  The quantitative convergence to equilibrium for reaction-diffusion systems
arising from complex balanced chemical reaction networks with mass action
kinetics is studied by using the so-called entropy method. In the first part of
the paper, by deriving explicitly the entropy dissipation, we show that for
complex balanced systems without boundary equilibria, each trajectory converges
exponentially fast to the unique complex balance equilibrium. Moreover, a
constructive proof is proposed to explicitly estimate the rate of convergence
in the special case of a cyclic reaction. In the second part of the paper,
complex balanced systems with boundary equilibria are considered. We
investigate two specific cases featuring two and three chemical substances
respectively. In these cases, the boundary equilibria are shown to be unstable
in some sense, so that exponential convergence to the unique strictly positive
equilibrium can also be proven.
