521 research outputs found
Escape of stars from gravitational clusters in the Chandrasekhar model
We study the evaporation of stars from globular clusters using the simplified
Chandrasekhar model. This is an analytically tractable model giving reasonable
agreement with more sophisticated models that require complicated numerical
integrations. In the Chandrasekhar model: (i) the stellar system is assumed to
be infinite and homogeneous (ii) the evolution of the velocity distribution of
stars f(v,t) is governed by a Fokker-Planck equation, the so-called
Kramers-Chandrasekhar equation (iii) the velocities |v| that are above a
threshold value R>0 (escape velocity) are not counted in the statistical
distribution of the system. In fact, high velocity stars leave the system, due
to free evaporation or to the attraction of a neighboring galaxy (tidal
effects). Accordingly, the total mass and energy of the system decrease in
time. If the star dynamics is described by the Kramers-Chandrasekhar equation,
the mass decreases to zero exponentially rapidly. Our goal is to obtain
non-perturbative analytical results that complement the seminal studies of
Chandrasekhar, Michie and King valid for large times and large
escape velocities . In particular, we obtain an exact
semi-explicit solution of the Kramers-Chandrasekhar equation with the absorbing
boundary condition f(R,t)=0. We use it to obtain an explicit expression of the
mass loss at any time t when . We also derive an exact integral
equation giving the exponential evaporation rate , and the
corresponding eigenfunction , when for any
sufficiently large value of the escape velocity R. For , we
obtain an explicit expression of the evaporation rate that refines the
Chandrasekhar results
Kinetic equations with Maxwell boundary conditions
We prove global stability results of {\sl DiPerna-Lions} renormalized
solutions for the initial boundary value problem associated to some kinetic
equations, from which existence results classically follow. The (possibly
nonlinear) boundary conditions are completely or partially diffuse, which
includes the so-called Maxwell boundary conditions, and we prove that it is
realized (it is not only a boundary inequality condition as it has been
established in previous works). We are able to deal with Boltzmann,
Vlasov-Poisson and Fokker-Planck type models. The proofs use some trace
theorems of the kind previously introduced by the author for the Vlasov
equations, new results concerning weak-weak convergence (the renormalized
convergence and the biting -weak convergence), as well as the
Darroz\`es-Guiraud information in a crucial way
From Vlasov-Poisson and Vlasov-Poisson-Fokker-Planck Systems to Incompressible Euler Equations: the case with finite charge
We study the asymptotic regime of strong electric fields that leads from the
Vlasov-Poisson system to the Incompressible Euler equations. We also deal with
the Vlasov-Poisson-Fokker-Planck system which induces dissipative effects. The
originality consists in considering a situation with a finite total charge
confined by a strong external field. In turn, the limiting equation is set in a
bounded domain, the shape of which is determined by the external confining
potential. The analysis extends to the situation where the limiting density is
non-homogeneous and where the Euler equation is replaced by the Lake Equation,
also called Anelastic Equation.Comment: 39 pages, 3 figure
Kinetic theory of point vortices in two dimensions: analytical results and numerical simulations
We develop the kinetic theory of point vortices in two-dimensional
hydrodynamics and illustrate the main results of the theory with numerical
simulations. We first consider the evolution of the system "as a whole" and
show that the evolution of the vorticity profile is due to resonances between
different orbits of the point vortices. The evolution stops when the profile of
angular velocity becomes monotonic even if the system has not reached the
statistical equilibrium state (Boltzmann distribution). In that case, the
system remains blocked in a sort of metastable state with a non standard
distribution. We also study the relaxation of a test vortex in a steady bath of
field vortices. The relaxation of the test vortex is described by a
Fokker-Planck equation involving a diffusion term and a drift term. The
diffusion coefficient, which is proportional to the density of field vortices
and inversely proportional to the shear, usually decreases rapidly with the
distance. The drift is proportional to the gradient of the density profile of
the field vortices and is connected to the diffusion coefficient by a
generalized Einstein relation. We study the evolution of the tail of the
distribution function of the test vortex and show that it has a front
structure. We also study how the temporal auto-correlation function of the
position of the test vortex decreases with time and find that it usually
exhibits an algebraic behavior with an exponent that we compute analytically.
We mention analogies with other systems with long-range interactions
Weakly collisional Landau damping and three-dimensional Bernstein-Greene-Kruskal modes: New results on old problems
Landau damping and Bernstein-Greene-Kruskal (BGK) modes are among the most
fundamental concepts in plasma physics. While the former describes the
surprising damping of linear plasma waves in a collisionless plasma, the latter
describes exact undamped nonlinear solutions of the Vlasov equation. There does
exist a relationship between the two: Landau damping can be described as the
phase-mixing of undamped eigenmodes, the so-called Case-Van Kampen modes, which
can be viewed as BGK modes in the linear limit. While these concepts have been
around for a long time, unexpected new results are still being discovered. For
Landau damping, we show that the textbook picture of phase-mixing is altered
profoundly in the presence of collision. In particular, the continuous spectrum
of Case-Van Kampen modes is eliminated and replaced by a discrete spectrum,
even in the limit of zero collision. Furthermore, we show that these discrete
eigenmodes form a complete set of solutions. Landau-damped solutions are then
recovered as true eigenmodes (which they are not in the collisionless theory).
For BGK modes, our interest is motivated by recent discoveries of electrostatic
solitary waves in magnetospheric plasmas. While one-dimensional BGK theory is
quite mature, there appear to be no exact three-dimensional solutions in the
literature (except for the limiting case when the magnetic field is
sufficiently strong so that one can apply the guiding-center approximation). We
show, in fact, that two- and three-dimensional solutions that depend only on
energy do not exist. However, if solutions depend on both energy and angular
momentum, we can construct exact three-dimensional solutions for the
unmagnetized case, and two-dimensional solutions for the case with a finite
magnetic field. The latter are shown to be exact, fully electromagnetic
solutions of the steady-state Vlasov-Poisson-Amp\`ere system
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