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

Exact diffusion coefficient of self-gravitating Brownian particles in two dimensions

We derive the exact expression of the diffusion coefficient of a self-gravitating Brownian gas in two dimensions. Our formula generalizes the usual Einstein relation for a free Brownian motion to the context of two-dimensional gravity. We show the existence of a critical temperature T_{c} at which the diffusion coefficient vanishes. For T<T_{c} the diffusion coefficient is negative and the gas undergoes gravitational collapse. This leads to the formation of a Dirac peak concentrating the whole mass in a finite time. We also stress that the critical temperature T_{c} is different from the collapse temperature T_{*} at which the partition function diverges. These quantities differ by a factor 1-1/N where N is the number of particles in the system. We provide clear evidence of this difference by explicitly solving the case N=2. We also mention the analogy with the chemotactic aggregation of bacteria in biology, the formation of ``atoms'' in a two-dimensional (2D) plasma and the formation of dipoles or supervortices in 2D point vortex dynamics

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. The equilibrium states correspond to polytropic distributions. 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 Smoluchowski-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 points arguments and analytical methods. When no equilibrium state exists, we investigate self-similar solutions describing the collapse. These results can be relevant for astrophysical systems, two-dimensional vortices and for the chemotaxis of bacterial populations. Above all, this model constitutes a prototypical dynamical model of systems with long-range interactions which possesses a rich structure and which can be studied in great detail.Comment: Submitted to Phys. Rev.

Lynden-Bell and Tsallis distributions for the HMF model

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

Curious behaviour of the diffusion coefficient and friction force for the strongly inhomogeneous HMF model

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

Quasilinear theory of the 2D Euler equation

We develop a quasilinear theory of the 2D Euler equation and derive an integro-differential equation for the evolution of the coarse-grained vorticity. This equation respects all the invariance properties of the Euler equation and conserves angular momentum in a circular domain and linear impulse in a channel. We show under which hypothesis we can derive a H-theorem for the Fermi-Dirac entropy and make the connection with statistical theories of 2D turbulence.Comment: 4 page

Thermodynamics of self-gravitating systems

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