Results 1  10
of
52
The distribution of free path lengths in the periodic Lorentz gas and related lattice point problems
, 2008
"... The periodic Lorentz gas describes the dynamics of a point particle in a periodic array of spherical scatterers, and is one of the fundamental models for chaotic diffusion. In the present paper we investigate the BoltzmannGrad limit, where the radius of each scatterer tends to zero, and prove the ..."
Abstract

Cited by 45 (19 self)
 Add to MetaCart
The periodic Lorentz gas describes the dynamics of a point particle in a periodic array of spherical scatterers, and is one of the fundamental models for chaotic diffusion. In the present paper we investigate the BoltzmannGrad limit, where the radius of each scatterer tends to zero, and prove the existence of a limiting distribution for the free path length. We also discuss related problems, such as the statistical distribution of directions of lattice points that are visible from a fixed position.
Drift in Phase Space: A New Variational Mechanism With Optimal Diffusion Time
"... We consider nonisochronous, nearly integrable, apriori unstable Hamiltonian systems with a (trigonometric polynomial) O()perturbation which does not preserve the unperturbed tori. We prove the existence of Arnold diffusion with diffusion time T d = O((1/) log(1/)) by a variational method which do ..."
Abstract

Cited by 28 (3 self)
 Add to MetaCart
We consider nonisochronous, nearly integrable, apriori unstable Hamiltonian systems with a (trigonometric polynomial) O()perturbation which does not preserve the unperturbed tori. We prove the existence of Arnold diffusion with diffusion time T d = O((1/) log(1/)) by a variational method which does not require the existence of "transition chains of tori" provided by KAM theory. We also prove that our estimate of the diffusion time T d is optimal as a consequence of a general stability result derived from classical perturbation theory.
The distribution of the free path lengths in the periodic twodimensional Lorentz gas in the smallscatterer limit
, 2003
"... We study the free path length and the geometric free path length in the model of the periodic twodimensional Lorentz gas (Sinai billiard). We give a complete and rigorous proof for the existence of their distributions in the smallscatterer limit and explicitly compute them. As a corollary one get ..."
Abstract

Cited by 27 (6 self)
 Add to MetaCart
(Show Context)
We study the free path length and the geometric free path length in the model of the periodic twodimensional Lorentz gas (Sinai billiard). We give a complete and rigorous proof for the existence of their distributions in the smallscatterer limit and explicitly compute them. As a corollary one gets a complete proof for the existence of the constant term c = 2 − 3ln 2 + 27ζ(3) 2π2 in the asymptotic formula h(T) = −2ln ε + c + o(1) of the KS entropy of the billiard map in this model.
Some considerations on the derivation of the nonlinear quantum Boltzmann equation
 J. Stat. Phys
, 2004
"... Abstract. In this paper we analyze a system of N identical quantum particles in a weakcoupling regime. The time evolution of the Wigner transform of the oneparticle reduced density matrix is represented by means of a perturbative series. The expansion is obtained upon iterating the Duhamel formula ..."
Abstract

Cited by 26 (8 self)
 Add to MetaCart
(Show Context)
Abstract. In this paper we analyze a system of N identical quantum particles in a weakcoupling regime. The time evolution of the Wigner transform of the oneparticle reduced density matrix is represented by means of a perturbative series. The expansion is obtained upon iterating the Duhamel formula. For short times, we rigorously prove that a subseries of the latter, converges to the solution of the Boltzmann equation which is physically relevant in the context. In particular, we recover the transition rate as it is predicted by Fermi's Golden Rule. However, we are not able to prove that the quantity neglected while retaining a subseries of the complete original perturbative expansion, indeed vanishes in the limit: we only give plausibility arguments in this direction. The present study holds in any space dimension d ≥ 2.
Kinetic transport in the twodimensional periodic Lorentz gas
 NONLINEARITY 21 (2008), 1413–1422
, 2008
"... The periodic Lorentz gas describes an ensemble of noninteracting point particles in a periodic array of spherical scatterers. We have recently shown that, in the limit of small scatterer density (BoltzmannGrad limit), the macroscopic dynamics converges to a stochastic process, whose kinetic trans ..."
Abstract

Cited by 20 (12 self)
 Add to MetaCart
(Show Context)
The periodic Lorentz gas describes an ensemble of noninteracting point particles in a periodic array of spherical scatterers. We have recently shown that, in the limit of small scatterer density (BoltzmannGrad limit), the macroscopic dynamics converges to a stochastic process, whose kinetic transport equation is not the linear Boltzmann equation—in contrast to the Lorentz gas with a disordered scatterer configuration. The present paper focuses on the twodimensional setup, and reports an explicit, elementary formula for the collision kernel of the transport equation.
The statistics of the trajectory of a certain billiard in a flat twotorus
 Commun. Math. Phys
"... Abstract. We consider a billiard in the punctured torus obtained by removing a small disk of radius ε> 0 from the flat torus T 2, with trajectory starting from the center of the puncture. In this case the phase space is given by the range of the velocity ω only. Let ˜τε(ω), and respectively ˜ Rε( ..."
Abstract

Cited by 19 (4 self)
 Add to MetaCart
(Show Context)
Abstract. We consider a billiard in the punctured torus obtained by removing a small disk of radius ε> 0 from the flat torus T 2, with trajectory starting from the center of the puncture. In this case the phase space is given by the range of the velocity ω only. Let ˜τε(ω), and respectively ˜ Rε(ω), denote the first exit time (length of the trajectory), and respectively the number of collisions with the side cushions when T 2 is being identified with [0, 1) 2. We prove that the probability measures on [0, ∞) associated with the random variables ε˜τε and ε ˜ Rε are weakly convergent as ε → 0 + and explicitly compute the densities of the limits. 1. Introduction and
Hyperbolic billiards and statistical physics, in
 International Congress of Mathematicians. Vol. II, Eur. Math. Soc., Zürich
"... Abstract. Mathematical theory of billiards is a fascinating subject providing a fertile source of new problems as well as conjecture testing in dynamics, geometry, mathematical physics and spectral theory. This survey is devoted to planar hyperbolic billiards with emphasis on their applications in ..."
Abstract

Cited by 19 (2 self)
 Add to MetaCart
(Show Context)
Abstract. Mathematical theory of billiards is a fascinating subject providing a fertile source of new problems as well as conjecture testing in dynamics, geometry, mathematical physics and spectral theory. This survey is devoted to planar hyperbolic billiards with emphasis on their applications in statistical physics, where they provide many physically interesting and mathematically tractable models. 1.
The BoltzmannGrad limit of the periodic Lorentz gas in two space dimensions
, 2007
"... The periodic Lorentz gas is the dynamical system corresponding to the free motion of a point particle in a periodic system of fixed spherical obstacles of radius r centered at the integer points, assuming all collisions of the particle with the obstacles to be elastic. In this Note, we study this mo ..."
Abstract

Cited by 16 (4 self)
 Add to MetaCart
The periodic Lorentz gas is the dynamical system corresponding to the free motion of a point particle in a periodic system of fixed spherical obstacles of radius r centered at the integer points, assuming all collisions of the particle with the obstacles to be elastic. In this Note, we study this motion on time intervals of order 1/r as r → 0 +.
The average length of a trajectory in a certain billiard in a flat twotorus
, 2003
"... ..."
(Show Context)
ON THE PERIODIC LORENTZ GAS AND THE LORENTZ KINETIC EQUATION
, 2007
"... Abstract. We prove that the BoltzmannGrad limit of the Lorentz gas with periodic distribution of scatterers cannot be described with a linear Boltzmann equation. This is at variance with the case of a Poisson distribution of scatterers, for which the convergence to the linear Boltzmann equation was ..."
Abstract

Cited by 14 (1 self)
 Add to MetaCart
(Show Context)
Abstract. We prove that the BoltzmannGrad limit of the Lorentz gas with periodic distribution of scatterers cannot be described with a linear Boltzmann equation. This is at variance with the case of a Poisson distribution of scatterers, for which the convergence to the linear Boltzmann equation was proved by Gallavotti [Phys. Rev. (2) 185, 308 (1969)]. The arguments presented here complete the analysis in [GolseWennberg, M2AN