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.

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

We develop the suggestion that dark matter could be a Bose-Einstein condensate. We determine the mass-radius relation of a Newtonian self-gravitating Bose-Einstein condensate with short-range interactions described by the Gross-Pitaevskii-Poisson system. We numerically solve the equation of hydrostatic equilibrium describing the balance between the gravitational attraction and the pressure due to quantum effects (Heisenberg's uncertainty principle) and short-range interactions (scattering). We connect the non-interacting limit to the Thomas-Fermi limit. We also consider the case of attractive self-interaction. We compare the exact mass-radius relation obtained numerically with the approximate analytical relation obtained with a Gaussian ansatz. An overall good agreement is found

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&

Random transitions described by the stochastic Smoluchowski-Poisson system and by the stochastic Keller-Segel model

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. 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 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 $\Delta F$ divided by $k_B T$. 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 $e^N$ and is considerable for $N\gg 1$. 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 $T_c$, 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

Relativistic stars with a linear equation of state: analogy with classical isothermal spheres and black holes

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.Comment: A minor mistake in calculation has been corrected in the second version (v2