Abstract
We study the statistical mechanics of classical Coulomb systems in a low coupling regime (Debye--Hückel regime) in a confined geometry with Dirichlet boundary conditions for the electric potential. We use a method recently developed by the authors which relates the grand partition function of a Coulomb system in a confined geometry with a certain regularization of the determinant of the Laplacian on that geometry with Dirichlet boundary conditions. We study several examples of fully confining geometry in two and three dimensions and semi-confined geometries where the system is confined only in one or two directions of the space. We also generalize the method to study systems confined in arbitrary geometries with smooth boundary. We find a relation between the expansion for small argument of the heat kernel of the Laplacian and the large-size expansion of the grand potential of the Coulomb system. This allow us to find the finite-size expansion of the grand potential of the system in general. We recover known results for the bulk grand potential (in two and three dimensions) and the surface tension (for two-dimensional systems). We find the surface tension for three-dimensional systems. For two-dimensional systems our general calculation of the finite-size expansion gives a proof of the existence a universal logarithmic finite-size correction predicted some time ago, at least in the low coupling regime. For three-dimensional systems we obtain a prediction for the curvature correction to the grand potential of a confined system.
Similar content being viewed by others
References
B. Jancovici G. Téllez (1996) J. Stat. Phys. 82 609
P. J. Forrester B. Jancovici G. Téllez (1996) J. Stat. Phys. 84 359
B. Jancovici G. Manificat C. Pisani (1994) J. Stat. Phys. 76 307
B Jancovici (1995) J. Stat. Phys 80 445
B Jancovici G Téllez (1996) Phys. Rev. E 70 011508
G Téllez (2004) Phys. Rev. E. 70 011508
J. L. Lebowitz Ph. A. Martin (1984) J. Stat. Phys. 34 287
J Cardy I Peschel (1988) Nucl. Phys. B 300 337
J. Cardy, in Fields, Strings and Critical Phenomena, Les Houches, E. Brézin and J. Zinn-Justin, eds. (North-Holland, Amsterdam, 1990).
P. Di Francesco, P. Mathieu, and D. Senechal, Conformal Field Theory (Springer, 1999).
P. J. Forrester B. Jancovici J. Madore (1992) J. Stat. Phys. 69 179
G Téllez (2001) J. Stat. Phys 104 945
G. Téllez P. J. Forrester (1999) J. Stat. Phys. 97 489
L. Šamaj B. Jancovici (2002) J. Stat. Phys 106 323
B. Jancovici E. Trizac (2000) Physica A 284 241
B. Jancovici (2000) J. Stat. Phys. 100 201
L. Šamaj (2001) Physica A 297 142
B. Jancovici L. Samaj (2001) J. Stat. Phys. 104 753
P. Kalinay P. Markos L. Samaj I. Travenec (2000) J. Stat. Phys. 98 639
B. Jancovici P. Kalinay L. Samaj (2000) Physica A 279 260
A. Torres G. Téllez (2004) J. Phys. A: Math. Gen. 37 2121
G. Téllez, Debye–Hückel theory for two-dimensional Coulomb systems living on a finite surface without boundaries, e-print cond-mat/0409093 (2004).
M. Kac (1966) Am. Math. Monthly 73 1
H. P. McKean SuffixJr. I. M. Singer (1967) J. Diff. Geom. 1 43
S. Samuel (1978) Phys. Rev. D 18 1916
T. Kennedy (1983) Comm. Math. Phys. 92 269
C. Deutsch H. E. Dewitt Y. Furutani (1979) Phys. Rev. A 20 2631
C. Deutsch M. Lavaud (1974) Phys. Rev. A 9 2598
L. Šamaj I. Travěnec (2000) J. Stat. Phys. 101 713
S. Gradshteyn I. M. Ryzhik (1965) Table of Integrals, Series, and Products Academic New York
L. Šamaj B. Jancovici (2001) J. Stat. Phys. 103 717
M. Abramowitz and I. S. Stegun, Handbook of Mathematical Functions, 9th edition (Dover Publications, 1972).
P. J. Forrester (1992) J. Stat. Phys. 67 433
H. Weyl (1946) The Classical Groups Princeton University Press Princeton
A. Voros (1987) Comm. Math. Phys. 110 439
M. M. Lipschutz, Theory and Problems of Differential Geometry (McGraw-Hill, 1969).
E. T. Whittaker and G. N. Watson, A Course of Modern Analysis (Cambridge University Press, 1902).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Torres, A., Téllez, G. General Considerations on the Finite-Size Corrections for Coulomb Systems in the Debye--Hückel Regime. J Stat Phys 118, 735–765 (2005). https://doi.org/10.1007/s10955-004-8827-z
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/s10955-004-8827-z