Thermal work and minimum free energy in a heat conductor with memory

A rigid linear heat conductor with memory effects is considered. Thus, the behaviour of the material is characterized by a constitutive equation which relates the heat flux to the history of the temperature gradient. A given thermal history is considered and its prolongation via an assigned process is defined. Then, the notion of equivalence is introduced to single out and associate together all those different thermal histories which correspond to the same heat flux. Notably, whenever the heat flux is the same the related thermal work is also the same. An explicit expression of the minimum free energy, related to the maximum recoverable work, is obtained in the frequency domain.