Exact expressions are derived for the molecular distribution functions in a one-dimensional fluid whose particles interact with a nearest neighbor pair potential. The pair distribution function for rigid spheres is found to be identical with Frenkel's result. The one-dimensional form of a new set of integral equations for the molecular distribution functions is examined. The superposition principle is found to be exact in a one-dimensional fluid with nearest neighbor interactions.
Under certain conditions, an asymptotic expression for the equation of state of a classical mechanical system of N ν-dimensional (ν=1, 2, or 3) hard spheres confined in a volume V is obtained in the form pVNkT=ν(1–1/N)(V/V0)−1+O(1).This expression agrees with the leading term in V/V0—1 of the usual free-volume approximation for N = ∞. The conditions under which this conclusion is established are a restriction to a finite number of molecules N with periodic boundary conditions, and the requirement that as V→V0 the accessible configuration states approach a close-packed configuration whose coordination number c satisfies the requirement c≥2ν−2(ν−1)/N.Such a limiting configuration, from which only an infinitesimal region of configuration space is accessible under an infinitesimal expansion, is called a stable configuration; the above restriction on the coordination number is a necessary condition for stability. The difficulties which appear as N→∞ are indicated.
The Monte Carlo numerical method for obtaining statistical mechanical averages in the petite canonical ensemble has been applied to the two-dimensional triangular lattice-gas model. The Monte Carlo procedure described by Rosenbluth et al. and by Wood and Parker has been extended slightly to include the calculation of the partition function. The results are compared with the exact solutions for the two-dimensional lattice and the agreement is well within the fluctuation displayed by the numerical method. The calculations are found to be relatively insensitive to a change in the pseudo-random number sequence and to display no exceptional behavior in the region of the first-order phase transition for small lattices.
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