  The dynamics near the top of a potential barrier is studied in the
temperature region where quantum effects become important. The time evolution
of the density matrix of a system that deviates initially from equilibrium in
the vicinity of the barrier top but is in local equilibrium away from the
barrier top is determined. Explicit results are given for a range of parameters
where the nonequilibrium state is not affected by anharmonicities of the
barrier potential except for the barrier height. In particular, for a system
confined initially to one side of the barrier the relaxation to a
quasi--stationary flux state is determined. The associated rate constant is
evaluated and the relation to other rate formulas is discussed in detail.
