  An E_0-semigroup acting on B(H) is called pure if the intersection of the
ranges $\alpha_t(B(H))$, $t>0$, is the algebra of scalars. We determine all
pure E_0-semigroups which have a weakly continuous invariant state $\omega$ and
which are minimal in an appropriate sense. In such cases, for every normal
state $\rho$ there is convergence to equilibrium in the trace norm
$\lim_{t\to\infty}\| \rho\circ\alpha_t - \omega \| = 0.$ A normal state
$\omega$ with this property is called an absorbing state. Such E_0-semigroups
mest be cocycle perturbations of CAR/CCR flows, and we develop systematic
methods for constructing those perturbations which have absorbing states with
prescribed finite eigenvalue lists.
