  The no-pumping theorem states that seemingly natural driving cycles of
stochastic machines fail to generate directed motion. Initially derived for
single particle systems, the no-pumping theorem was recently extended to
many-particle systems with zero-range interactions. Interestingly, it is known
that the theorem is violated by systems with exclusion interactions. These two
paradigmatic interactions differ by two qualitative aspects: the range of
interactions, and the dependence of branching fractions on the state of the
system. In this work two different models are studied in order to identify the
qualitative property of the interaction that leads to breakdown of no-pumping.
A model with finite-range interaction is shown analytically to satisfy
no-pumping. In contrast, a model in which the interaction affects the
probabilities of reaching different sites, given that a particle is making a
transition, is shown numerically to violate the no-pumping theorem. The results
suggest that systems with interactions that lead to state-dependent branching
fractions do not satisfy the no-pumping theorem.
