419 research outputs found
Tanaka Theorem for Inelastic Maxwell Models
We show that the Euclidean Wasserstein distance is contractive for inelastic
homogeneous Boltzmann kinetic equations in the Maxwellian approximation and its
associated Kac-like caricature. This property is as a generalization of the
Tanaka theorem to inelastic interactions. Consequences are drawn on the
asymptotic behavior of solutions in terms only of the Euclidean Wasserstein
distance
Kinematics of Tycho-2 Red Giant Clump Stars
Based on the Ogorodnikov-Milne model, we analyze the proper motions of 95 633
red giant clump (RGC) stars from the Tycho-2 Catalogue. The following Oort
constants have been found: A = 15.9+-0.2 km/s/kpc and B = -12.0+-0.2 km/s/kpc.
Using 3632 RGC stars with known proper motions, radial velocities, and
photometric distances, we show that, apart from the star centroid velocity
components relative to the Sun, only the model parameters that describe the
stellar motions in the XY plane differ significantly from zero. We have studied
the contraction (a negative K-effect) of the system of RGC stars as a function
of their heliocentric distance and elevation above the Galactic plane. For a
sample of distant (500--1000 pc) RGC stars located near the Galactic plane
(|Z|<200 pc) with an average distance of d=0.7 kpc, the contraction velocity is
shown to be Kd= -3.5+-0.9 km/s; a noticeable vertex deviation, lxy = 9.1+-0.5
degrees, is also observed for them. For stars located well above the Galactic
plane (|Z|>=200 pc), these effects are less pronounced, Kd = -1.7+-0.5 km/s and
lxy = 4.9+-0.6 degrees. Using RGC stars, we have found a rotation around the
Galactic X axis directed toward the Galactic center with an angular velocity of
-2.5+-0.3 km/s/kpc, which we associate with the warp of the Galactic
stellar-gaseous disk.Comment: 23 pages, 7 figures, 4 table
Solving the Boltzmann equation in N log N
In [C. Mouhot and L. Pareschi, "Fast algorithms for computing the Boltzmann
collision operator," Math. Comp., to appear; C. Mouhot and L. Pareschi, C. R.
Math. Acad. Sci. Paris, 339 (2004), pp. 71-76], fast deterministic algorithms
based on spectral methods were derived for the Boltzmann collision operator for
a class of interactions including the hard spheres model in dimension three.
These algorithms are implemented for the solution of the Boltzmann equation in
two and three dimension, first for homogeneous solutions, then for general non
homogeneous solutions. The results are compared to explicit solutions, when
available, and to Monte-Carlo methods. In particular, the computational cost
and accuracy are compared to those of Monte-Carlo methods as well as to those
of previous spectral methods. Finally, for inhomogeneous solutions, we take
advantage of the great computational efficiency of the method to show an
oscillation phenomenon of the entropy functional in the trend to equilibrium,
which was suggested in the work [L. Desvillettes and C. Villani, Invent. Math.,
159 (2005), pp. 245-316].Comment: 32 page
Probabilistic study of the speed of approach to equilibrium for an inelastic Kac model
This paper deals with a one--dimensional model for granular materials, which
boils down to an inelastic version of the Kac kinetic equation, with
inelasticity parameter . In particular, the paper provides bounds for
certain distances -- such as specific weighted --distances and the
Kolmogorov distance -- between the solution of that equation and the limit. It
is assumed that the even part of the initial datum (which determines the
asymptotic properties of the solution) belongs to the domain of normal
attraction of a symmetric stable distribution with characteristic exponent
\a=2/(1+p). With such initial data, it turns out that the limit exists and is
just the aforementioned stable distribution. A necessary condition for the
relaxation to equilibrium is also proved. Some bounds are obtained without
introducing any extra--condition. Sharper bounds, of an exponential type, are
exhibited in the presence of additional assumptions concerning either the
behaviour, near to the origin, of the initial characteristic function, or the
behaviour, at infinity, of the initial probability distribution function
Galactic Rotation Parameters from Data on Open Star Clusters
Currently available data on the field of velocities Vr, Vl, Vb for open star
clusters are used to perform a kinematic analysis of various samples that
differ by heliocentric distance, age, and membership in individual structures
(the Orion, Carina--Sagittarius, and Perseus arms). Based on 375 clusters
located within 5 kpc of the Sun with ages up to 1 Gyr, we have determined the
Galactic rotation parameters
Wo =-26.0+-0.3 km/s/kpc,
W'o = 4.18+-0.17 km/s/kpc^2,
W''o=-0.45+-0.06 km/s/kpc^3, the system contraction parameter K = -2.4+-0.1
km/s/kpc, and the parameters of the kinematic center Ro =7.4+-0.3 kpc and lo =
0+-1 degrees. The Galactocentric distance Ro in the model used has been found
to depend significantly on the sample age. Thus, for example, it is 9.5+-0.7
kpc and 5.6+-0.3 kpc for the samples of young (50 Myr)
clusters, respectively. Our study of the kinematics of young open star clusters
in various spiral arms has shown that the kinematic parameters are similar to
the parameters obtained from the entire sample for the Carina-Sagittarius and
Perseus arms and differ significantly from them for the Orion arm. The
contraction effect is shown to be typical of star clusters with various ages.
It is most pronounced for clusters with a mean age of 100 Myr, with the
contraction velocity being Kr = -4.3+-1.0 km/s.Comment: 14 pages, 4 figures, 2 table
Cooling process for inelastic Boltzmann equations for hard spheres, Part II: Self-similar solutions and tail behavior
We consider the spatially homogeneous Boltzmann equation for inelastic hard
spheres, in the framework of so-called constant normal restitution
coefficients. We prove the existence of self-similar solutions, and we give
pointwise estimates on their tail. We also give general estimates on the tail
and the regularity of generic solutions. In particular we prove Haff 's law on
the rate of decay of temperature, as well as the algebraic decay of
singularities. The proofs are based on the regularity study of a rescaled
problem, with the help of the regularity properties of the gain part of the
Boltzmann collision integral, well-known in the elastic case, and which are
extended here in the context of granular gases.Comment: 41 page
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