In real fractals the experimentally observed scaling range is usually much smaller than the maximum theoretical one, which is bounded from below and from above by atomic or molecular lengths and by system size respectively. Denoting with l_m and l_M the lower and upper bound of the scaling region, the typical value of δ = log₁₀ (l_M / l_m) is found to be smaller than 2 [1]. This experimental evidence recently stimulated theoretical investigations on the physical reasons leading to such a narrow bound. In particular non-equilibrium dynamical mechanism has been proposed [2] based on fractal growth holding in a regime where growth times are much shorter than thermalization times.

Here we propose an alternative thermodynamic explanation which holds in the opposite limit of thermalization times much shorter than the characteristic times of all the other dynamical mechanisms (including growth). Our approach is based on an extension to general discrete structures (fractals, glasses, polymers) of the classical Peierls-Landau result for thermodynamic instability of regular solids of arbitrary size in d ≤ 2. As an application, we calculate explicitely the fractal range in the case of a deterministic fractal with spectral dimension d ≤ 2 and we apply our model to discuss the experimental phenomenology in a classic example of real fractal systems, namely silica aerogels.