Extended thermodynamics of optimal networks.

We introduce a thermodynamic formalism for nonequilibrium transport networks based on generalized transport potentials (GTPs). When endowed with Legendre duality and the associated contact structure, this framework quantifies the distance from nonequilibrium operating points in analogy with thermodynamic availability, and clarifies how different optimality principles emerge under distinct constraints. Classical extremizations of dissipation and entropy production are recovered as special cases, requiring power-law resistances and, for entropy production, isothermal conditions. Stability analysis shows that changes between multiple operating points can be interpreted as nonequilibrium phase transitions governed by generalized equations of state. Finally, cost-based optimization of transport properties reveals new connections to generalized dissipation, particularly for power-law costs and linear resistance laws, though typically associated with unstable states that give rise to branched optimal transport.