Quantitative analysis of the correlations in the Boltzmann-Grad limit for hard spheres

2014 ◽  
Author(s):  
M. Pulvirenti
Keyword(s):  
Quantitative Analysis ◽  
Hard Spheres ◽  
Grad Limit

scholarly journals Boltzmann-Grad limit of a hard sphere system in a box with isotropic boundary conditions

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Corentin Le Bihan
Keyword(s):  
Boltzmann Equation ◽  
Boundary Conditions ◽  
Hard Sphere ◽  
Hard Spheres ◽  
Compact Domain ◽  
The Boltzmann Equation ◽  
Grad Limit ◽  
Random Direction ◽  
Time Restriction ◽  
Short Time

<p style='text-indent:20px;'>In this paper we present a rigorous derivation of the Boltzmann equation in a compact domain with {isotropic} boundary conditions. We consider a system of <inline-formula><tex-math id="M1">\begin{document}$ N $\end{document}</tex-math></inline-formula> hard spheres of diameter <inline-formula><tex-math id="M2">\begin{document}$ \epsilon $\end{document}</tex-math></inline-formula> in a box <inline-formula><tex-math id="M3">\begin{document}$ \Lambda : = [0, 1]\times(\mathbb{R}/\mathbb{Z})^2 $\end{document}</tex-math></inline-formula>. When a particle meets the boundary of the domain, it is instantaneously reinjected into the box with a random direction, {but} conserving kinetic energy. We prove that the first marginal of the process converges in the scaling <inline-formula><tex-math id="M4">\begin{document}$ N\epsilon^2 = 1 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M5">\begin{document}$ \epsilon\rightarrow 0 $\end{document}</tex-math></inline-formula> to the solution of the Boltzmann equation, with the same short time restriction of Lanford's classical theorem.</p>

Existence of the Boltzmann-Grad limit for an infinite system of hard spheres

1990 ◽  
Vol 83 (1) ◽  
pp. 402-418 ◽  
Author(s):  
V. I. Gerasimenko ◽  
D. Ya. Petrina
Keyword(s):  
Infinite System ◽  
Hard Spheres ◽  
Grad Limit

From hard spheres dynamics to the Stokes–Fourier equations: An L2 analysis of the Boltzmann–Grad limit

2015 ◽  
Vol 353 (7) ◽  
pp. 623-627 ◽  
Author(s):  
Thierry Bodineau ◽  
Isabelle Gallagher ◽  
Laure Saint-Raymond
Keyword(s):  
Hard Spheres ◽  
Grad Limit

scholarly journals On the validity of the Boltzmann equation for short range potentials

2014 ◽  
Vol 26 (02) ◽  
pp. 1450001 ◽  
Author(s):  
M. Pulvirenti ◽  
C. Saffirio ◽  
S. Simonella
Keyword(s):  
Boltzmann Equation ◽  
Short Range ◽  
Hard Spheres ◽  
Classical System ◽  
Short Range Potential ◽  
Point Particles ◽  
Smooth Potential ◽  
Grad Limit ◽  
Repulsive Potentials ◽  
Short Times

We consider a classical system of point particles interacting by means of a short range potential. We prove that, in the low-density (Boltzmann–Grad) limit, the system behaves, for short times, as predicted by the associated Boltzmann equation. This is a revisitation and an extension of the thesis of King [9] (that appeared after the well-known result of Lanford [10] for hard spheres) and of a recent paper by Gallagher et al. [5]. Our analysis applies to any stable and smooth potential. In the case of repulsive potentials (with no attractive parts), we estimate explicitly the rate of convergence.

An Improved Quantitative Image Analyser

1977 ◽  
Vol 35 ◽  
pp. 226-227
Author(s):  
J.P. Fallon ◽  
P.J. Gregory ◽  
C.J. Taylor
Keyword(s):  
Feature Extraction ◽  
Quantitative Analysis ◽  
Frequency Content ◽  
General Sense ◽  
Physical Measurements ◽  
Quantitative Image ◽  
Image Analyser ◽  
Automatic Methods ◽  
Quantitative Results

Quantitative image analysis systems have been used for several years in research and quality control applications in various fields including metallurgy and medicine. The technique has been applied as an extension of subjective microscopy to problems requiring quantitative results and which are amenable to automatic methods of interpretation.Feature extraction. In the most general sense, a feature can be defined as a portion of the image which differs in some consistent way from the background. A feature may be characterized by the density difference between itself and the background, by an edge gradient, or by the spatial frequency content (texture) within its boundaries. The task of feature extraction includes recognition of features and encoding of the associated information for quantitative analysis.Quantitative Analysis. Quantitative analysis is the determination of one or more physical measurements of each feature. These measurements may be straightforward ones such as area, length, or perimeter, or more complex stereological measurements such as convex perimeter or Feret's diameter.

Quantification of Silica Uptake by Alveolar Macrophages - An Emperical Scanning Electron Microprobe Method

1982 ◽  
Vol 40 ◽  
pp. 332-333
Author(s):  
V. V. Damiano ◽  
R. P. Daniele ◽  
H. T. Tucker ◽  
J. H. Dauber
Keyword(s):  
Quantitative Analysis ◽  
Alveolar Macrophages ◽  
Bulk Sample ◽  
Silica Particles ◽  
Empirical Method ◽  
Phagocytic Cells ◽  
Calibration Standards ◽  
X Ray ◽  
Accurate Quantitative Analysis ◽  
Scanning Electron

An important example of intracellular particles is encountered in silicosis where alveolar macrophages ingest inspired silica particles. The quantitation of the silica uptake by these cells may be a potentially useful method for monitoring silica exposure. Accurate quantitative analysis of ingested silica by phagocytic cells is difficult because the particles are frequently small, irregularly shaped and cannot be visualized within the cells. Semiquantitative methods which make use of particles of known size, shape and composition as calibration standards may be the most direct and simplest approach to undertake. The present paper describes an empirical method in which glass microspheres were used as a model to show how the ratio of the silicon Kα peak X-ray intensity from the microspheres to that of a bulk sample of the same composition correlated to the mass of the microsphere contained within the cell. Irregular shaped silica particles were also analyzed and a calibration curve was generated from these data.

Lateral resolution of the electron microbeam analysis

1978 ◽  
Vol 36 (1) ◽  
pp. 542-543
Author(s):  
H.J. Dudek
Keyword(s):  
Quantitative Analysis ◽  
Depth Resolution ◽  
Lateral Resolution ◽  
Cylindrical Particle ◽  
Correction Procedure ◽  
X Ray ◽  
Element Concentrations ◽  
Microbeam Analysis ◽  
The Matrix

The chemical inhomogenities in modern materials such as fibers, phases and inclusions, often have diameters in the region of one micrometer. Using electron microbeam analysis for the determination of the element concentrations one has to know the smallest possible diameter of such regions for a given accuracy of the quantitative analysis.In th is paper the correction procedure for the quantitative electron microbeam analysis is extended to a spacial problem to determine the smallest possible measurements of a cylindrical particle P of high D (depth resolution) and diameter L (lateral resolution) embeded in a matrix M and which has to be analysed quantitative with the accuracy q. The mathematical accounts lead to the following form of the characteristic x-ray intens ity of the element i of a particle P embeded in the matrix M in relation to the intensity of a standard S

An example of spectrum imaging used for comparison of EELS quantitative analysis techniques on Al-Li

1991 ◽  
Vol 49 ◽  
pp. 726-727
Author(s):  
John A. Hunt
Keyword(s):  
Quantitative Analysis ◽  
Large Data ◽  
Difference Spectrum ◽  
Large Data Sets ◽  
Foil Thickness ◽  
Data Sets ◽  
Analysis Techniques ◽  
Spectrum Imaging ◽  
Normal Spectrum ◽  
Electron Energy Loss

Spectrum-imaging is a useful technique for comparing different processing methods on very large data sets which are identical for each method. This paper is concerned with comparing methods of electron energy-loss spectroscopy (EELS) quantitative analysis on the Al-Li system. The spectrum-image analyzed here was obtained from an Al-10at%Li foil aged to produce δ' precipitates that can span the foil thickness. Two 1024 channel EELS spectra offset in energy by 1 eV were recorded and stored at each pixel in the 80x80 spectrum-image (25 Mbytes). An energy range of 39-89eV (20 channels/eV) are represented. During processing the spectra are either subtracted to create an artifact corrected difference spectrum, or the energy offset is numerically removed and the spectra are added to create a normal spectrum. The spectrum-images are processed into 2D floating-point images using methods and software described in [1].

The use of a Bent Grid to Improve the take-Off Angle for Electron Probe Analysis

1976 ◽  
Vol 34 ◽  
pp. 360-361
Author(s):  
Delbert E. Philpott ◽  
David Leaffer
Keyword(s):  
Quantitative Analysis ◽  
Tilt Angle ◽  
Count Rate ◽  
Electron Probe ◽  
Open Area ◽  
Electron Probe Analysis ◽  
Background Ratio ◽  
Probe Analysis

There are certain advantages for electron probe analysis if the sample can be tilted directly towards the detector. The count rate is higher, it optimizes the geometry since only one angle need be taken into account for quantitative analysis and the signal to background ratio is improved. The need for less tilt angle may be an advantage because the grid bars are not moved quite as close to each other, leaving a little more open area for observation. Our present detector (EDAX) and microscope (Philips 300) combination precludes moving the detector behind the microscope where it would point directly at the grid. Therefore, the angle of the specimen was changed in order to optimize the geometry between the specimen and the detector.

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