  We study the convergence analysis for general degenerate and non-reversible
stochastic differential equations (SDEs). We apply the Lyapunov method to
analyze the Fokker-Planck equation, in which the Lyapunov functional is chosen
as a weighted relative Fisher information functional. We derive a structure
condition and formulate the Lyapunov constant explicitly. We prove the
exponential convergence result for the probability density function towards its
invariant distribution in the $L_1$ distance. Two examples are presented:
underdamped Langevin dynamics with variable diffusion matrices and three
oscillator chain models with nearest-neighbor couplings.
