Langevin dynamics beyond the weak coupling limit

Many popular results of non-equilibrium statistical mechanics hold only in leading order in a small parameter $\lambda$ which controls the strength of the system-environment coupling. In this approximation the equations for the first two moments $\langle v\rangle$ and $\langle v^2\rangle$ of the Brownian particle's velocity are closed and describe exponential relaxation to thermal equilibrium. To higher orders in $\lambda$ these equations are not closed but coupled to higher moments $\langle v^n\rangle$. This may result in much richer dynamics (both transient and stationary) and non-trivial ergodic properties. Generalized fluctuation-dissipation relations are derived microscopically and shown to ensure convergence to thermal equilibrium to any order in $\lambda$. One exception is the regime of superlinear diffusion, characterized by zero integral friction (vanishing integral of the memory kernel), when the generalized Langevin equation may have non-ergodic solutions that do not relax to equilibrium values. Also, for specific memory kernels the equation may have non-dissipative (non-stationary) solutions even if the integral friction is finite and diffusion is normal.