From variational to bracket formulations in nonequilibrium thermodynamics of simple systems

François Gay-Balmaz*, Hiroaki Yoshimura

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

5 Citations (Scopus)


A variational formulation for nonequilibrium thermodynamics was recently proposed in Gay-Balmaz and Yoshimura (2017a, 2017b) for both discrete and continuum systems. This formulation extends the Hamilton principle of classical mechanics to include irreversible processes. In this paper, we show that this variational formulation yields a constructive and systematic way to derive, from a unified perspective, several bracket formulations for nonequilibrium thermodynamics proposed earlier in the literature, such as the single generator bracket and the double generator bracket. In the case of a linear relation between the thermodynamic fluxes and the thermodynamic forces, the metriplectic or GENERIC bracket is recovered. We also show how the processes of reduction by symmetry can be applied to these brackets. In the reduced setting, we also consider the case in which the coadjoint orbits are preserved and explain the link with double bracket dissipation. A similar development has been presented for continuum systems in Eldred and Gay-Balmaz (2020) and applied to multicomponent fluids.

Original languageEnglish
Article number103812
JournalJournal of Geometry and Physics
Publication statusPublished - 2020 Dec


  • Bracket formulation
  • Metriplectic and GENERIC brackets
  • Nonequilibrium thermodynamics
  • Variational formulation

ASJC Scopus subject areas

  • Mathematical Physics
  • Physics and Astronomy(all)
  • Geometry and Topology


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