Second order exact derivatives to perform optimization on self-consistent integral equations problems
by
I. Charpentier
LMC, projet Idopt, BP 53, F-38041 Grenoble Cedex 9
Coauthors: N. Jakse (Universite de Metz)

The integral equation theory is a powerful semi-analytic method to predict microscopic and macroscopic properties of dense fluids via the pair-correlation function g(r) describing the local arrangement of the molecules. Let's consider a fluid of density \rho and temperature T whose molecules interact by means of a pair potential u(r). The formal exact expression for g(r) (Morita and Hiroike, 1960) shows that g(r) depends on \rho, T, u(r) as well as on an intractable so-called``bridge'' function B(r). Great efforts have been done in the last fifteen years to search for semi-empirical of it in order to find reliable andaccurate solutions for g(r). These bridge functions appear to be functionals of g(r) and contain p parameters to ensure the thermodynamic self-consistency of the resulting solution. From a physical point of view, we will discuss three well-known bridge functions B(r)=B_p depending on 1, 3 and 6 parameters respectively. For given \rho, T, u(r) and an expression for B_p, a unique bridge function is obtained by fixing the values of the p parameters to satisfy an objective thermodynamic consistency involving explicit derivatives of the pair-correlation function with respect to the density and temperature. These derivatives were exactly computed in a previous study using Odyssée. On one hand, the objective function S implies the computation of the derivatives \partialg(r)/\partial\rho and \partialg(r)/\partialT and on the other hand an optimization process requires derivatives with respect to the parameters \alpha_p entering the bridge function. As a consequence, the code of the integral equation method that computes g(r) has to be differentiated two times. The presentation will discuss optimal differentiation stages and physical results will be given.

http://www-lmc.imag.fr/lmc-edp/Isabelle.Charpentier/diff.html

Date received: November 29, 1999