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 $p>0$. In particular, the paper provides bounds for certain distances -- such as specific weighted $\chi$--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

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

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

On a kinetic model for a simple market economy

In this paper, we consider a simple kinetic model of economy involving both exchanges between agents and speculative trading. We show that the kinetic model admits non trivial quasi-stationary states with power law tails of Pareto type. In order to do this we consider a suitable asymptotic limit of the model yielding a Fokker-Planck equation for the distribution of wealth among individuals. For this equation the stationary state can be easily derived and shows a Pareto power law tail. Numerical results confirm the previous analysis

On the speed of approach to equilibrium for a collisionless gas

We investigate the speed of approach to Maxwellian equilibrium for a collisionless gas enclosed in a vessel whose wall are kept at a uniform, constant temperature, assuming diffuse reflection of gas molecules on the vessel wall. We establish lower bounds for potential decay rates assuming uniform $L^p$ bounds on the initial distribution function. We also obtain a decay estimate in the spherically symmetric case. We discuss with particular care the influence of low-speed particles on thermalization by the wall.Comment: 22 pages, 1 figure; submitted to Kinetic and Related Model

Kinetic models with randomly perturbed binary collisions

We introduce a class of Kac-like kinetic equations on the real line, with general random collisional rules, which include as particular cases models for wealth redistribution in an agent-based market or models for granular gases with a background heat bath. Conditions on these collisional rules which guarantee both the existence and uniqueness of equilibrium profiles and their main properties are found. We show that the characterization of these stationary solutions is of independent interest, since the same profiles are shown to be solutions of different evolution problems, both in the econophysics context and in the kinetic theory of rarefied gases

Open Clusters IC 4665 and Cr 359 and a Probable Birthplace of the Pulsar PSR B1929+10

Based on the epicyclic approximation, we have simulated the motion of the young open star clusters IC 4665 and Collinder 359. The separation between the cluster centers is shown to have been minimal 7 Myr ago, 36 pc. We have established a close evolutionary connection between IC 4665 and the Scorpius-Centaurus association -- the separation between the centers of these structures was $\approx200$ pc 15 Myr ago. In addition, the center of IC 4665 at this time was near two well-known regions of coronal gas: the Local Bubble and the North Polar Spur. The star HIP 86768 is shown to be one of the candidates for a binary (in the past) with the pulsar PSR B1929+10. At the model radial velocity of the pulsar $V_r= 2\pm50$ km s$^{-1}$, a close encounter of this pair occurs in the vicinity of IC 4665 at a time of -1.1 Myr. At the same time, using currently available data for the pulsar B1929+10 at its model radial velocity $V_r=200\pm50$ km s$^{-1}$, we show that the hypothesis of Hoogerwerf et al. (2001) about the breakup of the $\zeta$Oph--B1929+10 binary in the vicinity of Upper Scorpius (US) about 0.9 Myr ago is more plausible.Comment: 19 pages, 8 figure

Evolution of the Velocity Ellipsoids in the Thin Disk of the Galaxy and the Radial Migration of Stars

Data from the revised Geneva--Copenhagen catalog are used to study the influence of radial migration of stars on the age dependences of parameters of the velocity ellipsoids for nearby stars in the thin disk of the Galaxy, assuming that the mean radii of the stellar orbits remain constant. It is demonstrated that precisely the radial migration of stars, together with the negative metallicity gradient in the thin disk,are responsible for the observed negative correlation between the metallicities and angular momenta of nearby stars, while the angular momenta of stars that were born at the same Galactocentric distances do not depend on either age or metallicity. (abridged)Comment: Astronomy Reports, Vol. 86 No. 9, P.1117-1126 (2009

Analysis of Peculiarities of the Stellar Velocity Field in the Solar Neighborhood

Based on a new version of the Hipparcos catalogue and an updated Geneva-Copenhagen survey of F and G dwarfs, we analyze the space velocity field of about 17000 single stars in the solar neighborhood. The main known clumps, streams, and branches (Pleiades, Hyades, Sirius, Coma Berenices, Hercules, Wolf 630-alpha Ceti, and Arcturus) have been identified using various approaches. The evolution of the space velocity field for F and G dwarfs has been traced as a function of the stellar age. We have managed to confirm the existence of the recently discovered KFR08 stream. We have found 19 Hipparcos stars, candidates for membership in the KFR08 stream, and obtained an isochrone age estimate for the stream, 13 Gyr. The mean stellar ages of the Wolf 630-alpha Ceti and Hercules streams are shown to be comparable, 4--6 Gyr. No significant differences in the metallicities of stars belonging to these streams have been found. This is an argument for the hypothesis that these streams owe their origin to a common mechanism.Comment: 23 pages, 9 figure

Relationship between the Velocity Ellipsoids of Galactic-Disk Stars and their Ages and Metallicities

The dependences of the velocity ellipsoids of F-G stars of the thin disk of the Galaxy on their ages and metallicities are analyzed based on the new version of the Geneva-Copenhagen Catalog. The age dependences of the major, middle, and minor axes of the ellipsoids, and also of the dispersion of the total residual veltocity, obey power laws with indices 0.25,0.29,0.32, and 0.27 (with uncertainties \pm 0.02). Due to the presence of thick-disk objects, the analogous indices for all nearby stars are about a factor of 1.5 larger. Attempts to explain such values are usually based on modeling relaxation processes in the Galactic disk. With increasing age, the velocity ellipsoid increases in size and becomes appreciably more spherical, turns toward the direction of the Galactic center, and loses angular momentum. The shape of the velocity ellipsoid remains far from equilibrium. With increasing metallicity, the velocity ellipsoid for stars of mixed age increases in size, displays a weak tendency to become more spherical, and turns toward the direction of the Galactic center (with these changes occurring substantially more rapidly in the transition through the metallicity [Fe/H]= -0.25). Thus, the ellipsoid changes similarly to the way it does with age; however, with decreasing metallicity, the rotational velocity about the Galactic center monotonically increases, rather than decreases(!). Moreover, the power-law indices for the age dependences of the axes depend on the metallicity, and display a maximum near [Fe/H]=-0.1. The age dependences of all the velocity-ellipsoid parameters for stars with equal metallicity are roughly the same. It is proposed that the appearance of a metallicity dependence of the velocity ellipsoids for thin-disk stars is most likely due to the radial migration of stars.Comment: 15 pages, 6 figures, accepted 2009, Astronomy Reports, Vol. 53 No. 9, P.785-80