  Analytical expressions for radial distribution function (RDF) are of critical
importance for various applications, such as development of the perturbation
theories for equilibrium properties. Theoretically, RDF expressions for
odd-dimensional fluids can be obtained by solving the Percus-Yevick integral
equations. But for even-dimensional cases, such as the hard disk (2D) fluid,
analytical expressions are infeasible. The only 2D RDF is a heuristic
expression proposed by Yuste et al. (J. Chem. Phys. 99, 2020,1993), which
approximates the 2D RDF with an interpolation of the RDFs for hard-rod (1D) and
hard-sphere (3D) fluids and provides acceptable estimations for an intermediate
and low density range. In this work, we employ a simple and empirical
expression for the 2D RDF and the 3D RDF based on the approach proposed by
Trokhymchuk et al. for the 3D RDF (J. Chem. Phys., 123, 024501, 2005). The
parameters are determined in such a way that the final RDF expressions are
thermodynamically consistent, namely the pressure constraint and the isothermal
compressibility constraint are both satisfied. The new RDFs for the 2D and 3D
hard spheres are highly accurate for the entire density range up to the
first-order phase transition points. The predictions of the first coordination
numbers are consistent with simulation results for 3D fluid. Finally, by using
the 2D RDF with a primitive second-order perturbation theory, the
pressure-volume-temperature relation and vapor-liquid equilibrium are
calculated for the 2D Lennard-Jones fluid. Comparisons with the simulation data
show promising results.
