Dedicated to Richard Lemmer on the occasion of his 60 th birthday
A statistical theory of the mean field is developed. It is based on the proposition that the mean field can be obtained as an energy average. Moreover, it is assumed that the matrix elements of the residual interaction, obtained after the average interaction is removed, are random with the average value of zero. With these two assumptions one obtains explicit expressions for the mean field and the fluctuation away from the average. The fluctuation is expanded in terms of more and more complex excitations. Using the randomness of the matrix elements one can then obtain formulas for the contribution to the error from each class of complex excitations and a general condition for the convergence of the expansion is derived. It is to be emphasized that no conditions on the nature of the system being studied are made. Making some simplifying assumptions a schematic model is developed. This model is applied to the problem of nuclear matter. The model yields a measure of the strength of the effective interaction. It turns out to be three orders of magnitude less than that calculated using a potential which gives a binding energy of about &7 MeVΓnucleon demonstrating the strong damping of the interaction strength induced by the averaging process.
1998 Academic Press
I. INTRODUCTION
Remarkably, the mean field provides a good description of many systems. The underlying dynamics may be very different, the interparticle interaction may be long range or short range; it may be weak or strong. Nevertheless in each case a mean field description agrees closely with experiment. The question naturally arises: article no. PH975731 105 0003-4916Γ98 25.00