Free energies for singleton minimal states

It is assumed that any free energy function exhibits strict periodic behavior for histories that have been periodic for all past times. This is not the case for the work function, which, however, has the usual defining properties of a free energy. Forms given in fairly recent years for the minimum and related free energies of linear materials with memory have this property. Materials for which the minimal states are all singletons are those for which at least some of the singularities of the Fourier transform of the relaxation function are not isolated. For such materials, the maximum free energy is the work function, and free energies intermediate between the minimum free energy and the work function should be given by a linear relation involving these two quantities. All such functionals, except the minimum free energy, therefore do not have strict periodic behavior for periodic histories, which contradicts our assumption. A way out of the difficulty is explored which involves approximating the relaxation function by a form for which the minimal states are no longer singletons. A representation can then be given of an arbitrary free energy as a linear combination of the minimum, maximum and intermediate free energies derived in earlier work. This representation obeys our periodicity assumption. Numerical data are presented, supporting the consistency of this approach.