Calculation of some thermodynamic quantities for the Ising model on a kth order Cayley tree

This paper focuses on the Ising model with first-and second-neighbor interactions on a kth order Cayley tree (k & GE; 2). We generalize the Ising-Vannimenus model's Gibbs measures on a Cayley tree with any order by means of Kolmogorov consistency condition. We classify the fixed points of the dynamical system associated with the model under given conditions. We demonstrate how the chaoticity of the corresponding system is influenced by the Cayley tree's order. Depending on the number k, we obtain a new formula to calculate the free energy of the model on Cayley tree of any order under given boundary conditions. We determine the system's entropy, which is the derivative of the free energy function with respect to temperature T.