  A quasi-nonlinear field theory which describes how to take ensemble averages
that are unique to the Collisionless Boltzmann Equation is described. The
assumption that the ensemble average of the distribution function is equal to
the extremum entropy state, $\langle f \rangle = f_0$ is taken apart and shown
to be wrong. An application describes the nonlinear saturation of Jeans'
instability, and the gravitational amplification of Poisson noise.
