Asymptotic Material Properties via Padé Approximants

 

Isaac C. Sanchez

Chemical Engineering Dept.

University of Texas

Austin, TX 78712

 

Padé approximants have been used to estimate the asymptotic equation of state properties of the following:  (1) condensed matter at extreme pressures such as those encountered deep within the earth, (2) a metastable hard sphere fluid (both in 2- and 3-dimensions), and (3) a polymer chain with excluded volume.  The latter example is analogous to a non-intersecting random walk in 3-dimensions in the infinite size limit.  In these applications there is always information available about behavior at low density that by Padé analysis is used to estimate behavior at high density. This low density information is always in the form of an series or virial expansion in density or some other parameter.  In the polymer chain case, it consists of a divergent series in the excluded volume parameter.  It is remarkable, if not mysterious, why a Padé analysis of a divergent series yields an accurate value for the asymptotic limit of a random walk in 3-dimensions.