A statistical approach to the theory of the mean field

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.