The Coulomb Gas at low temperature and low density

Conlon, Joseph G.; Lieb, Élliott H.; Yau, Horng‐Tzer

doi:10.1007/bf01217775

Cited by 30 publications

(55 citation statements)

References 10 publications

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“…This limit is called the molecular (or atomic) limit. It was first formulated in precise terms by (Fefferman 1985) and further studied by (Conlon, Lieb & Yau 1989, Graf & Schenker 1995b. Equilibrium ionization phases in this context are discussed in (Macris & Martin 1990).…”

Section: The Atomic and Molecular Limitmentioning

confidence: 99%

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Brydges

Martin

1999

Journal of Statistical Physics

142

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Results on the correlations of low density classical and quantum Coulomb systems at equilibrium in three dimensions are reviewed. The exponential decay of particle correlations in the classical Coulomb system -Debye-Hückel screening -is compared and contrasted with the quantum case where strong arguments are presented for the absence of exponential screening. Results and techniques for detailed calculations that determine the asymptotic decay of correlations for quantum systems are discussed. Theorems on the existence of molecules in the Saha regime are reviewed. Finally, new combinatoric formulas for the coefficients of Mayer expansions are presented and their role in proofs of results on Debye-Hückel screening is discussed.Typeset using REVT E X III-3 C. The scaling limit of the lattice dipole gas III-5 IV. Semi-classical Coulomb gas IV-1 A. The Feynman-Kac representation IV-1 B. The gas of charged filaments IV-3 C. Quantum fluctuations destroy exponential screening IV-7 D. Origin of the van der Waals forces IV-9 E. Semi-classical analysis of Coulombic correlations IV-11 F. Breakdown of exponential screening in the Quantum Sine-Gordon representation IV-15 V. The emergence of thermodynamics and of various physical laws from the more fundamental levels of atomic theory and statistical mechanics were high points in our education in physics. But by the time we reached textbook treatments of the complex world of the Coulomb interaction, for example, formation of atoms and molecules and equilibria between them, we learned to demand less from theory and to be more content with reasoning by analogy consistent with thermodynamics. We know experimentally that atoms and molecules form, that they can behave like mixtures of ideal gases, so phenomenologically they have chemical potentials and we no longer insist that this emerge in some limit from N-body quantum Coulomb systems. We also know that Coulomb systems can have screening and dielectric phases. But when screening is expected to occur, it is quite a common practice to rather bluntly replace the Coulomb potential by a screened potential inherited from meanfield theories without inquiring too deeply into the legitimacy of this change. Likewise, there does not exist a first principle theory of the dielectric constant that does not presuppose the existence of atoms and molecules. At best, assuming that atoms form, one uses several laws, such as the Clausius-Mosotti formula, whose microscopic foundations have to be elucidated. In all these cases, this is a sensible attitude since most of non-relativistic physics is reputed to be hidden within the N-particle Coulomb Hamiltonian. There are however basic facts that can be cleanly formulated as limiting theories and shown to persist near the limit as well.This review is devoted to these types of results on Coulomb systems at low density. At low density the most famous properties of Coulomb systems are related to screening. The classical Debye-Hückel theory and its quantum analogue the random phase approximation have...

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Section: The Atomic and Molecular Limitmentioning

confidence: 99%

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Brydges

Martin

1999

Journal of Statistical Physics

142

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“…Put differently, a gas of elementary particles is effectively described in this thermodynamic regime in terms of an ideal gas of composite particles. Different mathematical formulations and verifications of this fact have been given by Fefferman [7], by Conlon, Lieb and Yau [2], and by Macris and Martin [15]. See also [16,17] for suitable sequences {A} (e.g.…”

Section: The Molecular Limit Of Coulomb Gasesmentioning

confidence: 99%

“…A basic version of the result [2,7] states that for suitable values of the chemical potentials fiQ and for low enough temperature f3~1 the pressure of the S species is to good accuracy that of a classical free gas of specific 'molecules^. In this picture, molecules are non-interacting particles with no internal degrees of freedom.…”

Section: The Molecular Limit Of Coulomb Gasesmentioning

confidence: 99%

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An electrostatic inequality with applications to the constitution of matter

Graf¹

1996

Journées Équations Aux Dérivées Partielles

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“…The convergence results and derivation of the linear Boltzmann equation for a quantum Lorentz gas in the low density and weak coupling limits are presented in [27,28]. The Coulomb gas at low density is considered in [29].…”

Section: Introductionmentioning

confidence: 99%

Quantum stochastic equation for a test particle interacting with a dilute Bose gas

Pechen

2003

Journal of Mathematical Physics

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Abstract. We use the stochastic limit method to study long time quantum dynamics of a test particle interacting with a dilute Bose gas. The case of arbitrary form-factors and an arbitrary, not necessarily equilibrium, quasifree low density state of the Bose gas is considered. Starting from microscopic dynamics we derive in the low density limit a quantum white noise equation for the evolution operator. This equation is equivalent to a quantum stochastic equation driven by a quantum Poisson process with intensity S − 1, where S is the one-particle S matrix. The novelty of our approach is that the equations are derived directly in terms of correlators, without use of a FockantiFock (or Gel'fand-Naimark-Segal) representation. Advantages of our approach are the simplicity of derivation of the limiting equation and that the algebra of the master fields and the Ito table do not depend on the initial state of the Bose gas. The notion of a causal state is introduced. We construct master fields (white noise and number operators) describing the dynamics in the low density limit and prove the convergence of chronological (causal) correlators of the field operators to correlators of the master fields in the causal state. † permanent address:

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The Coulomb Gas at low temperature and low density

Cited by 30 publications

References 10 publications

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An electrostatic inequality with applications to the constitution of matter

Quantum stochastic equation for a test particle interacting with a dilute Bose gas

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