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arXiv:math-ph/0410005 · Communications on Pure and Applied Mathematics

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

László Erdős, Benjamin Schlein, Horng‐Tzer Yau

Published Feb 2, 2006• 171 citations

Abstract

Abstract 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 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.

DOI: 10.1002/cpa.20123

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Abstract 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 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.

Consider a system of

N

bosons in three dimensions interacting via a repulsive short range pair potential

N^2V(N(x_i-x_j))

, where $\bx=(x_1, >..., x_N)$ denotes the positions of the particles. Let

H_N

denote the Hamiltonian of the system and let

ψ_{N,t}

be the solution to the Schrödinger equation. Suppose that the initial data

ψ_{N,0}

satisfies the energy condition

< ψ_{N,0}, H_N^k ψ_{N,0} > \leq C^k N^k

for $k=1,2,... $. We also assume that the

k

-particle density matrices of the initial state are asymptotically factorized as

N\to\infty

. We prove that the

k

-particle density matrices of

ψ_{N,t}

are also asymptotically factorized and the one particle orbital wave function solves the Gross-Pitaevskii equation, a cubic non-linear Schrödinger equation with the coupling constant given by the scattering length of the potential

V

. We also prove the same conclusion if the energy condition holds only for

k=1

but the factorization of

ψ_{N,0}

is assumed in a stronger sense.

We derive rigorously, for both R^2 and [-L, L]^2, the cubic nonlinear Schrodinger equation in a suitable scaling limit from the two-dimensional many-body Bose systems with short-scale repulsive pair interactions. We first prove convergence of the solution of the BBGKY hierarchy, corresponding to the many-body systems, to a solution of the infinite Gross-Pitaevskii hierarchy, corresponding to the cubic NLS; and then we prove uniqueness for the infinite hierarchy, which requires number-theoretical techniques in the periodic case.

We derive rigorously, for both

{\Bbb R}^2

and

[{-}L,L]^{\times 2}

, the cubic nonlinear Schr\"odinger equation in a suitable scaling limit from the two-dimensional many-body Bose systems with short-scale repulsive pair interactions. We first prove convergence of the solution of the BBGKY hierarchy, corresponding to the many-body systems, to a solution of the infinite Gross-Pitaevskii hierarchy, corresponding to the cubic NLS; and then we prove uniqueness for the infinite hierarchy, which requires number-theoretical techniques in the periodic case.

Consider a system of

N

bosons in three dimensions interacting via a repulsive short range pair potential

N^2V(N(x_i-x_j))

, where $\bx=(x_1, >..., x_N)$ denotes the positions of the particles. Let

H_N

denote the Hamiltonian of the system and let

\psi_{N,t}

be the solution to the Schrodinger equation. Suppose that the initial data

\psi_{N,0}

satisfies the energy condition

\leq C N >.

and that the one-particle density matrix converges to a projection as

N \to \infty

. Then, we prove that the

k

-particle density matrices of

\psi_{N,t}

factorize in the limit

N \to \infty

. Moreover, the one particle orbital wave function solves the time-dependent Gross-Pitaevskii equation, a cubic non-linear Schrodinger equation with the coupling constant proportional to the scattering length of the potential

V

. In \cite{ESY}, we proved the same statement under the condition that the interaction potential

N \to \infty

is sufficiently small; in the present work we extend this result to large potentials.

Derivation of the Quintic NLS from many-body quantum dynamics in

T^2

In this paper, we investigate the dynamics of a boson gas with three-body interactionsin

T^2

. We prove that when the particlenumber

N

tends to infinity, the BBGKY hierarchy of

k

-particle marginals converges to a infiniteGross-Pitaevskii(GP) hierarchy for which we prove uniqueness of solutions,and for the asymptotically factorized

N

-body initial datum, we show that this

N\rightarrow\infty

limitcorresponds to the quintic nonlinear Schrödinger equation.Thus, the Bose-Einstein condensation is preserved in time.

Starting from first‐principle many‐body quantum dynamics, we show that the dynamics of Bose‐Einstein condensates can be approximated by the time‐dependent nonlinear Gross‐Pitaevskii equation, giving a bound on the rate of the convergence. Initial data are constructed on the bosonic Fock space applying an appropriate Bogoliubov transformation on a coherent state with expected number of particles N. The Bogoliubov transformation plays a crucial role; it produces the correct microscopic correlations among the particles. Our analysis shows that, on the level of the one‐particle reduced density, the form of the initial data is preserved by the many‐body evolution, up to a small error that vanishes as N−1/2 in the limit of large N.© 2015 Wiley Periodicals, Inc.