  We study the non-equilibrium dynamics of a one-dimensional interacting
particle system that is a mixture of the voter model and the exclusion process.
With the process started from a finite perturbation of the ground state
Heaviside configuration consisting of 1's to the left of the origin and 0's
elsewhere, we study the relaxation time $\tau$, that is, the first hitting time
of the ground state configuration (up to translation). We give conditions for
$\tau$ to be finite and for certain moments of $\tau$ to be finite or infinite,
and prove a result that approaches a conjecture of Belitsky et al. (Bernoulli 7
(2001) 119--144). Ours are the first non-existence-of-moments results for
$\tau$ for the mixture model. Moreover, we give almost sure asymptotics for the
evolution of the size of the hybrid (disordered) region. Most of our results
pertain to the discrete-time setting, but several transfer to continuous-time.
As well as the mixture process, some of our results also cover pure exclusion.
We state several significant open problems.
