Problems of Liquids Theory

Problem of Virial Expansion:

Virial expansion is a polynomial in density $\rho=N/V$. It is a continuous non-singular function of the density. We know however, that the density cannot exceed the value of the closed packing 0.74, therefore the equation of state should diverge. Empirical recipe: let's cook some divergent function and fit it with the known behavior at low densities.

Padé Approximants:

Let us write

$$P(\rho) = \rho kT\left(1 + B(T) \frac{1+c_1\rho+c_2\rho^2+... ...c_n\rho^n}{1 +d_1\rho+d_2\rho^2+\ldots+d_m\rho^m} \right)$$

The expansion at small $\rho$ should coincide with virial expansion! If we know N=n+m virial coefficients, (n,m)-Padé approximant can be constructed. (3,3) approximant describes the simulation (P,V,T) data for hard spheres a lot better, than the corresponding virial expansion with 6 virial coefficients. This is however empirical and ambiguous approach. Alternative description is necessary.