Abstract
In the framework of the grand-canonical ensemble of statistical mechanics, we give an exact diagrammatic representation of the density profiles in a classical multicomponent plasma near a dielectric wall. By a reorganization of Mayer diagrams for the fugacity expansions of the densities, we exhibit how the long-range of both the self-energy and pair interaction are exponentially screened at large distances from the wall. However, the self-energy due to Coulomb interaction with images still diverges in the vicinity of the dielectric wall and the variation of the density is drastically different at short or large distances from the wall. This variation is involved in the inhomogeneous Debye–Hückel equation obeyed by the screened pair potential. Then the main difficulty lies in the determination of the latter potential at every distance. We solve this problem by devising a systematic expansion with respect to the ratio of the fundamental length scales involved in the two coulombic effects at stake. (The application of this method to a plasma confined between two ideally conducting plates and to a quantum plasma will be presented elsewhere). As a result we derive the exact analytical perturbative expressions for the density profiles up to first order in the coupling between charges. The mean-field approach displayed in Paper I is then justified.
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REFERENCES
A. Alastuey, F. Cornu, and A. Perez, Virial expansions for quantum plasmas: Diagrammatic resummations, Phys. Rev. E 49:1077 (1994).
F. Cornu, Correlations in quantum plasmas, I. Resummations in Mayer-like diagrammatics, Phys. Rev. E 53:4562 (1996).
F. Cornu, Quantum plasmas with or without a uniform field. II. Exact low-density free energy, Phys. Rev. E 58:5293 (1998).
R. L. Guernsey, Correlation effects in semi-infinite plasmas, Phys. Fluids 13:2089 (1970).
E. Haga, On Mayer's theory of dilute ionic solutions, J. of the Physical Society of Japan 8:714 (1953).
J. P. Hansen and I. R. Mac Donalds, Theory of Simple Liquids (Academic Press, London, 1986).
J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1962).
B. Jancovici, Classical Coulomb systems near a plane wall. I. J. Stat. Phys. 28:43 (1982).
L. D. Landau and E. M. Lifshitz, Course of Theoretical Physics. Electrodynamics of Continuous Media, Vol. 8 (Pergamon Press, Oxford, 1985).
E. Lieb and J. L. Lebowitz, The constitution of matter: Existence of thermodynamics for systems composed of electrons and nuclei, Adv. Math. 9:316 (1972).
E. Meeron, Theory of potentials of average force and radial distribution functions in ionic solutions, J. Chem. Phys. 28:630 (1958).
A. Messiah, Quantum Mechanics (Wiley, New York, 1958).
A. S. Usenko and I. P. Yakimenko, Interaction energy of stationary charges in a bonded plasma, Sov. Techn. Phys. Lett. 5:549 (1979).
D. Zwillinger, Handbook of Differential Equations (Academic Press, 1989).
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Aqua, JN., Cornu, F. Density Profiles in a Classical Coulomb Fluid Near a Dielectric Wall. II. Weak-Coupling Systematic Expansions. Journal of Statistical Physics 105, 245–283 (2001). https://doi.org/10.1023/A:1012290228733
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DOI: https://doi.org/10.1023/A:1012290228733