Citation Context

...emma 6.2 follows then by applying a Gronwall Lemma. 6.2. Convergence to the brownian motion. To prove the convergence of the rescaled tagged particle Ξ to a brownian motion, one needs to check (see =-=[5]-=-) • the convergence of the marginals of the tagged particle sampled at different times (6.16) lim N→∞ E ( h1 ( Ξ(τ1) ) . . . h` ( Ξ(τ`) )) = E ( h1 ( B(τ1) ) . . . h` ( B(τ`) )) . We stress that these...

Citation Context

...a th. AP ]s15sM ays20 13 2 THIERRY BODINEAU, ISABELLE GALLAGHER AND LAURE SAINT-RAYMOND best result concerning this low density limit which is due to Lanford in the case of hardspheres [22] (see also =-=[9, 33, 14]-=- for a complete proof) is indeed valid only for short times, i.e. breaks down before any relaxation can be observed. Theorem 1.1. Consider a system of N particles interacting • either as hard-spheres ...

Citation Context

...sentera toujours l’intérêt le plus grand quant aux principes.” A huge literature has been devoted to these asymptotic problems, but up to now they remain still largely open. Important breakthroughs =-=[13, 2]-=- have allowed for a complete study of some hydrodynamic limits of the Boltzmann equation, especially in incompressible viscous regimes leading to the Navier-Stokes equations (see [17] for instance). N...

Citation Context

...5. 33 97 v1s[ ma th. AP ]s15sM ays20 13 2 THIERRY BODINEAU, ISABELLE GALLAGHER AND LAURE SAINT-RAYMOND best result concerning this low density limit which is due to Lanford in the case of hardspheres =-=[22]-=- (see also [9, 33, 14] for a complete proof) is indeed valid only for short times, i.e. breaks down before any relaxation can be observed. Theorem 1.1. Consider a system of N particles interacting • e...

Citation Context

... heat equation. We are going to derive the convergence of λdNf (1) N to (2.16). This boils down to proving that gN can be approximated at large scale by a diffusion which is a standard procedure (see =-=[3]-=-). For the sake of completeness, we recall the salient features of the proof and sketch how they can be adapted to our framework. Step 1: Approximation by the linear Boltzmann equation. The scaling (2...

Citation Context

...sentera toujours l’intérêt le plus grand quant aux principes.” A huge literature has been devoted to these asymptotic problems, but up to now they remain still largely open. Important breakthroughs =-=[13, 2]-=- have allowed for a complete study of some hydrodynamic limits of the Boltzmann equation, especially in incompressible viscous regimes leading to the Navier-Stokes equations (see [17] for instance). N...

Citation Context

...tion of particles, and more generally for all marginals of the N -particle distribution (see Proposition 3.2). This will be the key to control the collision process, and to prove (like in Kac’s model =-=[20]-=- for instance) that dynamics for which a very large number of collisions occur over a short time interval, are of vanishing probability. Note that a similar strategy, based on the existence of the inv...

Citation Context

... metals, is the core of a number of works, and the corresponding literature includes a large variety of contributions. We do not intend to be exhaustive here and refer the reader to the book by Spohn =-=[32]-=- (Chapter 8) for a survey on this topic. We state one basic result due to Gallavotti [15] in the low density limit and then indicate some of the many important research directions. Theorem 2.1. Consid...

Citation Context

...le on the set of functions{ g ∈ L2(Rd, aMβdv), ∫ g(v)Mβ(v)dv = 0 } . The Lemma follows by noticing the decomposition L = a(v) Id − K, where Id stands for the identity and K is a compact operator; see =-=[19]-=- for a proof in the case of the linearized Boltzmann equation, which can be easily adapted to our situation. We define the vector b(v) = ( bk(v) ) k≤d by (6.8) ∀k ≤ d, Lbk(v) = vk , and the diffusion ...

Citation Context

...sphere cross-section, in L∞([0, T ], L1(R2d)). A refinement of this result can be found for instance in [30] in terms of convergence of path measures (and not only of the mean density), as well as in =-=[6]-=- where the convergence is proven for typical scatterer configurations (and not only in average). These convergence statements lead naturally to various questions concerning • the assumptions on the mi...

Citation Context

...ny time T > 0, fN converges to the solution f of the linear Boltzmann equation (1.2), with hard-sphere cross-section, in L∞([0, T ], L1(R2d)). A refinement of this result can be found for instance in =-=[30]-=- in terms of convergence of path measures (and not only of the mean density), as well as in [6] where the convergence is proven for typical scatterer configurations (and not only in average). These co...

Citation Context

...tributed randomly according to some Poisson law; • the obstacles have no dynamics, in particular they do not feel the effect of collisions with the tagged particle. This problem, suggested by Lorentz =-=[25]-=- at the beginning of the twentieth century to study the motion of electrons in metals, is the core of a number of works, and the corresponding literature includes a large variety of contributions. We ...

Citation Context

...of metal are distributed on a periodic network. In this case, the Boltzmann-Grad limit of the Lorentz gas with a periodic distribution of scatterers cannot be described by a linear Boltzmann equation =-=[16, 8, 26, 27]-=-. 2.2. An alternative strategy based on the maximum principle. We adopt here a slightly different point of view, and consider a deterministic system of N hard-spheres, meaning that the tagged particle...

Citation Context

...e one hand, the problem seems more difficult insofar as the background has its own dynamics, which is coupled with the tagged particle. But, on the other hand, pathological situations as described in =-=[16, 7, 8]-=- are not stable: because of the dynamics of the scatterers, we expect the situation to be better since some ergodicity could be retrieved from the additional degrees of freedom. In particular, there a...

Citation Context

...breakthroughs [13, 2] have allowed for a complete study of some hydrodynamic limits of the Boltzmann equation, especially in incompressible viscous regimes leading to the Navier-Stokes equations (see =-=[17]-=- for instance). Note that other regimes such as the compressible Euler limit (which is the most immediate from the formal point of view) are still far from being understood. But, at this stage, the ma...

Citation Context

... ( β 2pi ) d 2 exp ( −β 2 |v|2 ) , β > 0 , for times diverging as N → ∞. Indeed, in diffusive regime, the convergence of the linear Boltzmann process towards the Brownian is by now a classical result =-=[21]-=-. 2. Strategy and main results The good notion of fluctuation is obtained by considering the motion of a tagged particle (or possibly a finite set of tagged particles) in a gas of N particles at equil...

Citation Context

...of metal are distributed on a periodic network. In this case, the Boltzmann-Grad limit of the Lorentz gas with a periodic distribution of scatterers cannot be described by a linear Boltzmann equation =-=[16, 8, 26, 27]-=-. 2.2. An alternative strategy based on the maximum principle. We adopt here a slightly different point of view, and consider a deterministic system of N hard-spheres, meaning that the tagged particle...

Citation Context

...) = ( Dk,`(v) ) k,`≤d (6.9) ∀k, ` ≤ d, LDk,`(v) = vkb`(v)− ∫ Rd vkb`(v)Mβ(v)dv . Note that by construction ∫ Rd bk(v)Mβ(v)dv = ∫ Rd Dk,`Mβ(v)dv = 0. From the symmetry of the model, one can check (see =-=[10]-=- for instance) that there is a function γ such that bk(v) = γ(|v|)vk, with |v|2 = d∑ `=1 v2` . The diffusion coefficient is given by (6.10) κ := ∫ Rd vL−1v Mβ(v)dv = 1 d ∫ Rd γ(|v|)|v|2Mβ(v)dv , where...

Citation Context

... the long time behavior of the system, in particular the relaxation towards thermodynamic equilibrium and hydrodynamic limits. The first point has been addressed by Desvillettes, Pulvirenti and Ricci =-=[11, 12]-=-. Their goal was to derive “singular ” kinetic equations such as the linear Boltzmann equation without angular cut-off or the Fokker-Planck equation, from a system of particles with long-range interac...

Citation Context

... the long time behavior of the system, in particular the relaxation towards thermodynamic equilibrium and hydrodynamic limits. The first point has been addressed by Desvillettes, Pulvirenti and Ricci =-=[11, 12]-=-. Their goal was to derive “singular ” kinetic equations such as the linear Boltzmann equation without angular cut-off or the Fokker-Planck equation, from a system of particles with long-range interac...

Citation Context

...es can be considered by this process), Hilbert further suggested to use Boltzmann’s kinetic equation as an intermediate step to understand the appearance of irreversibility and dissipative mechanisms =-=[18]-=-: “Quant aux principes de la Mécanique, nous possédons déjà au point de vue physique des recherches d’une haute portée; je citerai, par exemple, les écrits de MM. Mach, Hertz, Boltzmann et Volkm...

Citation Context

...t our main results still hold for the low density limit of system of interacting particles with a compactly supported interaction potential (typically satisfying the same assumptions as in [14] or in =-=[29]-=-). The microscopic model is therefore given by the following system of ODEs: for 1 ≤ i 6= j ≤ N , (2.1) dxi dt = vi, dvi dt = 0 as long as |xi(t)− xj(t)| > ε , vi(t +) = vi(t −)− 1 ε2 (vi − vj) · (xi ...

Citation Context

...of metal are distributed on a periodic network. In this case, the Boltzmann-Grad limit of the Lorentz gas with a periodic distribution of scatterers cannot be described by a linear Boltzmann equation =-=[16, 8, 26, 27]-=-. 2.2. An alternative strategy based on the maximum principle. We adopt here a slightly different point of view, and consider a deterministic system of N hard-spheres, meaning that the tagged particle...

Citation Context

...a th. AP ]s15sM ays20 13 2 THIERRY BODINEAU, ISABELLE GALLAGHER AND LAURE SAINT-RAYMOND best result concerning this low density limit which is due to Lanford in the case of hardspheres [22] (see also =-=[9, 33, 14]-=- for a complete proof) is indeed valid only for short times, i.e. breaks down before any relaxation can be observed. Theorem 1.1. Consider a system of N particles interacting • either as hard-spheres ...

Citation Context

...of metal are distributed on a periodic network. In this case, the Boltzmann-Grad limit of the Lorentz gas with a periodic distribution of scatterers cannot be described by a linear Boltzmann equation =-=[16, 8, 26, 27]-=-. 2.2. An alternative strategy based on the maximum principle. We adopt here a slightly different point of view, and consider a deterministic system of N hard-spheres, meaning that the tagged particle...

Citation Context

...ber of collisions occur over a short time interval, are of vanishing probability. Note that a similar strategy, based on the existence of the invariant measure, was already used by Lebowitz and Spohn =-=[24]-=- to derive the linear Boltzmann equation for long times. Let us now give the precise framework of our study. As explained above, the idea is to improve the result by Lanford by considering fluctuation...

Citation Context

... · (xi − xj) < 0 then fN (t, Z out N (i, j)) = fN (t, Z in N (i, j)) 6 THIERRY BODINEAU, ISABELLE GALLAGHER AND LAURE SAINT-RAYMOND with the notation introduced in (2.2)-(2.3). We recall, as shown in =-=[1]-=- for instance, that the set of initial configurations leading to an ill-defined problem (clustering of collision times, or collisions involving more than two particles) is of measure zero in TdNλ ×RdN...

Citation Context

...a th. AP ]s15sM ays20 13 2 THIERRY BODINEAU, ISABELLE GALLAGHER AND LAURE SAINT-RAYMOND best result concerning this low density limit which is due to Lanford in the case of hardspheres [22] (see also =-=[9, 33, 14]-=- for a complete proof) is indeed valid only for short times, i.e. breaks down before any relaxation can be observed. Theorem 1.1. Consider a system of N particles interacting • either as hard-spheres ...