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

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>

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.

The influence of third-order interactions on the density profile of associating hard spheres

1997 ◽  
Vol 91 (4) ◽  
pp. 761-767 ◽  
Author(s):  
D. HENDERSON ◽  
S. SOKOŁOWSKI ◽  
R. ZAGORSKI ◽  
A. TROKHYMCHUK
Keyword(s):  
Density Profile ◽  
Hard Spheres ◽  
Third Order

Glass transition for dipolar hard spheres: a mode-coupling approach

1998 ◽  
Vol 77 (2) ◽  
pp. 305-311 ◽  
Author(s):  
Thomas Scheidsteger, Rolf Schilling
Keyword(s):  
Glass Transition ◽  
Mode Coupling ◽  
Hard Spheres ◽  
Coupling Approach

Quantum effects in a system of Boltzmann hard spheres

2018 ◽  
Vol 189 (06) ◽  
pp. 659-664
Author(s):  
Sergei M. Stishov
Keyword(s):  
Hard Spheres ◽  
Quantum Effects
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