Some Topics in the Theory of Fluids

It is shown how certain thermodynamic functions, and also the radial distribution function, can be expressed in terms of the potential energy distribution in a fluid. A miscellany of results is derived from this unified point of view. (i) With g(r) the radial distribution function and Φ(r) the pair potential, it is shown that g exp (Φ/kT) may be written as a Fourier integral, or as a power series in r2 the terms of which alternate in sign. (ii) A potential-energy distribution which is independent of the temperature implies an equation of state which is a generalization of a number of well-known approximations. (iii) The grand partition function of the one-dimensional lattice gas is obtained from thermodynamic arguments without evaluating a sum over states. (iv) If in a two-dimensional honeycomb (three-coordinates) lattice gas fr(r=0, 1, 2, 3) is the fraction of all the empty sites which at equilibrium are neighbored by exactly r filled sites, then at the critical density the values of all four of the f's as functions of temperature follow from previously known properties of this system; in particular, at the critical point, f0 = 3/8+5√3/24, f1 = 1/8+√3/24, f2 = 1/8—√3/24, f3 = 3/8–5√3/24.