Direct variational approach to the calculation of dynamic structure functions

Abstract

We consider the problem of calculating the dynamic structure function $S(k,\omega )$ via a properly posed initial-value problem for a linear kinetic equation. A variational principle is introduced and shown to give a direct bounded estimate of $S(k,\omega )$. The variational trial functions for the problem are the space-time transforms of the phase-space correlation function. The choice of such trial functions are discussed and general expressions for $S(k,\omega )$ are developed which are applicable to simple gases. Numerical results are presented for the specific case of hard spheres, these results being compared to previous first-principle calculations based upon kinetic equations with and without the inclusion of memory effects. Excellent agreement with previous results, at all ratios of fluctuation wavelength to mean free path, is obtained with considerably less computational complexity. This computational efficiency of the variational approach suggests that it should be a valuable technique, from the standpoint of feasibility, in attempts at first-principle calculations for more complex many-body systems. In this regard, further study and improvement of the variational techniques themselves appear to be warranted.