cosmological density in the late universe. They represent fundamental upper and lower density bounds differing by 122 orders of magnitude. We add the contribution of baryonic matter and dark matter considered as independent species and obtain a simple cosmological model describing the whole evolution of the universe. We study the evolution of the scale factor, density, and temperature. This model gives the same results as the standard CDM model for , where is the Planck time and completes it by incorporating the phase of early inflation in a natural manner. Furthermore, this model does not present any singularity at and exists eternally in the past (although it may be incorrect to extrapolate the solution to the infinite past). Our study suggests that vacuum energy, radiation, and dark energy may be the manifestation of a unique form of “generalized radiation.” By contrast, the baryonic and dark matter components of the universe are treated as different species. This is at variance with usual models (quintessence, Chaplygin gas, ...) trying to unify dark matter and dark energy. 1. Introduction The evolution of the universe may be divided into four main periods [1]. In the vacuum energy era (Planck era), the universe undergoes a phase of early inflation that brings it from the Planck size ？m to an almost “macroscopic” size ？m in a tiniest fraction of a second [2–5]. The universe then enters in the radiation era and, when the temperature cools down below approximately 103？K, in the matter era [6]. Finally, in the dark energy era (de Sitter era), the universe undergoes a phase of late inflation [7]. The early inflation is

Abstract:
We introduce stochastic models of chemotaxis generalizing the deterministic Keller-Segel model. These models include fluctuations which are important in systems with small particle numbers or close to a critical point. Following Dean's approach, we derive the exact kinetic equation satisfied by the density distribution of cells. In the mean field limit where statistical correlations between cells are neglected, we recover the Keller-Segel model governing the smooth density field. We also consider hydrodynamic and kinetic models of chemotaxis that take into account the inertia of the particles and lead to a delay in the adjustment of the velocity of cells with the chemotactic gradient. We make the connection with the Cattaneo model of chemotaxis and the telegraph equation.

Abstract:
We complete classical investigations concerning the dynamical stability of an infinite homogeneous gaseous medium described by the Euler-Poisson system or an infinite homogeneous stellar system described by the Vlasov-Poisson system (Jeans problem). To determine the stability of an infinite homogeneous stellar system with respect to a perturbation of wavenumber k, we apply the Nyquist method. We first consider the case of single-humped distributions and show that, for infinite homogeneous systems, the onset of instability is the same in a stellar system and in the corresponding barotropic gas, contrary to the case of inhomogeneous systems. We show that this result is true for any symmetric single-humped velocity distribution, not only for the Maxwellian. If we specialize on isothermal and polytropic distributions, analytical expressions for the growth rate, damping rate and pulsation period of the perturbation can be given. Then, we consider the Vlasov stability of symmetric and asymmetric double-humped distributions (two-stream stellar systems) and determine the stability diagrams depending on the degree of asymmetry. We compare these results with the Euler stability of two self-gravitating gaseous streams. Finally, we determine the corresponding stability diagrams in the case of plasmas and compare the results with self-gravitating systems.

Abstract:
We derive the proper form of Virial theorem for a system of rotating self-gravitating Brownian particles. We show that, in the two-dimensional case, it takes a very simple form that can be used to obtain general results about the dynamics of the system without being required to solve the Smoluchowski-Poisson system explicitly. We also develop the analogy between self-gravitating systems and two-dimensional point vortices and derive a Virial-like relation for the vortex system.

Abstract:
We propose a formal extension of thermodynamics and kinetic theories to a larger class of entropy functionals. Kinetic equations associated to Boltzmann, Fermi, Bose and Tsallis entropies are recovered as a special case. This formalism first provides a unifying description of classical and quantum kinetic theories. On the other hand, a generalized thermodynamical framework is justified to describe complex systems exhibiting anomalous diffusion. Finally, a notion of generalized thermodynamics emerges in the context of the the violent relaxation of collisionless stellar systems and two-dimensional vortices due to the existence of Casimir invariants and incomplete relaxation. A thermodynamical analogy can also be developed to analyze the nonlinear dynamical stability of stationary solutions of the Vlasov and 2D Euler-Poisson systems. On general grounds, we suggest that generalized entropies arise due to the existence of ``hidden constraints'' that modify the form of entropy that we would naively expect. Generalized kinetic equations are therefore ``effective'' equations that are introduced heuristically to describe complex systems.

Abstract:
We discuss a general class of nonlinear mean-field Fokker-Planck equations [P.H. Chavanis, Phys. Rev. E, 68, 036108 (2003)] and show their applications in different domains of physics, astrophysics and biology. These equations are associated with generalized entropic functionals and non-Boltzmannian distributions (Fermi-Dirac, Bose-Einstein, Tsallis,...). They furthermore involve an arbitrary binary potential of interaction. We emphasize analogies between different topics (two-dimensional turbulence, self-gravitating systems, Debye-H\"uckel theory of electrolytes, porous media, chemotaxis of bacterial populations, Bose-Einstein condensation, BMF model, Cahn-Hilliard equations,...) which were previously disconnected. All these examples (and probably many others) are particular cases of this general class of nonlinear mean-field Fokker-Planck equations.

Abstract:
We complete our previous investigation concerning the structure and the stability of "isothermal" spheres in general relativity. This concerns objects that are described by a linear equation of state $P=q\epsilon$ so that the pressure is proportional to the energy density. In the Newtonian limit $q\to 0$, this returns the classical isothermal equation of state. We consider specifically a self-gravitating radiation (q=1/3), the core of neutron stars (q=1/3) and a gas of baryons interacting through a vector meson field (q=1). We study how the thermodynamical parameters scale with the size of the object and find unusual behaviours due to the non-extensivity of the system. We compare these scaling laws with the area scaling of the black hole entropy. We also determine the domain of validity of these scaling laws by calculating the critical radius above which relativistic stars described by a linear equation of state become dynamically unstable. For photon stars, we show that the criteria of dynamical and thermodynamical stability coincide. Considering finite spheres, we find that the mass and entropy as a function of the central density present damped oscillations. We give the critical value of the central density, corresponding to the first mass peak, above which the series of equilibria becomes unstable. Finally, we extend our results to d-dimensional spheres. We show that the oscillations of mass versus central density disappear above a critical dimension d_{crit}(q). For Newtonian isothermal stars (q=0) we recover the critical dimension d_{crit}=10. For the stiffest stars (q=1) we find d_{crit}=9 and for a self-gravitating radiation (q=1/d) we find d_{crit}=9.96404372... very close to 10. Finally, we give analytical solutions of relativistic isothermal spheres in 2D gravity.

Abstract:
We study the growth of correlations in systems with weak long-range interactions. Starting from the BBGKY hierarchy, we determine the evolution of the two-body correlation function by using an expansion of the solutions of the hierarchy in powers of 1/N in a proper thermodynamic limit $N\to +\infty$. These correlations are responsible for the ``collisional'' evolution of the system beyond the Vlasov regime due to finite $N$ effects. We obtain a general kinetic equation that can be applied to spatially inhomogeneous systems and that takes into account memory effects. These peculiarities are specific to systems with unshielded long-range interactions. For spatially homogeneous systems with short memory time like plasmas, we recover the classical Landau (or Lenard-Balescu) equations. An interest of our approach is to develop a formalism that remains in physical space (instead of Fourier space) and that can deal with spatially inhomogeneous systems. This enlightens the basic physics and provides novel kinetic equations with a clear physical interpretation. However, unless we restrict ourselves to spatially homogeneous systems, closed kinetic equations can be obtained only if we ignore some collective effects between particles. General exact coupled equations taking into account collective effects are also given. We use this kinetic theory to discuss the processes of violent collisionless relaxation and slow collisional relaxation in systems with weak long-range interactions. In particular, we investigate the dependence of the relaxation time with the system size and provide a coherent discussion of all the numerical results obtained for these systems.

Abstract:
We develop the kinetic theory of Hamiltonian systems with weak long-range interactions. Starting from the Klimontovich equation and using a quasilinear theory, we obtain a general kinetic equation that can be applied to spatially inhomogeneous systems and that takes into account memory effects. This equation is valid at order 1/N in a proper thermodynamic limit and it coincides with the kinetic equation obtained from the BBGKY hierarchy. For N tending to infinity, it reduces to the Vlasov equation describing collisionless systems. We describe the process of phase mixing and violent relaxation leading to the formation of a quasi stationary state (QSS) on the coarse-grained scale. We interprete the physical nature of the QSS in relation to Lynden-Bell's statistical theory and discuss the problem of incomplete relaxation. In the second part of the paper, we consider the relaxation of a test particle in a thermal bath. We derive a Fokker-Planck equation by directly calculating the diffusion tensor and the friction force from the Klimontovich equation. We give general expressions of these quantities that are valid for possibly spatially inhomogeneous systems with long correlation time. We show that the diffusion and friction terms have a very similar structure given by a sort of generalized Kubo formula. We also obtain non-markovian kinetic equations that can be relevant when the auto-correlation function of the force decreases slowly with time. An interest of our approach is to develop a formalism that remains in physical space (instead of Fourier space) and that can deal with spatially inhomogeneous systems.