295 research outputs 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&
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 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.
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