  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
MeV/nucleon demonstrating the strong damping of the interaction strength
induced by the averaging process.
