Thermodynamics of fractal networks

Optimal channel networks are fractal structures that bear a striking resemblance to real rivers. They are obtained by minimizing an energy functional associated with spanning trees. We show that large network development effectively occurs at zero temperature since the entropy scales subdominantly with system size compared to the energy. Thus these networks develop under generic conditions and freeze into a static scale-free structure. We suggest a link of optimal channel networks with self-organized critical systems and critical phenomena which exhibit spatial and temporal fractality, the former under generic conditions and the latter on fine tuning.