  We consider non-linear evolution equations arising from mean-field limits of
particle systems on discrete spaces. We investigate a notion of curvature
bounds for these dynamics based on convexity of the free energy along
interpolations in a discrete transportation distance related to the gradient
flow structure of the dynamics. This notion extends the one for linear Markov
chain dynamics studied by Erbar and Maas. We show that positive curvature
bounds entail several functional inequalities controlling the convergence to
equilibrium of the dynamics. We establish explicit curvature bounds for several
examples of mean-field limits of various classical models from statistical
mechanics.
