Temperature-based ascendancy derived from a cost or reward function

Ulanowicz in [1] defines ascendancy in terms of departure from maximum-entropy (proportional) flow; however he does not explain what may cause this departure. Here the ascendancy is derived by minimizing a cost function F= ∑ i,j=1ntij log tij+α ∑ i,j=1ntijcijwhere tij is the fraction of the total flow from input i to output j, cij is the corresponding cost of such flow, and α is a parameter (inversely proportional to temperature); the flow being subject to marginal flow constraints ∑ j=1ntij=ai for i=1 to nand ∑ i=1ntij=bi, for j=1 to n.Minimization of such a system is obtained by Evans in [2]. At high temperatures (small α>0) the first (min of negative entropy=max entropy) term dominates, but as α increases (temperature decreases), the cost function dominates, causing a departure from maximum entropy, or ascendancy. Riverbed analogy: At high temperature (fast flows) the flow is mostly uniform (max entropy) across the river bed, but at low temperatures (limited flow), the structure of the riverbed (cost f...