Thermodynamics and Legendre Duality in Optimal Networks

TL;DR

This paper develops a thermodynamic formalism for optimal transport networks using Legendre duality, linking stability, phase transitions, and cost-based optimization to generalized dissipation and entropy production.

Contribution

It introduces a thermodynamic framework with Legendre duality for analyzing optimal networks, connecting stability, phase transitions, and cost optimization.

Findings

Optimality principles relate to thermodynamic potentials and Legendre duality.

Stability changes are interpreted as phase transitions in non-equilibrium states.

Cost-based optimization reveals branched transport with unstable operating points.

Abstract

Optimality principles in nonequilibrium transport networks are linked to a thermodynamic formalism based on generalized transport potentials endowed with Legendre duality and related contact structure. This allows quantifying the distance from non-equilibrium operating points, analogously to thermodynamic availability as well as to shed light on optimality principles in relation to different imposed constraints. Extremizations of generalized dissipation and entropy production appear as special cases that require power-law resistances and -- for entropy production -- also isothermal conditions. Changes in stability of multiple operating points are interpreted as phase transitions based on non-equilibrium equations of state, while cost-based optimization of transport properties reveals connections to the generalized dissipation in the case of power law costs and linear resistance law, but now with typically unstable operating points which give rise to branched optimal transport.