论文标题
从多体到非线性schr {Ö} dinger的骨基地面状态的缩放限制
Scaling limits of bosonic ground states, from many-body to nonlinear Schr{ö}dinger
论文作者
论文摘要
假设所有粒子都是独立且分布相同的，如何描述许多粒子的相互作用系统？这个问题至少与统计力学本身一样古老。它的量子版本通过冷原子物理学的诞生而焕发了活力。特别是，玻色 - 因斯坦凝结物的实验创造直接询问以下变体：为什么以及如何组装非常冷的相互作用玻色子（被剥夺了Pauli排除原理的量子颗粒）都会占相同的量子状态？在本文中，我回顾了各种数学技术，允许证明骨系统的最低能量状态，以合理的宏观颗粒数（Bose-Einstein冷凝物）的合理宏观限制。这意味着，在相关的限制中，所有粒子的行为实际上都像独立且分布相同。这是统计力学中平均场近似值的理由的特定实例，从基本的多体schr {Ö} dinger hamiltonian开始。
How and why may an interacting system of many particles be described assuming that all particles are independent and identically distributed ? This question is at least as old as statistical mechanics itself. Its quantum version has been rejuvenated by the birth of cold atoms physics. In particular the experimental creation of Bose-Einstein condensates directly asks the following variant: why and how can a large assembly of very cold interacting bosons (quantum particles deprived of the Pauli exclusion principle) all populate the same quantum state ? In this text I review the various mathematical techniques allowing to prove that the lowest energy state of a bosonic system forms, in a reasonable macroscopic limit of large particle number, a Bose-Einstein condensate. This means that indeed in the relevant limit all particles approximately behave as if independent and identically distributed, according to a law determined by minimizing a non-linear Schr{ö}dinger energy functional. This is a particular instance of the justification of the mean-field approximation in statistical mechanics, starting from the basic many-body Schr{ö}dinger Hamiltonian.
