Bounds for the Derivatives of the Free Energy and the Pressure of a Hard‐Core System near Close Packing
1.This restriction of fixed orientations may be relaxed as indicated in Sec. 4.
2.The way in which may be defined with full mathematical rigor even if the close‐packed configuration is unknown has been shown by O. Penrose, Phys. Letters 11, 224 (1964). He considers spherical hard cores but his definitions do not depend on this at all.
3.See, for example, T. L. Hill, Statistical Mechanics (McGraw‐Hill Book Company, Inc., New York, 1956).
4.W. G. Hoover, “Bounds on the Configuration Integral for Hard Parallel Squares and Cubes,” J. Chem. Phys. (to be published).
5.The convexity of the canonical and grand canonical potentials (Helmholtz free energy and pressure) has been discussed rigorously by D. Ruelle (Ref. 6) and M. E. Fisher (Ref. 7). The mathematical properties of convex functions are described with great clarity in Ref. 8.
6.D. Ruelle, Helv. Phys. Acta 36, 183, 789 (1963).
7.M. E. Fisher, Arch. Ratl. Mech. Anal. 17, 377 (1964).
8.G. H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities (Cambridge University Press, New York, 1952), 2nd ed., pp. 91–96. For convenience we consider functions which are convex upwards or “concave” in the customary mathematical terminology.
9.A convex domain (in a d‐dimensional space) is defined by the condition that if and are points in the domain, then all the points on the straight line segment joining and also belong to the domain. A solid sphere and a solid cube are convex domains; a solid dumbbell is not.
10.See for example Ref. 7 or O. Penrose, J. Math. Phys. 4, 1312 (1963).
11.This may be proved, for example, by using the trace inequalities and arguments given in Ref. 7.
12.See, for example, Ref. 7.
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