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On the Distribution of Free Path Lengths for the Periodic Lorentz Gas
"... Consider the domain Z " = fx 2 Rn j dist(x; "Zn) ? "flg; and let the free path length be defined as ..."
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Consider the domain Z " = fx 2 Rn j dist(x; "Zn) ? "flg; and let the free path length be defined as
Quantum diffusion of the random Schrödinger evolution in the scaling limit II. The recollision diagrams
"... We consider random Schrödinger equations on R d for d ≥ 3 with a homogeneous AndersonPoisson type random potential. Denote by λ the coupling constant and ψt the solution with initial data ψ0. The space and time variables scale as x ∼ λ −2−κ/2,t ∼ λ −2−κ with 0 < κ < κ0(d). We prove that, in t ..."
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We consider random Schrödinger equations on R d for d ≥ 3 with a homogeneous AndersonPoisson type random potential. Denote by λ the coupling constant and ψt the solution with initial data ψ0. The space and time variables scale as x ∼ λ −2−κ/2,t ∼ λ −2−κ with 0 < κ < κ0(d). We prove that, in the limit λ → 0, the expectation of the Wigner distribution of ψt converges weakly to the solution of a heat equation in the space variable x for arbitrary L 2 initial data. The proof is based on analyzing the phase cancellations of multiple scatterings on the random potential by expanding the propagator into a sum of Feynman graphs. In this paper we consider the nonrecollision graphs and prove that the amplitude of the nonladder diagrams is smaller than their “naive size ” by an extra λ c factor per non(anti)ladder vertex for some c> 0. This is the first rigorous result showing that the
FokkerPlanck equations as scaling limits of reversible quantum systems
 J. Stat. Phys
"... We consider a quantum particle moving in a harmonic exterior potential and linearly coupled to a heat bath of quantum oscillators. Caldeira and Leggett [6] have derived the FokkerPlanck equation with friction for the Wigner distribution of the particle in the large temperature limit, however their ..."
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Cited by 25 (6 self)
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We consider a quantum particle moving in a harmonic exterior potential and linearly coupled to a heat bath of quantum oscillators. Caldeira and Leggett [6] have derived the FokkerPlanck equation with friction for the Wigner distribution of the particle in the large temperature limit, however their (nonrigorous) derivation was not free of criticism, especially since the limiting equation is not of Lindblad form. In this paper we recover the correct form of their result in a rigorous way. We also point out that the source of the diffusion is physically restrictive under this scaling. We investigate the model at a fixed temperature and in the large time limit, where the origin of the diffusion is a cumulative effect of many resonant collisions. We obtain a heat equation with a friction term for the
The meanfield limit for the dynamics of large particle systems, Journées “Équations aux Dérivées Partielles
 Exp. No. IX
, 2003
"... This short course explains how the usual meanfield evolution PDEs in Statistical Physics — such as the VlasovPoisson, SchrödingerPoisson or timedependent HartreeFock equations — are rigorously derived from first principles, i.e. from the fundamental microscopic models that govern the evolution ..."
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Cited by 19 (0 self)
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This short course explains how the usual meanfield evolution PDEs in Statistical Physics — such as the VlasovPoisson, SchrödingerPoisson or timedependent HartreeFock equations — are rigorously derived from first principles, i.e. from the fundamental microscopic models that govern the evolution of large, interacting particle systems. Mon cher Athos, je veux bien, puisque votre santé l’exige absolument, que vous vous reposiez quinze jours. Allez donc prendre les eaux de Forges ou telles autres qui vous conviendront, et rétablissezvous promptement. Votre affectionné
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 ..."
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Cited by 15 (3 self)
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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 linear Boltzmann equation as the low density limit of a random Schrödinger equation
 Rev. Math. Phys
, 2005
"... We study the long time evolution of a quantum particle interacting with a random potential in the BoltzmannGrad low density limit. We prove that the phase space density of the quantum evolution defined through the Husimi function converges weakly to a linear Boltzmann equation. The Boltzmann collis ..."
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We study the long time evolution of a quantum particle interacting with a random potential in the BoltzmannGrad low density limit. We prove that the phase space density of the quantum evolution defined through the Husimi function converges weakly to a linear Boltzmann equation. The Boltzmann collision kernel is given by the full quantum scattering cross section of the obstacle potential. 1 The Model and the Result The Schrödinger equation with a random potential describes the propagation of quantum particles in an environment with random impurities. In the first approximation one neglects the interaction between the particles and the problem reduces to a onebody Schrödinger equation. With high concentration of impurities the particle is localized, in particular no conduction occurs [1, 2, 3, 8, 11, 12]. In the low concentration regime conduction is expected to occur but there are no rigorous mathematical proof of the existence of the extended states except for the Bethe lattice [16, 17]. In this paper we study the long time evolution in the low concentration regime in a specific scaling limit, called the low density or BoltzmannGrad limit. Our model is the quantum analogue of the low density Lorentz gas. As the time increases, the concentration will be scaled down in such a way that the total interaction between the particle and the obstacles remains bounded for a typical configuration. Therefore our result is far from the extended states regime which requires to understand the behavior of the Schrödinger evolution for arbitrary long time, independently of the fixed (low) concentration of impurities. We start by defining our model and stating the main result.
The Brownian motion as the limit of a deterministic system of hardspheres. arXiv
"... Abstract. We provide a rigorous derivation of the brownian motion as the hydrodynamic limit of systems of hardspheres as the number of particles N goes to infinity and their diameter ε simultaneously goes to 0, in the fast relaxation limit Nεd−1 → ∞ (with a suitable scaling of the observation time ..."
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Abstract. We provide a rigorous derivation of the brownian motion as the hydrodynamic limit of systems of hardspheres as the number of particles N goes to infinity and their diameter ε simultaneously goes to 0, in the fast relaxation limit Nεd−1 → ∞ (with a suitable scaling of the observation time and length). As suggested by Hilbert in his sixth problem, we use the linear Boltzmann equation as an intermediate level of description for one tagged particle in a gas close to global equilibrium. Our proof relies on the fundamental ideas of Lanford. The main novelty here is the detailed study of the branching process, leading to explicit estimates on pathological collision trees. 1.
Towards the quantum Brownian motion
 the QMath9 Conference Proceedings, Giens
, 2004
"... Summary. We consider random Schrödinger equations on R d or Z d for d ≥ 3 with uncorrelated, identically distributed random potential. Denote by λ the coupling constant and ψt the solution with initial data ψ0. Suppose that the space and time variables scale as x ∼ λ −2−κ/2, t ∼ λ −2−κ with 0 < κ ..."
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Cited by 7 (4 self)
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Summary. We consider random Schrödinger equations on R d or Z d for d ≥ 3 with uncorrelated, identically distributed random potential. Denote by λ the coupling constant and ψt the solution with initial data ψ0. Suppose that the space and time variables scale as x ∼ λ −2−κ/2, t ∼ λ −2−κ with 0 < κ ≤ κ0, where κ0 is a sufficiently small universal constant. We prove that the expectation value of the Wigner distribution of ψt, EWψt(x,v), converges weakly to a solution of a heat equation in the space variable x for arbitrary L 2 initial data in the weak coupling limit λ → 0. The diffusion coefficient is uniquely determined by the kinetic energy associated to the momentum v. 1
ON THE BOLTZMANNGRAD LIMIT FOR THE TWO DIMENSIONAL PERIODIC LORENTZ GAS
"... Abstract. The twodimensional, periodic Lorentz gas, is the dynamical system corresponding with the free motion of a point particle in a planar system of fixed circular obstacles centered at the vertices of a square lattice in the Euclidian plane. Assuming elastic collisions between the particle a ..."
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Abstract. The twodimensional, periodic Lorentz gas, is the dynamical system corresponding with the free motion of a point particle in a planar system of fixed circular obstacles centered at the vertices of a square lattice in the Euclidian plane. Assuming elastic collisions between the particle and the obstacles, we consider this dynamical system in the BoltzmannGrad limit, i.e. assuming that the obstacle radius r and the reciprocal mean free path are asymptotically equivalent small quantities, and that the particle’s distribution function is slowly varying in the space variable. While it is known that the particle’s distribution function in that limit cannot be governed by a linear Boltzmann equation [F.Golse, Ann. Fac. Sci. Toulouse 17 (2008), 735–749], we propose a limiting theory involving an extended phase space larger than the usual phase space of a classical point particle in which the limiting particle’s distribution function evolves under an integrodifferential equation somewhat analogous to a linear Boltzmann equation. The particle’s distribution function in the classical phase space is obtained by averaging out the additional variables involved in the extended phase space. We derive explicit formulas for the coefficients appearing in that integrodifferential equation, and study its basic dynamical properties — especially, we establish an analogue of Boltzmann’s H theorem, provide a complete description of the equilibrium distribution functions and investigate the convergence to these equilibrium distribution functions in the long time limit. This article provides in particular complete proofs of the re