Thermodynamics from a scaling Hamiltonian

There are problems with defining the thermodynamic limit of systems with long-range interactions; as a result, the thermodynamic behavior of these types of systems is anomalous. In the present work, we review some concepts from both extensive and nonextensive thermodynamic perspectives. We use a model, whose Hamiltonian takes into account spins ferromagnetically coupled in a chain via a power law that decays at large interparticle distance $r$ as $1\mathrm{/}{r}^{\alpha }$ for $\alpha \ge 0$. Here, we review old nonextensive scaling. In addition, we propose a new Hamiltonian scaled by $2\frac{(N\mathrm{/}2{)}^{1-\alpha }-1}{1-\alpha }$ that explicitly includes symmetry of the lattice and dependence on the size, $N$, of the system. The new approach enabled us to improve upon previous results. A numerical test is conducted through Monte Carlo simulations. In the model, periodic boundary conditions are adopted to eliminate surface effects.