Scaling limits of bosonic ground states, from many-body to nonlinear Schrödinger

Nicolas Rougerie (1)

1 UMPA-ENSL - Unité de Mathématiques Pures et Appliquées

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 Schrödinger energy functional. This is a particular instance of the justification of the mean-field approximation in statistical mechanics, starting from the basic many-body Schrödinger Hamiltonian.

Domaines: Physique mathématique [math-ph]; Equations aux dérivées partielles [math.AP]; Théorie spectrale [math.SP]; Gaz Quantiques [cond-mat.quant-gas]; Mécanique statistique [cond-mat.stat-mech]

Nicolas Rougerie. Scaling limits of bosonic ground states, from many-body to nonlinear Schrödinger. EMS Surveys in Mathematical Sciences, 2020, 7 (2), pp. 253-408. ⟨hal-02468885v2⟩