A Rigorous Derivation¶of the Gross–Pitaevskii Energy Functional¶for a Two-dimensional Bose Gas

By Elliott H. Lieb, Robert Seiringer, J. Yngvason. 2000

198 citations recorded by the source corpus.

Abstract: We consider the ground state properties of an inhomogeneous two-dimensional Bose gas with a repulsive, short range pair interaction and an external confining potential. In the limit when the particle number N is large but ρ̅a2 is small, where ρ̅ is the average particle density and a the scattering length, the ground state energy and density are rigorously shown to be given to leading order by a Gross–Pitaevskii (GP) energy functional with a coupling constant g~1/|1n(ρ̅a2)|. In contrast to the 3D case the coupling constant depends on N through the mean density. The GP energy per particle depends only on Ng. In 2D this parameter is typically so large that the gradient term in the GP energy functional is negligible and the simpler description by a Thomas–Fermi type functional is adequate.

Subfields: Atomic and Molecular Physics, and Optics

View Paper: https://www.semanticscholar.org/paper/53eb0e067bba766d75a3a444b4b09cfd382c0de4

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In Atomic and Molecular Physics, and Optics

Citations in the corpus

On the Mass Concentration for Bose–Einstein Condensates with Attractive Interactions (2013, background): “1)), the GP functional can be rigorously derived from the quantum many-body problem in a suitable low-density limit [28, 29, 30, 27].”
Selection of the ground state for nonlinear schrödinger equations (2003, background): “At the most fundamental level the Gross–Pitaevskii equation (NLS with a potential) arises as a mean field limit model governing the interaction of a very large number of weakly interacting bosons [21, 51, 30, 9].”
Derivation of the Gross-Pitaevskii Equation for Rotating Bose Gases (2005, background): “In order to keep this paper manageable we do not discuss the 2D case, but the interested reader can easily combine the results in [15], [22] and the present paper.”
The quintic NLS as the mean field limit of a boson gas with three-body interactions (2008, background): “The mathematical study of systems of interacting Bose gases is a central research area in mathematical physics which has in recent years experienced remarkable progress.”
Second-Order Corrections to Mean Field Evolution of Weakly Interacting Bosons. I. (2009, methodology): “A mathematical proof of the Bose-Einstein condensation for the time-independent case was provided recently by Lieb, Seiringer, Solovej and Yngvanson [22–25].”