CHANNEL NETWORKS

This review proceeds from Luna Leopold's and Ronald Shreve's lasting a ccomplishments dealing with the study of random-walk and topologically random channel networks. According to the random perspective, which h as had a profound influence on the interpretation of natural landforms , nature's resiliency in producing recurrent networks and landforms wa s interpreted to be the consequence of chance. In fact, central to mod els of topologically random networks is the assumption of equal likeli hood of any tree-like configuration. However, a general framework of a nalysis exists that argues that all possible network configurations dr aining a fixed area are not necessarily equally likely. Rather, a prob ability P (s) is assigned to a particular spanning tree configuration, say s, which can be generally assumed to obey a Boltzmann distributio n: P(s) proportional to e(-H(s)/T), where T is a parameter and H (s) i s a global property of the network configuration s related to energeti c characters, i.e, its Hamiltonian. One extreme case is the random top ology model where all trees are equally likely, i.e. the limit case fo r T --> infinity. The other extreme case is T --> 0, and this correspo nds to network configurations that tend to minimize their total energy dissipation to improve their likelihood. Networks obtained in this ma nner are termed optimal channel networks (OCNs). Observational evidenc e suggests that the characters of real river networks are reproduced e xtremely well by OCNs. Scaling properties of energy and entropy of OCN s suggest that large network development is likely to effectively occu r at zero temperature(i.e, minimizing its Hamiltonian). We suggest a c orollary of dynamic accessibility of a network configuration and specu late towards a thermodynamics of critical self-organization. We thus c onclude that both chance and necessity are equally important ingredien ts for the dynamic origin of channel networks-and perhaps of the geome try of nature.