  With a statistical measure of distance, we derive a classical uncertainty
relation for processes traversing nonequilibrium states both transiently and
irreversibly. The geometric uncertainty associated with dynamical histories
that we define is an upper bound for the entropy production and flow rates, but
it does not necessarily correlate with the shortest distance to equilibrium.
For a model one-bit memory device, we find that expediting the erasure protocol
increases the maximum dissipated heat and geometric uncertainty. A driven
version of Onsager's three-state model shows that a set of dissipative,
high-uncertainty initial conditions, some of which are near equilibrium, scar
the state space.
