A gradient structure for reaction-diusion systems and for energy-drift-diu

In recent years the theory of the Wasserstein metric has opened up a new treat- ments of the diusion equations as gradient systems, where the free energy or the entropy take the role of the driving functional and where the space is equipped with the Wasserstein metric. We show on the formal level that this gradient structure can be generalized to reaction-diusion systems with reversible mass-action kinetic. The metric is constructed by using the dual dissipation potential, which is a quadratic functional of all chemical potentials including the mobilities as well as the reaction kinetics. The metric structure is obtained by Legendre transform from the dual dissipation potential. The same ideas extend to systems including electrostatic interactions or a correct energy balance via coupling to the heat equation. We show this by treating the semiconductor equations involving the electron and hole densities, the electrostatic potential, and the temperature. Thus, the models in (AGH02), which stimulated this work, have a gradient structure.