By Fraydoun Rezakhanlou
Entropy and entropy creation have lately turn into mathematical instruments for kinetic and hydrodynamic limits, whilst deriving the macroscopic behaviour of structures from the interplay dynamics in their many microscopic hassle-free ingredients on the atomic or molecular level.
During a distinct semester on Hydrodynamic Limits on the Centre Émile Borel in Paris, 2001 of the learn classes have been held by way of C. Villani and F. Rezakhanlou. either illustrate the foremost position of entropy and entropy construction in a mutual and complementary demeanour and feature been written up and up-to-date for joint e-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 versions and formulates 4 stochastic adaptations of this version, additionally reviewing recognized effects for these.
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Additional resources for Entropy Methods for the Boltzmann Equation: Lectures from a Special Semester at the Centre Émile Borel, Institut H. Poincaré, Paris, 2001
The ﬁrst EEP inequality for the Boltzmann operator was established by Carlen and Carvalho (1992), . This a very clever, complicated construction, and the function Θ appearing in the EEP inequality was very, very ﬂat close to 0. A huge improvement was then provided by Toscani and Villani , and recently I improved again on these results . In the sequel I shall describe a subset of what is presently known. 22 C. Villani Theorem 3 (). Let B satisfy B(v − v∗ , σ) ≥ KB (1 + |v − v∗ |)−β Then, for all for some β ≥ 0.
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. Consider the kernel B ⎛ = |v⎞− v∗ | ,⎛0 <⎞ 1 1 f (v) ⎝ v ⎠ dv = ⎝ 0 ⎠.
So (radius)2 Here I(f ) = I St+ 4n t n−1 f ≤ e−4nt I(St f ), and we have − d −4nt n I S(1+ 4n )t f (n) e D(St f (n) ) ≥ dt n−1 n−1 +∞ If one dt, one ﬁnds 0 D f (n) ≥ = n n−1 +∞ I S 0 ( 1+ 4n n−1 )t f (n) dt = n−1 n n − 1 5n − 1 +∞ I(St f ) 0 n H f (n) . 5n − 1 Physical interpretation. This model enhances the relaxation of fast particles, and this is precisely what the Boltzmann equation needs for the fast decrease of entropy. 2 The Central Limit Theorem for Maxwellian Molecules In the particular case when B(v − v∗ , σ) = b(cos θ) [θ = deviation angle as usual], then there is a strong analogy between the Q+ operator Q+ (f, f ) = dv∗ dσ b(cos θ)f f∗ 1 Entropy Production and Convergence to Equilibrium 45 and the rescaled convolution operator f1/2 ∗ f1/2 = 1 2N/2 · · 1 f √ ∗ N/2 f √ .