Functional Integral and Stochastic Representations for Ensembles of Identical Bosons on a Lattice

Salmhofer, Manfred

doi:10.1007/s00220-021-04010-4

Cited by 4 publications

(3 citation statements)

References 32 publications

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“…In the recent paper [29], a construction of regularized coherent-state functional integrals for ensembles of bosons on a lattice is given. As in [10], an important tool is the Hubbard-Stratonovich formula.…”

Section: Related Resultsmentioning

confidence: 99%

Interacting Loop Ensembles and Bose Gases

Fröhlich

Knowles

Schlein

et al. 2022

Ann. Henri Poincaré

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Section: Related Resultsmentioning

confidence: 99%

Interacting Loop Ensembles and Bose Gases

Fröhlich

Knowles

Schlein

et al. 2022

Ann. Henri Poincaré

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“…In the recent paper [27], a construction of regularized coherent-state functional integrals for ensembles of bosons on a lattice is given. As in [10], an important tool is the Hubbard-Stratonovich formula.…”

Section: Related Resultsmentioning

confidence: 99%

Interacting loop ensembles and Bose gases

Fröhlich¹,

Knowles²,

Schlein³

et al. 2020

Preprint

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We study interacting Bose gases in thermal equilibrium on a lattice. We establish convergence of the grand canonical Gibbs states of such gases to their mean-field (classical field) and largemass (classical particle) limits. The former is a classical field theory for a complex scalar field with quartic self-interaction. The latter is a classical theory of point particles with two-body interactions. Our analysis is based on representations in terms of ensembles of interacting random loops, the Ginibre loop ensemble for Bose gases and the Symanzik loop ensemble for classical scalar field theories. For small enough interactions, our results also hold in infinite volume. The results of this paper were previously sketched in [11].

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“…Results on time-dependent correlations were first considered when d = 1 in [20]. Related problems on the lattice were considered in [23,32,55]. For more details on the previously known literature, we refer the reader to the expository works [38] and [22], as well as to the introduction of [54].…”

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confidence: 99%

A Microscopic Derivation of Gibbs Measures for Nonlinear Schrödinger Equations with Unbounded Interaction Potentials

Sohinger

2021

International Mathematics Research Notices

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We study the derivation of the Gibbs measure for the nonlinear Schrödinger (NLS) equation from many-body quantum thermal states in the mean-field limit. In this paper, we consider the nonlocal NLS with defocusing and unbounded $L^p$ interaction potentials on $\mathbb{T}^d$ for $d=1,2,3$. This extends the author’s earlier joint work with Fröhlich et al. [ 45], where the regime of defocusing and bounded interaction potentials was considered. When $d=1$, we give an alternative proof of a result previously obtained by Lewin et al. [ 69]. Our proof is based on a perturbative expansion in the interaction. When $d=1$, the thermal state is the grand canonical ensemble. As in [ 45], when $d=2,3$, the thermal state is a modified grand canonical ensemble, which allows us to estimate the remainder term in the expansion. The terms in the expansion are analysed using a graphical representation and are resummed by using Borel summation. By this method, we are able to prove the result for the optimal range of $p$ and obtain the full range of defocusing interaction potentials, which were studied in the classical setting when $d=2,3$ in the work of Bourgain [ 15].

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Functional Integral and Stochastic Representations for Ensembles of Identical Bosons on a Lattice

Cited by 4 publications

References 32 publications

Interacting Loop Ensembles and Bose Gases

Interacting Loop Ensembles and Bose Gases

Interacting loop ensembles and Bose gases

A Microscopic Derivation of Gibbs Measures for Nonlinear Schrödinger Equations with Unbounded Interaction Potentials

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