Effective Dynamics for Weakly Interacting Bosons in an Iterated High-Density Thermodynamic Limit

We study the time evolution of weakly interacting Bose gases on a three-dimensional torus of arbitrary volume. The coupling constant is supposed to be inversely proportional to the density, which is considered to be large and independent of the particle number. We take into account a class of initial states exhibiting quasi-complete Bose-Einstein condensation. For each fixed time in a finite interval, we prove the convergence of the one-particle reduced density matrix towards the projection onto the normalised order parameter describing the condensate - evolving according to the Hartree equation - in the iterated limit where the volume (and therefore the particle number), and subsequently the density go to infinity. The rate of convergence depends only on the density and on the decay of both the expected number of particles and the energy of the initial quasi-vacuum state.

💡 Research Summary

The paper investigates the dynamics of a three‑dimensional Bose gas confined to a torus of side length L, in a regime where the interaction strength is the inverse of the particle density ρ. The number of particles N scales with the volume (N≈ρ L³), so that ρ remains fixed while L→∞, and only afterwards ρ→∞. The pair potential V_L is obtained by periodising a real, spherically symmetric, integrable function V_∞ that satisfies suitable decay estimates both in position and momentum space. The many‑body Hamiltonian reads

H_{N,ρ,L}= -∑{i=1}^N Δ{x_i} + (1/ρN)∑_{i<j} V_L(x_i-x_j),

which corresponds to a coupling constant ρ⁻¹. This scaling keeps the speed of sound (the Bogoliubov linear dispersion coefficient) fixed when the density is large, thereby mimicking the physical situation of a high‑density condensate.

Mathematically the authors work in the second‑quantised picture on the symmetric Fock space ℱ_s(L²(Λ_L)). They introduce creation and annihilation operators a⁺(f), a(f) and the Weyl operator W(Ψ_{ρ,L}) which implements a coherent shift by a one‑particle wave function Ψ_{ρ,L}∈H¹(Λ_L). The initial state is taken as

ϕ₀^{ρ,L}=W(Ψ_{ρ,L}) ξ_{ρ,L},

where ξ_{ρ,L} encodes the fluctuation (or “quasi‑vacuum”) part. By Definition 2.2 this state exhibits a quasi‑complete Bose‑Einstein condensation: the expected number of particles outside the condensate mode and the associated fluctuation energy are both small compared with ρ.

The main result (Theorem 2.5) states that for any fixed finite time interval