Scaling Laws for a System with Long-Range Interactions within Tsallis Statistics

We study the one-dimensional Ising model with long-range interactions in the context of Tsallis nonextensive statistics by computing numerically the number of states with a given energy. We find that the internal energy, magnetization, entropy, and free energy follow nontrivial scaling laws with the number of constituents $N$ and temperature $T$. Each of the scaling functions for the internal energy, the magnetization, and the free energy, adopts three different forms corresponding to $q>\mathrm{}1\mathrm{}$, $q=1\mathrm{}$, and $q<\mathrm{}1\mathrm{}$, $q$ being the nonextensivity parameter of Tsallis statistics.