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We derive the equations governing the protocols minimizing the heat released by a
continuous-time Markov jump process on a one-dimensional countable state space during a
transition between assigned initial and final probability distributions in a finite time horizon.
In particular, we identify the hypotheses on the transition rates under which the optimal
control strategy and the probability distribution of the Markov jump problem obey a system
of differential equations of Hamilton-Jacobi-Bellman-type. As the state-space mesh tends
to zero, these equations converge to those satisfied by the diffusion process minimizing the
heat released in the Langevin formulation of the same problem. We also show that in full
analogy with the continuum case, heat minimization is equivalent to entropy production
minimization. Thus, our results may be interpreted as a refined version of the second law of
thermodynamics
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