Derivation of the Gross‐Pitaevskii hierarchy for the dynamics of Bose‐Einstein condensate

By L. Erdős, B. Schlein, Horng-Tzer Yau. 2004

194 citations recorded by the source corpus.

Consider a system of N bosons on the three‐dimensional unit torus interacting via a pair potential N2V(N(xi − xj)) where x = (x1, …, xN) denotes the positions of the particles. Suppose that the initial data ψN, 0 satisfies the condition $$ \bigl< \psi_{N,0}, H_{N}^{2} \psi_{N,0}\bigr> \leq C N^{2}$$ where HN is the Hamiltonian of the Bose system. This condition is satisfied if ψN, 0 = WNϕN, 0 where WN is an approximate ground state to HN and ϕN, 0 is regular. Let ψN, t denote the solution to the Schrödinger equation with Hamiltonian HN. Gross and Pitaevskii proposed to model the dynamics of such a system by a nonlinear Schrödinger equation, the Gross‐Pitaevskii (GP) equation. The GP hierarchy is an infinite BBGKY hierarchy of equations so that if ut solves the GP equation, then the family of k‐particle density matrices ⊗k|ut〉〈ut| solves the GP hierarchy. We prove that as N → ∞ the limit points of the k‐particle density matrices of ψN, t are solutions of the GP hierarchy. Our analysis requires that the N‐boson dynamics be described by a modified Hamiltonian that cuts off the pair interactions whenever at least three particles come into a region with diameter much smaller than the typical interparticle distance. Our proof can be extended to a modified Hamiltonian that only forbids at least n particles from coming close together for any fixed n. © 2006 Wiley Periodicals, Inc.

Subfields: Atomic and Molecular Physics, and Optics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/64a26a3bc530c604c3cd6f72f4792a3d260531c9

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Citations in the corpus

The cubic nonlinear Schr\"odinger equation in two dimensions with radial data (2007, background): “It arises as a simplified model for studying Bose–Einstein condensates [22, 27, 51], Kerr media in nonlinear optics [33, 61], and even freak waves in the ocean [21, 29].”
Second-Order Corrections to Mean Field Evolution of Weakly Interacting Bosons. I. (2009, methodology): “This approach has been highly successful, even for very singular potentials, in the work of Elgart, Erdős, Schlein and Yau [6, 7, 8, 9, 10, 11].”
Kac’s program in kinetic theory (2011, methodology): “The BBGKY hierarchy method revisited 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, a”
Quantitative Derivation of the Gross‐Pitaevskii Equation (2012, methodology): “In fact, the following result was established in [6, 7, 8, 9, 10], and, in a slightly different form, in [21].”
A Simple Derivation of Mean Field Limits for Quantum Systems (2009, background): “Derivations of such mean field equations from the microscopic N body Schrödinger evolution are usually done for the reduced one particle density and are naturally based on hierarchies [16, 3, 2, 4, 5, 1].”