Hasan Akin

This study investigates the thermodynamic and information-theoretic properties of a nonhomogeneous q-state Akın–Potts hybrid model on an [Formula: see text]-lattice. We primarily utilize the transfer matrix formalism to construct the partition function for the one-dimensional lattice model. By decomposing the global transfer matrix into a product of submatrices that represent distinct interaction types, we analyze the equilibrium states of the system within a two-step transition framework using the Kolmogorov–Chapman relation. Furthermore, we employ a gauge transformation that maps the physical transfer matrix onto a row-stochastic transition matrix, which enables the calculation of the entropy rate [Formula: see text] as a measure of the information production in the system. Unlike traditional homogeneous frameworks that rely on a single competing interaction, our model incorporates successively distinct coupling constants to better mimic the structural heterogeneities in real-world materials. We examine the interplay between these multi-coupling parameters and an external magnetic field, demonstrating how they govern the crossover from deterministic ordering at low temperatures to maximal stochasticity [Formula: see text] in the high-temperature limit. Notably, our findings reveal that this sequential interaction framework provides a highly tunable environment for modulating complex thermodynamic anomalies and pseudo-transitions, thereby offering a new source of inspiration for decoding nonhomogeneous physical systems in modern condensed matter physics.

We investigate the hierarchical bifurcation and fixed-point structure of discrete-time dynamical systems arising from the q -state asymmetric Potts model on a Cayley tree of order k = 2 . For arbitrary q ⩾ 3 , the model reduces to a d = q − 1 -dimensional rational map. We derive a universal quadratic equation governing the fixed points on all synchronized invariant slices, yielding the critical thresholds K c ( d , r ) = 4 r ( d − r + 1 ) , where K = ( θ − 1 ) ( ρ − 1 ) with θ = e β J 1 and ρ = e β J 2 , and the corresponding double roots u 0 ( r ) = ( d − r + 1 ) / r . The duality K c ( d , r ) = K c ( d , d − r + 1 ) pairs complementary synchronization patterns and reduces the number of distinct critical thresholds to ⌈ ( q − 1 ) / 2 ⌉ . In the generic fully developed regime, K > max r K c ( d , r ) with K ≠ q 2 , the number of synchronized fixed points reaches 2 q − 1 , yielding the corresponding enumeration of synchronized translation-invariant splitting Gibbs measures under the boundary law correspondence. The exceptional value K = q 2 produces a degeneracy in which a quadratic root coincides with the persistent trivial fixed-point u = 1 . For even q , this value coincides with the critical threshold of the unique self-dual slice. The q = 5 model is analyzed in detail, exhibiting successive bifurcations at K = 16 and K = 24 , while the q = 6 and q = 7 cases further confirm the universal hierarchical structure and its synchronized Gibbs measure enumeration.

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 [ ]. 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.