It is the general form of dynamic equation for a system with both reversible and irreversible dynamics (generated by energy and entropy, respectively).
GENERIC formalism is the theory built around the GENERIC equation, which has been proposed in its final form in 1997 by Miroslav Grmela and Hans Christian Öttinger.
[1][2][3] The GENERIC equation is usually written as Here: In addition to the above equation and the properties of its constituents, systems that ought to be properly described by the GENERIC formalism are required to fulfill the degeneracy conditions which express the conservation of entropy under reversible dynamics and of energy under irreversible dynamics, respectively.
(antisymmetry and some others) express that the energy is reversibly conserved, and the condition on
(positive semidefiniteness) express that the entropy is irreversibly non-decreasing.