The general equation of nonequilibrium reversible-irreversible coupling (GENERIC) is studied in light of time-reversal transformation. It is shown that Onsager-Casimir reciprocal relations are implied by GENERIC in the near-equilibrium regime. A general structure which gives the reciprocal relations but which is valid also far from equilibrium is identified, and Onsager-Casimir reciprocal relations are generalized to far-from-equilibrium regime in this sense. Moreover, reversibility and irreversibility are carefully discussed and the results are illustrated in Hamiltonian dynamics, classical hydrodynamics, classical irreversible thermodynamics, the quantum master equation, and the Boltzmann equation.

Originally derived by Walther Nernst more than a century ago, the Nernst equation for the open-circuit voltage is a cornerstone in the analysis of electrochemical systems. Unfortunately, the assumptions behind its derivation are often overlooked in the literature, leading to incorrect forms of the equation when applied to complex systems (for example, those with ion-exchange membranes or involving mixed potentials). Such flaws can be avoided by applying a correct thermodynamic derivation independently of the form in which the electrochemical reactions are written. The proper derivation of the Nernst equation becomes important, for instance, in modeling of vanadium redox flow batteries or zinc-air batteries. The rigorous path towards the Nernst equation derivation starts in non-equilibrium thermodynamics.

Reduction of a mesoscopic level to a level with fewer details is made by the time evolution during which the entropy increases. An extension of a mesoscopic level is a construction of a level with more details. In particular, we discuss extensions in which extra state variables are found in the vector fields appearing on the level that we want to extend. Reductions, extensions, and compatibility relations among them are formulated first in an abstract setting and then illustrated in specific mesoscopic theories.