On gradient flows and reaction-diffusion systems

Gradient system can be understood as mathematical realizations of the Onsager principle in thermodynamics, which states that the flux is given by a positive definite operator, called Onsager operator, times the thermodynamic driving force. We show that reaction-diffusion systems satisfying a detailed-balance condition can be formulated as a gradient system for the relative entropy and an Onsager operator (inverse of the Riemannian tensor), which is given as a sum of a diffusion part (Wasserstein metric) and a reaction part. This approach allows us to connect gradient-flow formulations of discrete many-particle systems with their continuous limits. Moreover, well-established concepts for scalar equations, as geodesic lambda-convexity or exponential decay into equilibrium, can be generalized to these more general systems.