The thermodynamic limit of quantum Coulomb systems Part I. General theory

Hainzl, Christian; Lewin, Mathieu; Solovej, Jan Philip

doi:10.1016/j.aim.2008.12.010

Cited by 20 publications

(71 citation statements)

References 32 publications

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“…We observe here that such subadditivity methods have been used previously in similar settings (see, e.g. [HLS09a] or [HLS09b]). Note that, unlike [LLS17], we work in our statements with probability measures.…”

Section: Preliminaries and Notationsupporting

confidence: 65%

Next-order asymptotic expansion for N-marginal optimal transport with Coulomb and Riesz costs

Cotar

Petrache

2019

Advances in Mathematics

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Motivated by a problem arising from Density Functional Theory, we provide the sharp next-order asymptotics for a class of multimarginal optimal transport problems with cost given by singular, long-range pairwise interaction potentials. More precisely, we consider an N -marginal optimal transport problem with N equal marginals supported on R d and with cost of the form i =j |x i − x j | −s . In this setting we determine the second-order term in the N → ∞ asymptotic expansion of the minimum energy, for the long-range interactions corresponding to all exponents 0 < s < d. We also prove a small oscillations property for this second-order energy term. Our results can be extended to a larger class of models than power-law-type radial costs, such as non-rotationally-invariant costs. The key ingredient and main novelty in our proofs is a robust extension and simplification of the Fefferman-Gregg decomposition [F85], [Ge89], extended here to our class of kernels, and which provides a unified method valid across our full range of exponents. Our first result generalizes a recent work of Lewin, Lieb and Seiringer [LLS17], who dealt with the second-order term for the Coulomb case s = 1, d = 3 .1 and we will prove the following asymptotic expansion for F OT N,c (µ). For all 0 < s < d, if µ has density ρ ∈ L 1+s/d (R d ), then we have (see Theorem 1.1 below)Moreover we will prove that the strictly positive constant C(s, d) is independent of the choice of the marginal dµ(x) = ρ(x)dx and therefore can be interpreted as arising in an independent model problem. Furthermore, we will derive a small oscillations principle (with respect to N ) forwhich could also be interpreted as a rough third order asymptotic bound for F OT N,c (µ) with c as in (1.3). Our methods are extendable to more general costs, as non-rotationally-invariant ones. See Remark 1.3 below.For d = 3, s = 1 , the F OT N,c (µ) was introduced in the physics literature in the context of Density Functional Theory (DFT) by Seidl, Perdew, Levy, Gori-Giorgi, and Savin [Sd99, SPL99, SGGS07], without them being aware of its meaning in optimal transport. Namely, for s = 1, d = 3 , (1.1) is a natural semiclassical limit to the famous Hohenberg-Kohn (HK) functional from quantum mechanics, originally introduced by Hohenberg-Kohn in [HK64], and rigorously proved by Levy and Lieb in [Ly79], [Li83]. The connection to optimal transport was mathematically established later by [CFK11] and [BdPGG12] for N = 2 , later further extended to N = 3 in [BdP17], and recently independently proved for all N ≥ 2 by [CFK17] and [Lw17].In the process of establishing our two main results, we were required to prove, as key new tools for them, a set of additional secondary results of independent interest and of possible use to other settings, leading to generalized versions of the Fefferman-Gregg decomposition and positive definiteness criteria, as described in Section 4.1. What we use is a decomposition of positive definite kernels following the strategy established for s = 1, d = 3 by Fefferman [F85],...

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Section: Preliminaries and Notationsupporting

confidence: 65%

Next-order asymptotic expansion for N-marginal optimal transport with Coulomb and Riesz costs

Cotar

Petrache

2019

Advances in Mathematics

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“…This the case for the Coulomb potential in dimension d = 3 with two kinds of particles of opposite charge which has been shown to be stable for fermions [165,166,161] but not for bosons [73,60,163,164,227]. The existence of the thermodynamic limit in the Coulomb case is proved in various situations [156,94,84,121,122,27]. Correlation functions were studied in this case in [101,94].…”

Section: Quantum Modelsmentioning

confidence: 99%

The crystallization conjecture: a review

Blanc¹,

Lewin²

2015

EMS Surv. Math. Sci.

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134

191

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In this article we describe the crystallization conjecture. It states that, in appropriate physical conditions, interacting particles always place themselves into periodic configurations, breaking thereby the natural translation-invariance of the system. This famous problem is still largely open. Mathematically, it amounts to studying the minima of a real-valued function defined on R 3N where N is the number of particles, which tends to infinity. We review the existing literature and mention several related open problems, of which many have not been thoroughly studied.

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“…For the Hartree-Fock and Kohn-Sham type models (abbreviated as HF and KS models, respectively), the fundamental issues (i)-(ii) remain open. All that is known (see [39]) is for HF that a limit exists, but that limit is not identified. The two key difficulties are first that the latter models are not convex (convexity plays a crucial role in the analysis of the TFW model) and second that the number of one-body molecular orbitals to be dealt with, or equivalently the rank of the one-body density matrix, are also growing to infinity (in the TFW model, only the density ρ is relevant).…”

Section: Electronic Structure Of Perfect Crystalsmentioning

confidence: 99%

“…The latter approach was chosen for the Schrödinger model with quantum nuclei in the canonical and grand canonical ensembles [68] in the seminal paper [55] (see also [51]), where the existence of a limit for the energy per unit volume is proved. The crystal case in the Schrödinger model was addressed in [32] (see also [39]). …”

Section: Crystals With Local Defects In the Tfw Settingmentioning

confidence: 99%

Mathematical Modeling of Point Defects in Materials Science

Cancès

Bris

2013

Math. Models Methods Appl. Sci.

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We survey some recent mathematical works we have contributed to that are related to the modeling of defects in materials science at different scales. We emphasize the similarities (need of a reference, often periodic system; renormalization procedure; etc) shared by models arising in different contexts. Our illustrative examples are taken from electronic structure models, atomistic models, homogenization problems. The exposition is pedagogic and deliberately kept elementary. Both theoretical and numerical aspects are addressed. * Invited article for a special issue of M 3 AS.

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The thermodynamic limit of quantum Coulomb systems Part I. General theory

Cited by 20 publications

References 32 publications

Next-order asymptotic expansion for N-marginal optimal transport with Coulomb and Riesz costs

Next-order asymptotic expansion for N-marginal optimal transport with Coulomb and Riesz costs

The crystallization conjecture: a review

Mathematical Modeling of Point Defects in Materials Science

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