Thermodynamic analysis of the q -state Akin model on a Cayley tree

The q-state modified Potts model was originally introduced by Akin [H. Akin, arXiv:2504.17616 [math-ph]]. Unlike the classical Potts model, this modified version undergoes phase transitions solely in the antiferromagnetic regime, as rigorously demonstrated in [H. Akın, Chaos Solitons Fractals 199, 116746 (2025), doi:10.1016/j.chaos.2025.116746]. The formulation considered here constitutes a previously unexplored generalization within statistical mechanics. For clarity and consistency in future references, we refer to this construction as the q-state Akin model, thereby distinguishing it from conventional Potts-type models while recognizing its first rigorous definition in our earlier work. Building upon [H. Akın, Chaos Solitons Fractals 199, 116746 (2025), doi:10.1016/j.chaos.2025.116746], this study extends the thermodynamic analysis of the q-state Akin model on the Cayley tree. Employing the cavity method, we systematically derive the total partition function using recurrence relations that incorporate arbitrary q-spin configurations. From this framework, exact expressions for the free energy are obtained in terms of q, the external field h, the coupling constant J and the inverse temperature [Formula: see text]. Thermodynamic observables such as entropy, magnetization and susceptibility are then calculated as partial derivatives of the free energy. Through two- and three-dimensional graphical analyses, we identify critical phenomena including kinks, periodic oscillations and W-shaped peaks that arise at specific parameter values.