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  • Open Access

Extended thermodynamics of optimal networks

Amilcare Porporato*

Shashank Kumar Anand† and Salvatore Calabrese‡

Luca Ridolfi§

Lamberto Rondoni∥

  • Department of Civil and Environmental Engineering and High Meadows Environmental Institute, Princeton University, Princeton, New Jersey 08540, USA

  • Department of Biological and Agricultural Engineering, Texas A&M University, College Station, Texas, USA

  • Department of Environmental, Land, and Infrastructure Engineering, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Turin, Italy

  • *Contact author: aporpora@princeton.edu
  • †Contact author: skanand@tamu.edu
  • ‡Contact author: salvatore.salabrese@ag.tamu.edu
  • §Contact author: luca.ridolfi@polito.it
  • ∥Contact author: lamberto.rondoni@polito.it
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Phys. Rev. E 114, 014135 – Published 17 July, 2026

DOI: https://doi.org/10.1103/52xd-2wx7

Abstract

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.

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Phys. Rev. E 114, 014135– Published 17 July, 2026
  • Received 7 October 2025
  • Accepted 28 April 2026
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