1 99 9 Coulomb systems at low density

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 REVTEX –1 CONTENTS I. Coulomb systems I–1 A. Introduction I–1 B. The classical Coulomb gas I–4 C. The quantum Coulomb gas I–9 D. Neutrality I–12 II. Debye screening II–1 A. Debye Hückel Theory II–1 B. Theorems on Debye screening II–3 C. The Sine-Gordon transformation II–7 1. The Fourier transform of a Gaussian is a Gaussian II–7 2. Sum rules II–10 D. Debye spheres II–12 E. The Debye-Hückel limit II–14 F. Charge expulsion II–18 G. Symmetries and tunneling II–19 III. Dipoles III–1 A. The Dipole ensemble III–1 B. A no-screening theorem for dipoles 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 gas of charged loops V–1 A. The statistical mechanics of loops V–1 1. The magic formula V–1 2. Loop and particle correlations V–4 3. Loop interactions V–6 4. Derivation of the magic formula V–7 B. The loop Mayer expansion V–8 1. The loop Ursell function V–8 2. Summing the Coulomb chains V–9 3. The prototype diagrams V–11 4. Integrability of prototype diagrams V–13 5. The density expansion V–14 VI. Correlations and the equation of state VI–1 A. Asymptotic behavior of particle and charge correlations VI–1 1. Large distance behavior of the loop Ursell function VI–1 2. Large distance behavior of the two-particle correlation VI–3 3. Correlations of the total charge density VI–6 4. A comparison with the standard many-body perturbation theory VI–8 B. The virial equation of state VI–12 1. The Maxwell-Boltzmann pressure VI–12 2. The exchange contributions and the final result VI–15 VII. The atomic and molecular limit VII–1 A. The electron-proton gas in the Saha regime VII–1 B. The main theorem VII–5 C. Elements of proof VII–8 1. Towards a solution of problem 1 VII–9 2. Towards a solution of problem 2 VII–11 VIII. Convergent expansions VIII–1 A. Tree graph formulas and the Mayer expansion VIII–1 B. Convergence of Mayer expansions VIII–5 1. Repulsive potentials VIII–5 2. Debye spheres VIII–6 3. The Mayer expansion for a polymer gas VIII–7