Entropy production given constraints on the energy functions

We consider the problem of driving a finite-state classical system from some initial distribution p to some final distribution p′ with vanishing entropy production (EP), under the constraint that the driving protocols can only use some limited set of energy functions E. Assuming no other constraints on the driving protocol, we derive a simple condition that guarantees that such a transformation can be carried out, which is stated in terms of the smallest probabilities in {p,p′} and a graph-theoretic property defined in terms of E. Our results imply that a surprisingly small amount of control over the energy function is sufficient (in particular, any transformation p → p′ can be carried out as soon as one can control some one-dimensional parameter of the energy function, e.g., the energy of a single state). We also derive a lower bound on the EP under more general constraints on the transition rates, which is formulated in terms of a convex optimization problem.