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We study the long-time behaviour of both the classical second-order Langevindynamics and the nonlinear second-order Langevin dynamics of McKean-Vlasovtype. By a coupling approach, we establish global contraction in an $L^1$Wasserstein distance with an explicit dimension-free rate for pairwise weakinteractions. For external forces corresponding to a $\kappa$-strongly convexpotential, a contraction rate of order $\mathcal{O}(\sqrt{\kappa})$ is obtainedin certain cases. But the contraction result is not restricted to these forces.It rather includes multi-well potentials and non-gradient-type external forcesas well as non-gradient-type repulsive and attractive interaction forces. Theproof is based on a novel distance function which combines two contractionresults for large and small distances and uses a coupling approach adjusted tothe distance. By applying a componentwise adaptation of the coupling we provideuniform in time propagation of chaos bounds for the corresponding mean-fieldparticle system.

Global contractivity for Langevin dynamics with distribution-dependent forces and uniform in time propagation of chaos