GENERIC formalism

Remarks

These conditions give $dE/dt = \nabla E^T L \nabla E + \nabla E^T M \nabla S = 0$ and $dS/dt = \nabla S^T L \nabla E + \nabla S^T M \nabla S \ge 0$. Example: a particle of mass $m$ in a potential $V$ with friction coefficient $\gamma > 0$, whose lost energy becomes internal energy $e$ of a medium with entropy $S(e)$ and temperature $T(e) = 1/S'(e) > 0$. Take $x = (q, p, e)$, $E = p^2/(2m) + V(q) + e$, $L$ the canonical skew matrix on $(q, p)$ and zero on $e$, and $M = \gamma T\, w w^T$ with $w = (0, 1, -p/m)$. Then $L \nabla S = 0$ and $w^T \nabla E = p/m - p/m = 0$, so clauses 1 to 4 hold, and the equations are $\dot q = p/m$, $\dot p = -V'(q) - \gamma p/m$, $\dot e = \gamma p^2/m^2$. Non-example: the same system with $M = \gamma T \operatorname{diag}(0, 1, 0)$ fails clause 4, since $M \nabla E = (0, \gamma T p/m, 0) \ne 0$ when $p \ne 0$. In infinite dimensions, as for fluids, the gradients become functional derivatives and $L$, $M$ become operators.