One application using the above definition of dimension was to the extensiveness of statistical mechanics systems with a power law potential where the interaction varies with the distance r as 1 / r α α . In one dimension the system properties like the free energy do not behave extensively when 0 ≤ ≤ α α ≤ ≤ 1 , i.e., they increase faster than N as N → → ∞ ∞ , where N is the number of spins in the system.