Extremal entropy for a constrained probability density

The state of extremal entropy for a one-dimensional probability density is considered. This density is constrained by fixed values of the first and second moment. The grandcanonical distribution yields the extremum of the entropy within a certain range of values of the moments. A different type of density corresponds to an extremum of entropy when the moments are outside this range. The shape of this density is approximated with the Ritz variation method. The results are applied to the formation of breathers in the discrete nonlinear Schrödinger equation.