Derivation of the Schrödinger–Poisson equation from the quantum <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>𝐍</mml:mi></mml:math>-body problem

Bardos, Claude; Erdős, László; Golse, François; Mauser, Norbert J.; Yau, Horng‐Tzer

doi:10.1016/s1631-073x(02)02253-7

Cited by 97 publications

(123 citation statements)

References 8 publications

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“…The uniqueness for the infinite BBGKY hierarchy (II.15) in the case of the Coulomb potential is proved in [15] -see also [2]. The proof by Erdös and Yau follows the argument presented in part II of these lectures.…”

Section: Part IV Extensions Open Problems and Further Readingmentioning

confidence: 81%

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The mean-field limit for the dynamics of large particle systems

Golse¹

2003

Journées Équations Aux Dérivées Partielles

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This short course explains how the usual mean-field evolution PDEs in Statistical Physics -such as the Vlasov-Poisson, Schrödinger-Poisson or time-dependent Hartree-Fock equations -are rigorously derived from first principles, i.e. from the fundamental microscopic models that govern the evolution of large, interacting particle systems. A review of physical modelsThe subject matter of these lectures is the relation between "exact" microscopic models that govern the evolution of large particle systems and a certain type of approximate models known in Statistical Mechanics as "mean-field equations". This notion of mean-field equation is best understood by getting acquainted with the most famous examples of such equations, described below.

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Section: Part IV Extensions Open Problems and Further Readingmentioning

confidence: 81%

“…in the sense of distributions, with initial data 2 We denote by M(X) the space of Radon measures on a locally compact space X; its weak-* topology is the one defined by duality with test functions in C c (X).…”

Section: For the Topology Of Uniform Convergence On Compact Subsets Omentioning

confidence: 99%

The mean-field limit for the dynamics of large particle systems

Golse¹

2003

Journées Équations Aux Dérivées Partielles

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101

168

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“…Thus the system of (1.1)-(1.2) is usually called the Schrödinger-Poisson system (SPS) in the literature [12,50]. In fact, the corresponding rigorous derivation of this kind of "Hartree equations" was started from a Hartree ansatz for the many-body (e.g., N -body) wave-function by using a "weak coupling scaling" (i.e., a factor 1/N in front of the Coulomb interaction potential) and passing to the limit N → ∞ in the BBGKY hierarchy [13,14,29]. For detailed derivation of the above SPS (1.1)-(1.2) and its mathematical justification, we refer the reader to [13,14,29] and references therein.…”

Section: ⇐⇒ −δϕ(X T) = |ψ(X T)|mentioning

confidence: 99%

“…Since this system is numerically intractable, usually effective "one particle" models are used, by using approximations like the Hartree(-Fock) ansatz and mean field theory [2,13,14,15,28,29,30,32,41,45]. Thus, the original 3N -or 3N + 1-dimensional problem (stationary or time-dependent) can be reduced to one or a coupled system of nonlinear Schrödinger (NLS) equation(s) in 3D [13,14,15,28,29,36]. In some situations, the 3D "one particle" NLS equation can be further reduced to lower-dimensional NLS equations in one or two space dimensions (1D or 2D), which decreases the numerical effort once more.…”

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confidence: 99%

“…For electrons, again via the "mean field limit" [13,14,15,29], the linear Schrö-dinger equation for N electrons with binary Coulomb interaction between different electrons can be approximated by a dimensionless single NLS equation with Coulomb interaction in 3D: (1.1) i ∂ t ψ(x, t) = − 1 2 Δ + V (x) + κ ϕ ψ, x ∈ R 3 , t > 0, where (1.2) ϕ(x, t) = 1 4π|x| * |ψ| 2 …”

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Dimension Reduction of the Schrödinger Equation with Coulomb and Anisotropic Confining Potentials

Bao¹,

Jian²,

Mauser³

et al. 2013

SIAM J. Appl. Math.

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Abstract. We consider dimension reduction for the three-dimensional (3D) Schrödinger equation with the Coulomb interaction and an anisotropic confining potential to lower-dimensional models in the limit of infinitely strong confinement in one or two space dimensions and obtain formally the surface adiabatic model (SAM) or surface density model (SDM) in two dimensions (2D) and the line adiabatic model (LAM) in one dimension (1D). Efficient and accurate numerical methods for computing ground states and dynamics of the SAM, SDM, and LAM models are presented based on efficient and accurate numerical schemes for evaluating the effective potential in lower-dimensional models. They are applied to find numerically convergence and convergence rates for the dimension reduction from 3D to 2D and 3D to 1D in terms of ground state and dynamics, which confirm some existing analytical results for the dimension reduction in the literature. In particular, we explain and demonstrate that the standard Schrödinger-Poisson system in 2D is not appropriate to simulate a "2D electron gas" of point particles confined to a plane (or, more generally, a 2D manifold), whereas SDM should be the correct model to be used for describing the Coulomb interaction in 2D in which the square root of Laplacian operator is used instead of the Laplacian operator. Finally, we report ground states and dynamics of the SAM and SDM in 2D and LAM in 1D under different setups.

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Mean‐Field and Classical Limit for the N‐Body Quantum Dynamics with Coulomb Interaction

Golse

Paul

2021

Comm Pure Appl Math

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This paper proves the validity of the joint mean-field and classical limit of the quantum N -body dynamics leading to the pressureless Euler-Poisson system for factorized initial data whose first marginal has a monokinetic Wigner measure. The interaction potential is assumed to be the repulsive Coulomb potential. The validity of this derivation is limited to finite time intervals on which the Euler-Poisson system has a smooth solution that is rapidly decaying at infinity. One key ingredient in the proof is an inequality from [S. Serfaty, with an appendix of M. Duerinckx arXiv:1803.08345v3 [math.AP]].

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Derivation of the Schrödinger–Poisson equation from the quantum 𝐍-body problem

Cited by 97 publications

References 8 publications

The mean-field limit for the dynamics of large particle systems

The mean-field limit for the dynamics of large particle systems

Dimension Reduction of the Schrödinger Equation with Coulomb and Anisotropic Confining Potentials

Mean‐Field and Classical Limit for the N‐Body Quantum Dynamics with Coulomb Interaction

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