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A variational formulation for nonequilibrium thermodynamics was recentlyproposed in \cite{GBYo2017a,GBYo2017b} for both discrete and continuum systems.This formulation extends the Hamilton principle of classical mechanics toinclude irreversible processes. In this paper, we show that this variationalformulation yields a constructive and systematic way to derive from a unifiedperspective several bracket formulations for nonequilibrium thermodynamicsproposed earlier in the literature, such as the single generator bracket andthe double generator bracket. In the case of a linear relation between thethermodynamic fluxes and the thermodynamic forces, the metriplectic or GENERICbracket is recovered. We also show how the processes of reduction by symmetrycan be applied to these brackets. In the reduced setting, we also consider thecase in which the coadjoint orbits are preserved and explain the link withdouble bracket dissipation. A similar development has been presented forcontinuum systems in \cite{ElGB2019} and applied to multicomponent fluids.

From variational to bracket formulations in nonequilibrium thermodynamics of simple systems