Abstract
Within the Feynman–Kac path integral representation, the equilibrium quantities of a quantum plasma can be represented by Mayer graphs. The well known Coulomb divergencies that appear in these series are eliminated by partial resummations. In this paper, we propose a resummation scheme based on the introduction of a single effective potential φ that is the quantum analog of the Debye potential. A low density analysis of φ shows that it reduces, at short distances, to the bare Coulomb interaction between the charges (which is able to lead to bound states). At scale of the order of the Debye screening length κ −1 D, φ approaches the classical Debye potential and, at large distances, it decays as a dipolar potential (this large distance behaviour is due to the quantum nature of the particles). The prototype graphs that result from the resummation obey the same diagrammatical rules as the classical graphs of the Abe–Meeron series. We give several applications that show the usefulness of φ to account for Coulombic effects at all distances in a coherent way.
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REFERENCES
J. Ginibre, Some applications of funtional integration in statistical mechanics, in Statistical Mechanics and Quantum Field Theory, C. DeWitt and R. Stora, eds., Les Houches (Gordon and Breach, 1971), pp. 327–427.
L. Fetter and J. D. Walecka,Quantum Theory of Many Particle Systems (McGraw-Hill, New York, 1971).
H. E. DeWitt, Evaluation of the quantum mechanical ring sum with Boltzmann statistics, J. Math. Phys. 7:1216–1228 (1962); Statistical mechanics of high temperature quantum plasmas beyond the ring approximation, J. Math. Phys. 7:616–626 (1966).
H. E. DeWitt, M. Schlanges, A. Y. Sakakura, and W. D. Kraeft, Low density expansion of the equation of state for a quantum electron gas, Phys. Lett. 197:326–329 (1995).
E. W. Montroll and J. C. Ward, Quantum statistics of interacting particles: general theory and some remarks on properties of an electron gas, Phys. Fluids 1:55–72 (1958).
T. Morita, Equation of State of High Temperature Plasma, Prog. Theor. Phys. 22:757 (1959).
W. Ebeling, Statistische Thermodynamik der gebundenen Zustände in Plasmen, Ann. Phys. (Leipzig) 19:104–112 (1967).
W. D. Kraeft, D. Kremp, W. Ebeling, and G. Röpke, Quantum Statistics of Charged Particles (Plenum Press, New York, 1986).
A. Alastuey and A. Perez, Virial expansion of the equation of state of a quantum plasma, Eur. Phys. Lett. 20:19–24 (1992).
A. Alastuey, F. Cornu, and A. Perez, Virial expansion for quantum plasmas: Diagrammatic resummations, Phys. Rev. E 49:1077–1093 (1994); Virial expansion for quantum plasmas: Maxwell-Boltzmann statistics, Phys. Rev. E 51:1725–1744 (1995).
A. Alastuey and A. Perez, Virial expansion for quantum plasmas: Fermi-Bose statistics, Phys. Rev. E 53:5714–5728 (1996).
F. Cornu, Correlations in quantum plasmas. I. Resummations in Mayer-like diagrammatics, Phys. Rev. E 53:4562–4594 (1996); II. Algebraic tails, Phys. Rev. E 53:4595–4631 (1996).
F. Cornu, Exact algebraic tails of static correlations in quantum plasmas at low density, Phys. Rev. Lett. 78:1464–1467 (1997).
F. Cornu, Quantum plasma with or without uniform magnetic field. I. General formalism and algebraic tails of correlations, Phys. Rev. E 58:5268–5292 (1998); II. Exact low-density free energy, Phys. Rev. E 58:5293–5321 (1998); III. Exact low-density algebraic tails of correlations, Phys. Rev. E 58:5322–5346 (1998).
D. C. Brydges and Ph. A. Martin, Coulomb systems at low density: A review, J. Stat. Phys. 96:1163–1330 (1999).
E. Meeron, Theory of potentials of average force and radial distribution functions in ionic solutions, J. Chem. Phys. 28:630–643 (1958).
R. Abe, Giant cluster expansion theory and its application to high temperature plasma Prog. Theor. Phys. 22:213 (1959).
J.-P. Hansen and I. R. McDonald, Theory of simple liquids (Academic Press, London, 1986).
F. Cornu and Ph. A. Martin, Electron gas beyond the random phase approximation: algebraic screening, Phys. Rev. A 44:4893–4910 (1991).
N. Macris, Ph. A. Martin, and J. V. Pulé, Diamagnetic currents, Comm. Math. Phys. 117:215–241 (1988).
A. Alastuey and Ph. A. Martin, Absence of exponential clustering for static quantum correlations and time-displaced correlations in charged fluids, Eur. Phys. Lett. 6:385–390 (1988); Absence of exponential clustering in quantum Coulomb fluids, Phys. Rev. A 40:6485–6520 (1989).
A. Alastuey, F. Cornu, and Ph. A. Martin, The Hydrogen Plasma in the Atomic Limit: I. Diagrammatic Analysis of the Equation of State. (In preparation).
Ph. A. Martin, Sum rules in charged fluids, Rev. Mod. Phys. 60:1075–1127 (1988).
Ph. A. Martin and Ch. Gruber, Screening in quantum charged systems, Phys. Rev. A 30:512 (1984).
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Ballenegger, V., Martin, P.A. & Alastuey, A. Quantum Mayer Graphs for Coulomb Systems and the Analog of the Debye Potential. Journal of Statistical Physics 108, 169–211 (2002). https://doi.org/10.1023/A:1015443603197
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DOI: https://doi.org/10.1023/A:1015443603197