  We develop a statistical theory of the mean field. 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 are random
with the average value of zero. Explicit expressions for the mean field and the
fluctuation away from the average are obtained. 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. Making some simplifying assumptions a schematic
model is developed and applied to the problem of nuclear matter. It yields a
measure of the strength of the effective interaction. The latter 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.
