Gibbs Measures of Nonlinear Schrödinger Equations as Limits of Many-Body Quantum States in Dimensions $${d \leqslant 3}$$ d ⩽ 3

Abstract

We prove that Gibbs measures of nonlinear Schrödinger equations arise as high-temperature limits of thermal states in many-body quantum mechanics. Our results hold for defocusing interactions in dimensions d = 1, 2, 3. The many-body quantum thermal states that we consider are the grand canonical ensemble for d = 1 and an appropriate modification of the grand canonical ensemble for d = 2, 3. In dimensions d = 2, 3, the Gibbs measures are supported on singular distributions, and a renormalization of the chemical potential is necessary. On the many-body quantum side, the need for renormalization is manifested by a rapid growth of the number of particles. We relate the original many-body quantum problem to a renormalized version obtained by solving a counterterm problem. Our proof is based on ideas from field theory, using a perturbative expansion in the interaction, organized by using a diagrammatic representation, and on Borel resummation of the resulting series.

A Path-Integral Analysis of Interacting Bose Gases and Loop Gases

“…That this so-called mean-field limit makes sense mathematically is a key result discussed in this paper (see Section 5) and is proven in [1] and [8]; see [23,26] for earlier results. The Hamilton functional H is well defined on a complex phase space equal to the complex Sobolev space, H 1 , over the domain Λ equipped with the Poisson brackets…”

“…Solving the Hamiltonian equations of motion determined by the Hamilton functional in (14) for initial conditions in the support of the measure d P Λ is fairly easy in one dimension, because a typical sample field configuration φ in the support of d P Λ is Sobolev regular of index s < 1 2 ; see, e.g., [32,26]. But, in dimension d = 2, hard analysis is required to solve this problem, because typical sample configurations in the support of d P Λ are distributional; see [33,34] for relevant results.…”

“…We note that the trace appearing in the exponent on the RHS of (26) is finite, and the t-integration converges, for arbitrary N < ∞, ρ < ∞ and ν > 0. For λ = O( 1 N ), one can use saddle point methods to calculate the asymptotics of (26), as N → ∞. This yields what is known as the 1 N -expansion, which is a powerful tool to study the phase transition to a high-density phase exhibiting BEC in interacting Bose gases with a large number of particle species, as we will briefly discuss in Section 5.…”

“…When using expressions (24), (26) to study the approach to the mean-field limit, ν 0, we take λ to be given by λ 0 ν 2 N +1 , with λ 0 kept constant, and choose the parameter ρ in such a way that (up to a uniformly finite term) the phase θ(σ) defined in (22)…”

Derivation of renormalized Gibbs measures from equilibrium many-body quantum Bose gases

“…The first attempt to resolve the above conjecture is due to Fröhlich-Knowles-Schlein-Sohinger [10]. They proved (13) for d = 2, 3 and p = 2, but with the modified quantum state…”