A Toy Model Approach to Fractal Nature: Thermodynamics on a Cantor-Lattice Ising Model

Abstract

Thermodynamics on a Cantor-lattice Ising model is studied to clarify effects of fractal structure. Exact solutions based on the transfer matrix are investigated for finite size systems, and it is found that there is non-trivial relationship between entropy and fractal structure. In order to understand the nature in the thermodynamic limit, the renormalization method is applied. The results suggest a possibility of residual entropy due to the competition between non-uniformity of the fractal structure and uniform external field. These pave a simple way to approach general behaviors of non-uniform systems including fractal structures, such as quasicrystals.