The Thermodynamic Limit of Quantum Coulomb Systems: A New Approach

Advances in Mathematics

Solovej

2009

Self Cite

This article is the first in a series dealing with the thermodynamic properties of quantum Coulomb systems.In this first part, we consider a general real-valued function E defined on all bounded open sets of R 3 . Our aim is to give sufficient conditions such that E has a thermodynamic limit. This means that the limit E(Ω n )|Ω n | −1 exists for all 'regular enough' sequence Ω n with growing volume, |Ω n | → ∞, and is independent of the considered sequence.The sufficient conditions presented in our work all have a clear physical interpretation. In the next paper, we show that the free energies of many different quantum Coulomb systems satisfy these assumptions, hence have a thermodynamic limit.

“…by (12) and since Ω + w ∈ R for any w ∈ R 3 by (P1). The result is obtained by taking the limit L → ∞.…”

Section: Proofsmentioning

confidence: 87%

Section: Thermodynamic Limit For a Special Sequence Of Setsmentioning

confidence: 99%

Section: Thermodynamic Limit For a Special Sequence Of Setsmentioning

confidence: 99%

“…Proofs are given in Section 2. Let us mention that the results of this paper and of [13] have been summarized in [12].…”

Section: Introductionmentioning

confidence: 96%

See 2 more Smart Citations

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

Hainzl

Advances in Mathematics

Solovej

2009

Self Cite

“…It is now a classical tool in Quantum Field Theory. It was recently used by Hainzl, Solovej and the author of the present paper, to prove the existence of the thermodynamic limit for quantum Coulomb systems in the grand canonical picture, see Appendix A.1 in [23]. In this latter work, the strong subadditivity of the quantum entropy was also formulated using geometric localization.…”

Section: Definitionmentioning

confidence: 92%

Geometric methods for nonlinear many-body quantum systems

Journal of Functional Analysis

2011

143

Geometric techniques have played an important role in the seventies, for the study of the spectrum of many-body Schr\"odinger operators. In this paper we provide a formalism which also allows to study nonlinear systems. We start by defining a weak topology on many-body states, which appropriately describes the physical behavior of the system in the case of lack of compactness, that is when some particles are lost at infinity. We provide several important properties of this topology and use them to provide a simple proof of the famous HVZ theorem in the repulsive case. In a second step we recall the method of geometric localization in Fock space as proposed by Derezi\'nski and G\'erard, and we relate this tool to our weak topology. We then provide several applications. We start by studying the so-called finite-rank approximation which consists in imposing that the many-body wavefunction can be expanded using finitely many one-body functions. We thereby emphasize geometric properties of Hartree-Fock states and prove nonlinear versions of the HVZ theorem, in the spirit of works of Friesecke. In the last section we study translation-invariant many-body systems comprising a nonlinear term, which effectively describes the interactions with a second system. As an example, we prove the existence of the multi-polaron in the Pekar-Tomasevich approximation, for certain values of the coupling constant.Comment: Final version to appear in Journal of Functional Analysi

The thermodynamic limit of quantum Coulomb systems Part II. Applications

Hainzl

Advances in Mathematics

Solovej

2009

Self Cite

In a previous paper [19], we have developed a general theory of thermodynamic limits. We apply it here to three different Coulomb quantum systems, for which we prove the convergence of the free energy per unit volume.The first system is the crystal for which the nuclei are classical particles arranged periodically in space and only the electrons are quantum particles. We recover and generalize a previous result of Fefferman. In the second example, both the nuclei and the electrons are quantum particles, submitted to a periodic magnetic field. We thereby extend a seminal result of Lieb and Lebowitz. Finally, in our last example we take again classical nuclei but optimize their position. To our knowledge such a system was never treated before.The verification of the assumptions introduced in [19] uses several tools which have been introduced before in the study of large quantum systems. In particular, an electrostatic inequality of Graf and Schenker is one main ingredient of our new approach.