Gibbs measures based on 1D (an)harmonic oscillators as mean-field limits

Mathieu Lewin (1) , Phan Thành Nam (2) , Nicolas Rougerie (3)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
2 Department of Mathematics and Statistics
3 LPM2C - Laboratoire de physique et modélisation des milieux condensés

We prove that Gibbs measures based on 1D defocusing nonlinear Schrödinger functionals with sub-harmonic trapping can be obtained as the mean-field/large temperature limit of the corresponding grand-canonical ensemble for many bosons. The limit measure is supported on Sobolev spaces of negative regularity and the corresponding density matrices are not trace-class. The general proof strategy is that of a previous paper of ours, but we have to complement it with Hilbert-Schmidt estimates on reduced density matrices.

Domaines: Physique mathématique [math-ph]; Equations aux dérivées partielles [math.AP]; Probabilités [math.PR]

Mathieu Lewin, Phan Thành Nam, Nicolas Rougerie. Gibbs measures based on 1D (an)harmonic oscillators as mean-field limits. Journal of Mathematical Physics, 2018, 59, pp.041901. ⟨10.1063/1.5026963⟩. ⟨hal-01496137v2⟩