Abstract
The thermodynamic properties of the mean spherical (MSA), Percus-Yevick (PY), and hypernetted-chain (HNC) approximations are derived by a simple and unified approach by considering the RPA free-energy functionalF and employing an Ewald-type identity. It is demonstrated that with decreasing relative contribution of the hard-core insertion to the thermodynamic functions, the MSA changes its nature from PY-like to HNC-like, withF changing its role from excess pressure to excess free energy, respectively. It is found that the condition of continuity of the MSA pair functions is equivalent to a stationarity condition forF and leads to thermodynamic consistency between the “virial” and “energy” equations of state for the (thus defined) “soft”-MSA (SMSA), withF playing the role of the excess free energy. It is shown that the PY-“compressibility” and “virial” equations of state forD-dimensional hard spheres may be simply obtained one from the other without knowing any details of the solution of the model. Using this relation we find an indication that the PY approximation for hard spheres becomes less accurate with increasing dimensionality. A general variational formulation is presented for the application of the MSA for soft potentials, and results for the one-component plasma are discussed and extended.
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References
J. K. Percus and G. Yevick,Phys. Rev. 136:8290 (1964); J. L. Lebowitz and J. K. Percus,Phys. Rev. 144:251 (1966).
The pioneering methods for solving the MSA were established by M. S. Wertheim,J. Math. Phys. 5:643 (1964); and R. J. Baxter,Aust. J. Phys. 21:563 (1968).
References to earlier literature in the field are given by, e.g., L. Blum and J. S. Hoye,J. Stat. Phys. 19:317 (1978); direction to more recent works may be found in, e.g., J. S. Hoye and G. Stell,Mol. Phys. 11:1 (1984).
J. S. Hoye and G. Stell,J. Chem. Phys. 67:439 (1977).
Y. Rosenfeld,J. Chem. Phys. 76:1170 (1982).
M. J. Gillan,J. Phys. C: Solid State Phys. 7:Ll (1974); a closely related approach employing a “marriage” between a short range PY behavior and long range MSA behavior for the direct correlation functions is described by A. H. Narten, L. Blum, and R. H. Fowler,J. Chem. Phys. 60:3378 (1974), and see references therein.
R. G. Palmer and J. D. Weeks,J. Chem. Phys. 58:4171 (1973).
G. Pastore, C. Napi, U. DeAngelis, and A. Forlani,Phys. Lett. 78A:75 (1980).
J. P. Hansen and J. B. Hayter,Mol. Phys. 46:651 (1982).
Y. Rosenfeld and N. W. Ashcroft,Phys. Rev. A 20:1208 (1979).
Y. Rosenfeld and N. W. Ashcroft,Phys. Rev. A 20:2162 (1979).
R. G. Palmer,J. Chem. Phys. 73:2009 (1980).
U. DeAngelis, A. Forlani, and M. Giordano,J. Phys. C.: Solid State Phys. 13:3649 (1980).
Y. Rosenfeld,J. Phys. C.: Solid State Phys. 15:L125 (1982).
Y. Rosenfeld,Phys. Rev. A 25:1206 (1982).
D. MacGowan,J. Phys. C.: Solid State Phys. 16:59 (1983).
D. MacGowan,J. Phys. C.: Solid State Phys. 16:L7 (1983).
D. MacGowan,J. Stat. Phys. 32:123 (1983).
S. M. Foiles and N. W. Ashcroft,J. Chem. Phys. 75:3594 (1981).
Y. Rosenfeld,Phys. Rev. A 29:2877 (1984).
E. Waisman,Mol. Phys. 25:45 (1973).
See, e.g., the review article by J. A. Barker and D. Henderson,Rev. Mod. Phys. 48:587 (1976).
M. Baus and J. P. Hansen,Phys. Rep. 59:1 (1980).
H. C. Andersen and D. Chandler,J. Chem. Phys. 57:1918 (1972).
R. J. Baxter,J. Chem. Phys. 47:4855 (1967).
R. J. Baxter,J. Chem. Phys. 52:4559 (1970).
R. E. Caligaris, A. E. Rodriguez, and M. Silbert,J. Chem. Phys. 51:1016 (1969).
This result corrects an error in Ref. 12.
J. L. Lebowitz,Phys. Rev. 133:A895 (1964).
J. L. Lebowitz and D. Zomick,J. Chem. Phys. 54:3335 (1971).
B. C. Freasier and D. J. Isbister,Mol. Phys. 42:927 (1981).
D. G. Chae, F. H. Ree, and T. Ree,J. Chem. Phys. 50:1581 (1969).
K. C. Ng,J. Chem. Phys. 61:2680 (1974).
F. Lado,Phys. Rev. B 17:2827 (1978).
F. Rogers, private communication.
Y. Rosenfeld,Phys. Rev. 26:3622 (1982).
Y. Rosenfeld and W. M. Gelbart, On the Statistical Thermodynamics of Charged Particles of Arbitrary Shape and Concentration,J. Chem. Phys., in press.
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On sabbatical leave from the Nuclear Research Center-Negev, P.O. Box 9001, Beer Sheva, Israel.
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Rosenfeld, Y. Comments on the mean spherical approximation for hard-core and soft potentials and an application to the one-component plasma. J Stat Phys 37, 215–236 (1984). https://doi.org/10.1007/BF01012912
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DOI: https://doi.org/10.1007/BF01012912