Weak coupling limit of the $N$-particle Schrödinger equation

Golse, François; Bardos, Claude; Mauser, Norbert J.

doi:10.4310/maa.2000.v7.n2.a2

Cited by 155 publications

(241 citation statements)

References 10 publications

(14 reference statements)

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“…The so-called BBGKY hierarchy method (Bogoliubov, Born, Green, Kirkwood and Yvon) is very popular in physics and mathematics for studying many-particle systems: see for instance [3] where this approach is used for Kac's master equation for hard spheres, or see, among many other works, the recent series of papers [4,25,26,27] where this approach is used for the derivation of nonlinear mean-field Schrödinger equations in quantum physics. The basic ideas underlying this approach to mean-field limit could be summarized as:…”

Section: The Bbgky Hierarchy Methods Revisitedmentioning

confidence: 99%

“…Therefore, in the study of the long-time behavior we will often consider N -particle distributions that are supported on S N (E) for appropriate energy E. We shall discuss the construction of such chaotic initial data in Section 4. 4 By extensivity we mean here that the functional measuring the distance between two distributions should behave additively with respect to the tensor product.…”

Section: 3mentioning

confidence: 99%

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Kac’s program in kinetic theory

Mischler

Mouhot²

2012

Invent. math.

153

292

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Abstract. This paper is devoted to the study of propagation of chaos and mean-field limits for systems of indistinguable particles, undergoing collision processes. The prime examples we will consider are the many-particle jump processes of Kac and McKean [42,53] giving rise to the Boltzmann equation. We solve the conjecture raised by Kac [42], motivating his program, on the rigorous connection between the long-time behavior of a collisional many-particle system and the one of its mean-field limit, for bounded as well as unbounded collision rates.Motivated by the inspirative paper by Grünbaum [35], we develop an abstract method that reduces the question of propagation of chaos to that of proving a purely functional estimate on generator operators (consistency estimates), along with differentiability estimates on the flow of the nonlinear limit equation (stability estimates). This allows us to exploit dissipativity at the level of the mean-field limit equation rather than the level of the particle system (as proposed by Kac).Using this method we show: (1) Quantitative estimates, that are uniform in time, on the chaoticity of a family of states. (2) Propagation of entropic chaoticity, as defined in [10]. (3) Estimates on the time of relaxation to equilibrium, that are independent of the number of particles in the system. Our results cover the two main Boltzmann physical collision processes with unbounded collision rates: hard spheres and true Maxwell molecules interactions. The proof of the stability estimates for these models requires significant analytic efforts and new estimates.

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Section: The Bbgky Hierarchy Methods Revisitedmentioning

confidence: 99%

Section: 3mentioning

confidence: 99%

Kac’s program in kinetic theory

Mischler

Mouhot²

2012

Invent. math.

153

292

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“…Therefore, to conclude the proof of (1.7), it suffices to show that: 1) every limit point of the family { ( ) , } =1 is a solution of the infinite hierarchy (2.2), and 2) the solution to (2.2) is unique. This strategy has already been used to derive the nonlinear Hartree equations for the effective dynamics of so-called mean-field systems (see [27,13,4,9]) to derive the cubic nonlinear Schrödinger equation with different (and simpler) scalings of the interaction potential (see [8,11]) and to derive the nonlinear Schrödinger equation in a one-dimensional setting (see [1,2]). We remark that the first derivation of the Hartree equation was obtained using a different method in [17,14].…”

Section: Resolution Of the Correlation Structure For Large Potentialmentioning

confidence: 99%

Rigorous derivation of the Gross-Pitaevskii equation with a large interaction potential

Erdős¹,

Schlein

Yau

2009

J. Amer. Math. Soc.

143

247

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Consider a system of N N bosons in three dimensions interacting via a repulsive short range pair potential N 2 V ( N ( x i − x j ) ) N^2V(N(x_i-x_j)) , where x = ( x 1 , … , x N ) \mathbf {x}=(x_1, \ldots , x_N) denotes the positions of the particles. Let H N H_N denote the Hamiltonian of the system and let ψ N , t \psi _{N,t} be the solution to the Schrödinger equation. Suppose that the initial data ψ N , 0 \psi _{N,0} satisfies the energy condition \[ ⟨ ψ N , 0 , H N ψ N , 0 ⟩ ≤ C N \langle \psi _{N,0}, H_N \psi _{N,0} \rangle \leq C N \] and that the one-particle density matrix converges to a projection as N → ∞ N \to \infty . Then, we prove that the k k -particle density matrices of ψ N , t \psi _{N,t} factorize in the limit N → ∞ N \to \infty . Moreover, the one particle orbital wave function solves the time-dependent Gross-Pitaevskii equation, a cubic nonlinear Schrödinger equation with the coupling constant proportional to the scattering length of the potential V V . In a recent paper, we proved the same statement under the condition that the interaction potential V V is sufficiently small. In the present work we develop a new approach that requires no restriction on the size of the potential.

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“…In [4] the convergence (1.7) was proven for bosons with non-relativistic dispersion interacting through a Coulomb potential. Partial results for the non-relativistic Coulomb case were also established in [1]. In [2], a joint work with L. Erdős and H.-T. Yau, we consider a system of N non-relativistic bosons, interacting, in the mean-field scaling, through an N -dependent potential V N (x) = N 3β V (N β x), with 0 < β < 3/5, which converges to a delta-function in the limit N → ∞: for this potential we prove the convergence of solutions of the finite hierarchy (1.5) (with V (x) replaced by V N (x)) to solutions of the infinite hierarchy (1.6) (with V replaced by δ(x)).…”

Section: Introductionmentioning

confidence: 99%

Mean field dynamics of boson stars

Elgart

Schlein

2006

Comm Pure Appl Math

248

233

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We consider a quantum mechanical system of N bosons with relativistic dispersion interacting through a mean field Coulomb potential (attractive or repulsive). We choose the initial wave function to describe a condensate, where the N bosons are all in the same one-particle state. Starting from the N -body Schrödinger equation, we prove that, in the limit N → ∞, the time evolution of the one-particle density is governed by the relativistic nonlinear Hartree equation. This equation is used to describe the dynamics of boson stars (Chandrasekhar theory). The corresponding static problem was rigorously solved in [10].

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Weak coupling limit of the $N$-particle Schrödinger equation

Cited by 155 publications

References 10 publications

Kac’s program in kinetic theory

Kac’s program in kinetic theory

Rigorous derivation of the Gross-Pitaevskii equation with a large interaction potential

Mean field dynamics of boson stars

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