Dynamics of interacting bosons: a compact review

Napiórkowski, Marcin

doi:10.48550/arxiv.2101.04594

Cited by 5 publications

(5 citation statements)

References 134 publications

(158 reference statements)

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“…For other results in this direction, we refer to [8, 11, 14, 15, 17-20, 22, 41-43, 45-47, 49, 58, 59, 61]. More references concerning the dynamics of BECs in the above scaling limits can be found in the lecture notes [9] and in the review article [60].…”

Section: Background and Summarymentioning

confidence: 99%

Dynamics of mean-field bosons at positive temperature

Caporaletti¹,

Deuchert²,

Schlein³

2022

Preprint

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We study the time-evolution of an initially trapped weakly interacting Bose gas at positive temperature, after the trapping potential has been switched off. It has been recently shown in [24] that the one-particle density matrix of Gibbs states of the interacting trapped gas is given, to leading order in N, as N → ∞, by the one of the ideal gas, with the condensate wave function replaced by the minimizer of the Hartree energy functional. We show that this structure is stable with respect to the many-body evolution in the following sense: the dynamics can be approximated in terms of the time-dependent Hartree equation for the condensate wave function and in terms of the free evolution for the thermally excited particles. The main technical novelty of our work is the use of the Hartree-Fock-Bogoliubov equations to define a fluctuation dynamics.

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Section: Background and Summarymentioning

confidence: 99%

Dynamics of mean-field bosons at positive temperature

Caporaletti¹,

Deuchert²,

Schlein³

2022

Preprint

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“…mean-field) for quantum many-body systems, with contributions by many authors. As it is not our intention to review this body of literature and we are exclusively interested here in results pertaining to the LL model, we refer the reader to the review texts [51,75,5,31,74,59], and references therein, for more discussion on this general subject. We intend no offense by any omissions.…”

Section: Introductionmentioning

confidence: 99%

The mean-field limit of the Lieb-Liniger model

Rosenzweig

2022

DCDS

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<p style='text-indent:20px;'>We consider the well-known Lieb-Liniger (LL) model for <inline-formula><tex-math id="M1">\begin{document}$ N $\end{document}</tex-math></inline-formula> bosons interacting pairwise on the line via the <inline-formula><tex-math id="M2">\begin{document}$ \delta $\end{document}</tex-math></inline-formula> potential in the mean-field scaling regime. Assuming suitable asymptotic factorization of the initial wave functions and convergence of the microscopic energy per particle, we show that the time-dependent reduced density matrices of the system converge in trace norm to the pure states given by the solution to the one-dimensional cubic nonlinear Schrödinger equation (NLS) with an explict rate of convergence. In contrast to previous work [<xref ref-type="bibr" rid="b3">3</xref>] relying on the formalism of second quantization and coherent states and without an explicit rate, our proof is based on the counting method of Pickl [<xref ref-type="bibr" rid="b65">65</xref>,<xref ref-type="bibr" rid="b66">66</xref>,<xref ref-type="bibr" rid="b67">67</xref>] and Knowles and Pickl [<xref ref-type="bibr" rid="b44">44</xref>]. To overcome difficulties stemming from the singularity of the <inline-formula><tex-math id="M3">\begin{document}$ \delta $\end{document}</tex-math></inline-formula> potential, we introduce a new short-range approximation argument that exploits the Hölder continuity of the <inline-formula><tex-math id="M4">\begin{document}$ N $\end{document}</tex-math></inline-formula>-body wave function in a single particle variable. By further exploiting the <inline-formula><tex-math id="M5">\begin{document}$ L^2 $\end{document}</tex-math></inline-formula>-subcritical well-posedness theory for the 1D cubic NLS, we can prove mean-field convergence when the limiting solution to the NLS has finite mass, but only for a very special class of <inline-formula><tex-math id="M6">\begin{document}$ N $\end{document}</tex-math></inline-formula>-body initial states.</p>

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“…bounded) [Spo80,BGM00]. Many authors have contributed to the mean-field theory when V is not necessarily Coulomb, and we refer to the surveys [Gol16,Rou20,Nap21] and references therein for further information. Several works [NS81, Spo81, GMP03, FGS07, PP09, GMP16, GP17, GPP18, GP19, CLS21] have investigated the combined semiclassical and mean-field regimes for general potentials V , as the -uniformity of the quantum Nbody mean-field convergence rate implies mean-field convergence for the N -body classical dynamics.…”

Section: Introductionmentioning

confidence: 99%

From Quantum Many-Body Systems to Ideal Fluids

Rosenzweig

2021

Preprint

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We give a rigorous, quantitative derivation of the incompressible Euler equation from the many-body problem for N bosons on T d with binary Coulomb interactions in the semiclassical regime. The coupling constant of the repulsive interaction potential is 1/(ε 2 N ), where ε ≪ 1 and N ≫ 1, so that by choosing ε = N −λ , for appropriate λ > 0, the scaling is supercritical with respect to the usual mean-field regime. For approximately monokinetic initial states with nearly uniform density, we show that the density of the first marginal converges to 1 as N → ∞ and → 0, while the current of the first marginal converges to a solution u of the incompressible Euler equation on an interval for which the equation admits a classical solution. In dimension 2, the dependence of ε on N is essentially optimal, while in dimension 3, heuristic considerations suggest our scaling is optimal. To the best of our knowledge, our result is a new connection between quantum many-body systems and ideal hydrodynamics, complementing the previously known connection to compressible fluids. Our proof is based on a Gronwall relation for a quantum modulated energy with an appropriate corrector and is inspired by recent work of Golse and Paul [GP21] on the derivation of the pressureless Euler-Poisson equation in the classical and mean-field limits and of Han-Kwan and Iacobelli [HKI21] and the author [Ros21] on the derivation of the incompressible Euler equation from Newton's second law in the supercritical mean-field limit. As a byproduct of our analysis, we also derive the incompressible Euler equation from the Schrödinger-Poisson equation in the limit as + ε → 0, corresponding to a combined classical and quasineutral limit.

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Dynamics of interacting bosons: a compact review

Cited by 5 publications

References 134 publications

Dynamics of mean-field bosons at positive temperature

Dynamics of mean-field bosons at positive temperature

The mean-field limit of the Lieb-Liniger model

From Quantum Many-Body Systems to Ideal Fluids

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