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By Fraydoun Rezakhanlou
Entropy and entropy creation have lately turn into mathematical instruments for kinetic and hydrodynamic limits, whilst deriving the macroscopic behaviour of platforms from the interplay dynamics in their many microscopic undemanding components on the atomic or molecular level.
During a different semester on Hydrodynamic Limits on the Centre Émile Borel in Paris, 2001 of the study classes have been held via C. Villani and F. Rezakhanlou. either illustrate the key position of entropy and entropy construction in a mutual and complementary demeanour and feature been written up and up to date for joint book. Villani describes the mathematical concept of convergence to equilibrium for the Boltzmann equation and its relation to varied difficulties and fields, together with info idea, logarithmic Sobolev inequalities and fluid mechanics. Rezakhanlou discusses 4 conjectures for the kinetic behaviour of the not easy sphere types and formulates 4 stochastic diversifications of this version, additionally reviewing recognized effects for these.
Read Online or Download Entropy Methods for the Boltzmann Equation: Lectures from a Special Semester at the Centre Émile Borel, Institut H. Poincaré, Paris, 2001 PDF
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Additional info for Entropy Methods for the Boltzmann Equation: Lectures from a Special Semester at the Centre Émile Borel, Institut H. Poincaré, Paris, 2001
Example text
There may be a third dichotomy in the theory, depending on the behavior of the kernel at small relative velocities (singular or not). Nobody is sure... ). In these lectures, for simplicity I only consider the cutoff case, in the model situations B(v − v∗ , σ) = |v − v∗ |γ (γ > 0), B(v − v∗ , σ) = (1 + |v − v∗ |)−β (β ≥ 0). The most important case of application is certainly that of hard spheres: in dimension 3, B(v − v∗ , σ) = |v − v∗ | (up to a constant). 34 C. 2 Current State of Regularity Theory By gathering many results from different authors, one obtains γ γ ≤ 1, and let f be a Theorem 5.
However, the picture is still not so clear because the Bobylev–Cercignani counterexamples exploit quite well the structure of the Boltzmann operator, and cannot be directly adapted to the context of the Kac equation. In fact, after working on that question with Carlen, Carvalho and Loss I have come to suspect that linear entropy–entropy production inequalities may after all exist for the Kac equation under some moment conditions; there is something to understand here. Anyway, here is now some positive result: it was shown in my Habilitation (2000), that µ−1 n = O(n).
For instance, + + + + Q+ γ (f0 ) = Q (Q (f0 , Q (f0 , f0 )), Q (f0 , f0 )) stands for particles having undergone, first a collision with another particle having undergone no collision, second a collision with a particle who had previously twice collided a particle having undergone no collision before (Exercise: draw the corresponding tree). Finally, α(γ) is a combinatorial coefficient, recursively defined by α(γ) = α(γ1 )α(γ2 ) ; n−1 α(γ) = 1 if n = 1. (γ1 , γ2 are the subtrees of γ). Intuitively, each time a particle undergoes a collision, this will take it closer to equilibrium.