This review proceeds from Luna Leopold's and Ronald Shreve's lasting accomplishments dealing with the study of random-walk and topologically random channel networks. According to the random perspective, which has had a profound influence on the interpretation of natural landforms, nature's resiliency in producing recurrent networks and landforms was interpreted to be the consequence of chance. In fact, central to models of topologically random networks is the assumption of equal likelihood of any tree-like configuration. However, a general framework of analysis exists that argues that all possible network configurations draining a fixed area are not necessarily equally likely. Rather, a probability P (s) is assigned to a particular spanning tree configuration, say s, which can be generally assumed to obey a Boltzmann distribution: P(s) proportional to e(-H(s)/T), where T is a parameter and H (s) is a global property of the network configuration s related to energetic characters, i.e, its Hamiltonian. One extreme case is the random topology model where all trees are equally likely, i.e. the limit case for T --> infinity. The other extreme case is T --> 0, and this corresponds to network configurations that tend to minimize their total energy dissipation to improve their likelihood. Networks obtained in this manner are termed optimal channel networks (OCNs). Observational evidence suggests that the characters of real river networks are reproduced extremely well by OCNs. Scaling properties of energy and entropy of OCNs suggest that large network development is likely to effectively occur at zero temperature(i.e, minimizing its Hamiltonian). We suggest a corollary of dynamic accessibility of a network configuration and speculate towards a thermodynamics of critical self-organization. We thus conclude that both chance and necessity are equally important ingredients for the dynamic origin of channel networks-and perhaps of the geometry of nature.
Record created on 2009-10-07, modified on 2016-08-08