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Seventh Virial Coefficients for Hard Spheres and Hard Disks
1.J. E. Mayer and M. G. Mayer, Statistical Mechanics (John Wiley & Sons, Inc., New York, 1940).
2.We follow the graph theoretical terminologies used by G. E. Uhlenbeck and G. W. Ford, in Studies in Statistical Mechanics, J. de Boer and G. E. Uhlenbeck, Eds. (North‐Holland Publ. Co., Amsterdam, 1962), Vol. 1, Pt. B.
3.J. O. Hirschfelder, C. F. Curtiss, and R. B. Bird, Molecular Theory of Gases and Liquids (John Wiley & Sons, Inc., New York, 1954), Chap. 3.
4.E. A. Guggenheim, J. Chem. Phys. 13, 253 (1945).
5.R. N. Keeler, M. Van Thiel, and B. J. Alder, Physica 31, 1437 (1965).
6.For example, see L. Verlet, Physica 30, 95 (1964).
7.J. L. Lebowitz and O. Penrose, J. Math. Phys. 5, 841 (1964).
7.The bound given in this paper can be slightly improved for repulsive potentials [see F. H. Ree, Phys. Rev. 155, 84 (1967)].
8.F. H. Ree and W. G. Hoover, J. Chem. Phys. 40, 939 (1964).
9.B. J. Alder and T. E. Wainwright, J. Chem. Phys. 33, 1439 (1960).
10.W. W. Wood and J. D. Jacobson, J. Chem. Phys. 27, 1207 (1957).
11.See the references quoted in Ref. 8. The exact value of the hard‐disk is obtained recently by J. S. Rowlinson, Mol. Phys. 7, 593 (1963–1964);
11.and P. C. Hemmer, J. Chem. Phys. 42, 1116 (1965).
11.The hard‐sphere quoted in Eq. (4) agrees with the calculations of S. Katsura and Y. Abe, J. Chem. Phys. 39, 2068 (1963);
11.J. S. Rowlinson, Proc. Roy. Soc. (London) A279, 147 (1964);
11.and those of F. H. Ree, R. N. Keeler, and S. L. McCarthy, J. Chem. Phys. 44, 3407 (1966) within each other’s error bounds.
12.F. H. Ree and W. G. Hoover, J. Chem. Phys. 41, 1635 (1964).
13.For graphical notations used in this section, we refer to Ref. 12.
14.See p. 184 of Ref. 2.
15.F. H. Ree and W. G. Hoover, “Transformation Table for the Seven‐Point Stars,” Rept. UCRL‐14634, February 1966.
16.If a part of a wiggly line graph containing a triangular set of wiggly lines is not linked to some other part of the wiggly line graph by fewer than two intermediate wiggly lines, the wiggly line graph can be regarded as a subgraph (Ref. 8) of “Pls. see pdf. for this diagram.” Therefore, the corresponding integral also vanishes. For example, “Pls. see pdf. for this diagram,” .
17.W. G. Hoover and A. G. De Rocco, J. Chem. Phys. 36, 3141 (1962).
18.F. H. Ree, R. N. Keeler, and S. L. McCarthy, Ref. 11.
19.G. P. Hoel, Introduction to Mathematical Statistics (John Wiley & Sons, Inc., New York, 1954), 2nd ed., p. 198.
20.J. K. Percus and G. J. Yevick, Phys. Rev. 110, 1 (1958);
20.W. G. Hoover and J. C. Poirier, J. Chem. Phys. 38, 327 (1963).
21.M. S. Wertheim, Phys. Rev. Letters 10, 321 (1963);
21.E. Thiele, J. Chem. Phys. 39, 474 (1963);
21.H. Reiss, H. L. Frisch, and J. L. Lebowitz, J. Chem. Phys. 31, 369 (1959). , J. Chem. Phys.
21.For two‐dimensional hard disks, Reiss, Frisch, and Lebowitz’s theory leads to [E. Helfand, H. L. Frisch, and J. L. Lebowitz, J. Chem. Phys. 34, 1037 (1961)], with 0.5, 0.3125, 0.1875, and 0.109375 for These values of differ by no more than 6% from the exact values of quoted in (4), (5), and Table II. Furthermore, calculated from this equation at the predicted transition density is 10.49. This agrees fairly well with the observed value, 10.13 (see Table V), for a finite system of 870 hard disks.
22.J. Yvon, Actualités Scientifiques et Industrielles (Hermann et Cie., Paris, 1935), Vol. 203;
22.M. Born and H. S. Green, Proc. Roy. Soc. (London) A188, 10 (1946).
23.F. H. Ree and B. J. Alder (unpublished result).
24. given in Eq. (20) is obtained by Rowlinson (Ref. 11).
25.B. J. Alder and T. E. Wainwright, Phys. Rev. 127, 359 (1962).
26.For a general review of the Padé approximant, we refer to G. A. Baker, in Advances in Theoretical Physics, K. A. Brueckner, Ed. (Academic Press Inc., New York, 1965), Vol. 1, Chap. 1.
27.The present (revised) value is estimated from the molecular‐dynamics data for a system of 500 hard spheres. For 500 spheres the solid phase is unstable at The value earlier quoted in Ref. 8 is estimated from molecular‐dynamics data for 108 hard spheres. The molecular‐dynamics data were kindly furnished by B. J. Alder.
28.This predicted phase transition may contradict the proof of instability of a solid phase with a periodic singlet distribution function in a two‐dimensional continuum system [L. D. Landau and E. M. Lifshitz, Statistical Physics (Addison‐Wesley Publ. Co., Reading, Mass., 1958), Sec. 125,
28.cited in F. H. Stillinger, E. A. DiMarzio, and R. L. Kornegay, J. Chem. Phys. 40, 1564 (1964)].
28.Whether or not hard‐disk systems have a periodic singlet distribution function at high density is unknown. In a recent paper [W. G. Hoover and F. H. Ree, J. Chem. Phys. 45, 3649 (1966)],
28.we investigate the melting problem by employing a simple hard‐core model. For this model, two‐ and three‐dimensional systems behave quite differently. In particular, the two‐dimensional model system seems to show no phase transition to an ordered state, thus supporting the argument of Landau and Lifshitz. At the same time the hard‐disk transition seems well established. The number dependence of the tie line has been calculated to vary as lnN. This dependence agrees with the machine results and predicts only a small change from 870 disks to an infinite system [W. G. Hoover and B. J. Alder, J. Chem. Phys. 46, 686 (1967)].
29.The coefficients appearing in the Padé approximants in Table VI are sensitive functions of the values of the virial coefficients used to calculate them. However, no significant change in calculated values occurs for entire fluid range. For example, if the upper and lower Monte Carlo values of and quoted in Eqs. (4), (5), and Table II are used, the corresponding pressures from the approximants P(3,4) for hard disks are 10.34 for upper limits and 10.30 for lower limits of at which is close to the density at which the phase transition starts from the fluid phase. The values of the approximants P(4,3) for hard spheres, using the extremum Monte Carlo values of are similarly in good agreement with the values calculated from the approximant P(4,3) in Table VI.
30.M. Kac, G. E. Uhlenbeck, and P. C. Hemmer, J. Math. Phys. 4, 216 (1963).
31.R. J. Riddell and G. E. Uhlenbeck, J. Chem. Phys. 21, 2056 (1953).
32.K. F. Herzfeld and M. G. Mayer, J. Chem. Phys. 2, 38 (1934).
33.For a hard lattice gas, the coefficients given by D. S. Gaunt and M. F. Fisher [J. Chem. Phys. 43, 2840 (1965)] do not alternate in sign.
34.The elements for the graph complexity determinant defined in this manner have signs opposite to those used in Ref. 2.
35.The diagrams for these 468 Mayer stars are taken from Ref. 17.
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