Free energy and entropy for the constructive Gibbs measures of the Ising model on the Cayley tree of order three

In this paper, we identify non-translation-invariant constructive Gibbs measures for the Ising model on a third-order Cayley tree, which differ from known ones. We provide the conditions for the existence of at least two distinct Gibbs measures, which implies that a phase transition occurs. The free energies and entropies corresponding to the identified measures are calculated. These free energies and entropies are then compared with the known ones and shown to differ from them.