Entropy production at all scales

Spatially homogeneous systems are characterized by the simultaneous presence of a wide range of time scales. When the dynamics of such reactive systems develop very-slow and very-fast time scales separated by a range of active time scales, with large gaps in the fast/active and slow/active time scales, then it is possible to achieve multi-scale adaptive model reduction along-with the integration of the governing ordinary differential equations using the G-Scheme framework. The G-Scheme assumes that the dynamics is decomposed into active, slow, fast, and when applicable, invariant subspaces. We derive the expressions that express the direct link between time scales and entropy production by resorting to the estimates provided by the G-Scheme. With reference to a constant volume, adiabatic batch reactor, we compute the contribution to entropy production by the four subspaces. The numerical experiments show that, as indicated by the theoretical derivation, the contribution to entropy production of the fast subspace is of the same magnitude of the error threshold chosen for the numerical integration, and that the contribution of the slow subspace is generally much smaller than that of the active subspace.