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Abstract

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...

Thermodynamic Properties of Gaseous Plasmas in the Zero‐Temperature Limit

“…Examples of this limiting structure are exposed at figures 1 and 2 for thermal and caloric EOS of lithium and helium plasmas [4][5][6]. For rigorous theoretical proof of existing the limit, which is under discussion (Saha-limit) in the case of hydrogen see [7,8] and references therein.The same stepped structure is valid in the zero-temperature limit for any molecular gases, for example for hydrogen (Fig. 3) [4] [6].…”

Casimir force between two ideal-conductor walls revisited

“…To make the correspondence with the quantum-mechanical version of the model, the hard-core diameter of particles of type α has to be set equal to the thermal de Broglie wavelength λ α =h(2πβ/m α ) 1/2 [20]. We would like to emphasize that the present particle system represents a microscopic model of "living conductors" where the charge density and the corresponding electric potential/field fluctuate, even for extreme values of physical parameters like the particle density.…”

“…Supposing in what follows for the sake of simplicity that α Z 2 α n α /n is of order of unity and omitting irrelevant numerical factors, these inequalities can be rewritten in a more transparent form a 0 ≪ a ≪ κ −1 (20) where a 0 ∼h 2 /(me 2 ) with m being the "typical" particle mass. The lightest of the charges are the electrons for which the quantum microscopic scale a 0 attains its maximum value, equal to the Bohr radius ∼ 10 −10 m. The first "Bohr" inequality in (20) is a quantum upper bound for possible values of particle densities.…”

The Casimir force at high temperature

“…In this formalism a quantum point particle of species γ is represented by a closed Brownian path r + λ γ ξ(s), 0 ≤ s < 1, ξ(0) = ξ(1) = 0, starting at r and of extension λ γ : it can be viewed as a charged random wire at r. Thus the ensemble of wires can be treated as a classical-like system with phase space points (r i , ξ i ). The wire shape λ γ ξ(s) (the quantum fluctuation) plays the role of an internal degree of freedom; see [10], section IV, for more details on this formalism. Here, for simplicity, we use Maxwell-Boltzmann statistics for the particles.…”