Estimation of critical exponents from cluster coefficients and the application of this estimation to hard spheres

For a large class of repulsive interaction models, the Mayer cluster integrals can be transformed into a tridiagonal real symmetric matrix R(mn), whose elements converge to two constants with 1/n(2) correction. We find exact expressions in terms of these correction terms for the two critical exponents describing the density near the two singular termination points of the fluid phase. We apply the method to the hard-spheres model and find that the metastable fluid phase terminates at rho(t) = 0.751[5]. The density near the transition is given by rho(t)-rho approximately (z(t) - z)(sigma'), where the critical exponent is predicted to be sigma' = 0.0877[25]. Interestingly, the termination density is close to the observed glass transition; thus, the above critical behavior is expected to be associated with the onset of glassy behavior in hard spheres.