  For an overdamped colloidal particle diffusing in a fluid in a controllable,
virtual potential, we show that arbitrarily slow transformations, produced by
smooth deformations of a double-well potential, need not be reversible. The
arbitrarily slow transformations do need to be fast compared to the barrier
crossing time, but that time can be extremely long. We consider two types of
cyclic, isothermal transformations of a double-well potential. Both start and
end in the same equilibrium state, and both use the same basic operations---but
in different order. By measuring the work for finite cycle times and
extrapolating to infinite times, we found that one transformation required no
work, while the other required a finite amount of work, no matter how slowly it
was carried out. The difference traces back to the observation that when time
is reversed, the two protocols have different outcomes, when carried out
arbitrarily slowly. A recently derived formula relating work production to the
relative entropy of forward and backward path probabilities predicts the
observed work average.
