  We study long-time dynamical behaviors of weakly self-consistent
Vlasov-Fokker-Planck equations. We introduce Hessian matrix conditions on
mean-field kernel functions, which characterizes the exponential convergence of
solutions in $L^1$ distances. The matrix condition is derived from the
dissipation of a selected Lyapunov functional, namely auxiliary Fisher
information functional. We verify proposed matrix conditions in examples.
