Structure preserving schemes for mean-field equations of collective behavior

In this paper we consider the development of numerical schemes for mean-field equations describing the collective behavior of a large group of interacting agents. The schemes are based on a generalization of the classical Chang-Cooper approach and are capable to preserve the main structural properties of the systems, namely nonnegativity of the solution, physical conservation laws, entropy dissipation and stationary solutions. In particular, the methods here derived are second order accurate in transient regimes whereas they can reach arbitrary accuracy asymptotically for large times. Several examples are reported to show the generality of the approach.Comment: Proceedings of the XVI International Conference on Hyperbolic Problem

Characterization of multilayer stack parameters from X-ray reflectivity data using the PPM program: measurements and comparison with TEM results

Future hard (10 -100 keV) X-ray telescopes (SIMBOL-X, Con-X, HEXIT-SAT, XEUS) will implement focusing optics with multilayer coatings: in view of the production of these optics we are exploring several deposition techniques for the reflective coatings. In order to evaluate the achievable optical performance X-Ray Reflectivity (XRR) measurements are performed, which are powerful tools for the in-depth characterization of multilayer properties (roughness, thickness and density distribution). An exact extraction of the stack parameters is however difficult because the XRR scans depend on them in a complex way. The PPM code, developed at ERSF in the past years, is able to derive the layer-by-layer properties of multilayer structures from semi-automatic XRR scan fittings by means of a global minimization procedure in the parameters space. In this work we will present the PPM modeling of some multilayer stacks (Pt/C and Ni/C) deposited by simple e-beam evaporation. Moreover, in order to verify the predictions of PPM, the obtained results are compared with TEM profiles taken on the same set of samples. As we will show, PPM results are in good agreement with the TEM findings. In addition, we show that the accurate fitting returns a physically correct evaluation of the variation of layers thickness through the stack, whereas the thickness trend derived from TEM profiles can be altered by the superposition of roughness profiles in the sample image

HEXIT-SAT: a mission concept for X-ray grazing incidence telescopes from 0.5 to 70 keV

While the energy density of the Cosmic X-ray Background (CXB) provides a statistical estimate of the super massive black hole (SMBH) growth and mass density in the Universe, the lack, so far, of focusing instrument in the 20-60 keV (where the CXB energy density peaks), frustrates our effort to obtain a comprehensive picture of the SMBH evolutionary properties. HEXIT-SAT (High Energy X-ray Imaging Telescope SATellite) is a mission concept capable of exploring the hard X-ray sky with focusing/imaging instrumentation, to obtain an unbiased census of accreting SMBH up to the redshifts where galaxy formation peaks, and on extremely wide luminosity ranges. This will represent a leap forward comparable to that achieved in the soft X-rays by the Einstein Observatory in the late 70'. In addition to accreting SMBH, and very much like the Einstein Observatory, this mission would also have the capabilities of investigating almost any type of the celestial X-ray sources. HEXIT-SAT is based on high throughput (>400 cm2 @ 30 keV; >1200 cm2 @ 1 keV), high quality (15 arcsec Half Power Diameter) multi-layer optics, coupled with focal plane detectors with high efficiency in the full 0.5-70keV range. Building on the BeppoSAX experience, a low-Earth, equatorial orbit, will assure a low and stable particle background, and thus an extremely good sensitivity for faint hard X-ray sources. At the flux limits of 1/10 microCrab (10-30 keV) and 1/3 microCrab (20-40 keV) (reachable in one Msec observation) we should detect ~100 and ~40 sources in the 15 arcmin FWHM Field of View respectively, thus resolving >80% and ~65% of the CXB where its energy density peaks.Comment: to appear in Proceeedings of SPIE Vol. 5488, UV to Gamma Ray Space Telescope System

Design and advancement status of the Beam Expander Testing X-ray facility (BEaTriX)

The BEaTriX (Beam Expander Testing X-ray facility) project is an X-ray apparatus under construction at INAF/OAB to generate a broad (200 x 60 mm2), uniform and low-divergent X-ray beam within a small lab (6 x 15 m2). BEaTriX will consist of an X-ray source in the focus a grazing incidence paraboloidal mirror to obtain a parallel beam, followed by a crystal monochromation system and by an asymmetrically-cut diffracting crystal to perform the beam expansion to the desired size. Once completed, BEaTriX will be used to directly perform the quality control of focusing modules of large X-ray optics such as those for the ATHENA X-ray observatory, based on either Silicon Pore Optics (baseline) or Slumped Glass Optics (alternative), and will thereby enable a direct quality control of angular resolution and effective area on a number of mirror modules in a short time, in full X-ray illumination and without being affected by the finite distance of the X-ray source. However, since the individual mirror modules for ATHENA will have an optical quality of 3-4 arcsec HEW or better, BEaTriX is required to produce a broad beam with divergence below 1-2 arcsec, and sufficient flux to quickly characterize the PSF of the module without being significantly affected by statistical uncertainties. Therefore, the optical components of BEaTriX have to be selected and/or manufactured with excellent optical properties in order to guarantee the final performance of the system. In this paper we report the final design of the facility and a detailed performance simulation.Comment: Accepted paper, pre-print version. The finally published manuscript can be downloaded from http://dx.doi.org/10.1117/12.223895

Mesoscopic modelling of financial markets

We derive a mesoscopic description of the behavior of a simple financial market where the agents can create their own portfolio between two investment alternatives: a stock and a bond. The model is derived starting from the Levy-Levy-Solomon microscopic model (Econ. Lett., 45, (1994), 103--111) using the methods of kinetic theory and consists of a linear Boltzmann equation for the wealth distribution of the agents coupled with an equation for the price of the stock. From this model, under a suitable scaling, we derive a Fokker-Planck equation and show that the equation admits a self-similar lognormal behavior. Several numerical examples are also reported to validate our analysis

An asymptotic preserving scheme for the Kac model of the Boltzmann equation in the diffusion limit

International audienceIn this paper we propose a numerical scheme to solve the Kac model of the Boltzmann equation for multiscale rarefied gas dynamics. This scheme is uniformly stable with respect to the Knudsen number, consistent with the fluid-diffusion limit for small Knudsen numbers, and with the Kac equation in the kinetic regime. Our approach is based on the micro-macro decomposition which leads to an equivalent formulation of the Kac model that couples a kinetic equation with macroscopic ones. This method is validated with various test cases and compared to other standard methods

Science with Simbol-X

Simbol-X is a French-Italian mission, with a participation of German laboratories, for X-ray astronomy in the wide 0.5-80 keV band. Taking advantage of emerging technology in mirror manufacturing and spacecraft formation flying, Simbol-X will push grazing incidence imaging up to ~80 keV, providing an improvement of roughly three orders of magnitude in sensitivity and angular resolution compared to all instruments that have operated so far above 10 keV. This will open a new window in X-ray astronomy, allowing breakthrough studies on black hole physics and census and particle acceleration mechanisms. We describe briefly the main scientific goals of the Simbol-X mission, giving a few examples aimed at highlighting key issues of the Simbol-X design.Comment: Proc. of the workshop "Simbol-X: The hard X-ray universe in focus", Bologna 14-16 May, 200

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

Gravitational waves in dynamical spacetimes with matter content in the Fully Constrained Formulation

The Fully Constrained Formulation (FCF) of General Relativity is a novel framework introduced as an alternative to the hyperbolic formulations traditionally used in numerical relativity. The FCF equations form a hybrid elliptic-hyperbolic system of equations including explicitly the constraints. We present an implicit-explicit numerical algorithm to solve the hyperbolic part, whereas the elliptic sector shares the form and properties with the well known Conformally Flat Condition (CFC) approximation. We show the stability andconvergence properties of the numerical scheme with numerical simulations of vacuum solutions. We have performed the first numerical evolutions of the coupled system of hydrodynamics and Einstein equations within FCF. As a proof of principle of the viability of the formalism, we present 2D axisymmetric simulations of an oscillating neutron star. In order to simplify the analysis we have neglected the back-reaction of the gravitational waves into the dynamics, which is small (<2 %) for the system considered in this work. We use spherical coordinates grids which are well adapted for simulations of stars and allow for extended grids that marginally reach the wave zone. We have extracted the gravitational wave signature and compared to the Newtonian quadrupole and hexadecapole formulae. Both extraction methods show agreement within the numerical errors and the approximations used (~30 %).Comment: 17 pages, 9 figures, 2 tables, accepted for publication in PR