  We investigate the problem of minimizing the entropy production for a
physical process that can be described in terms of a Markov jump dynamics. We
show that, without any further constraints, a given time-evolution may be
realized at arbitrarily small entropy production, yet at the expense of
diverging activity. For a fixed activity, we find that the dynamics that
minimizes the entropy production is given in terms of conservative forces. The
value of the minimum entropy production is expressed in terms of the
graph-distance based Wasserstein distance between the initial and final
configuration. This yields a new kind of speed limit relating dissipation, the
average number of transitions and the Wasserstein distance. It also allows us
to formulate the optimal transport problem on a graph in term of a
continuous-time interpolating dynamics, in complete analogy to the continuous
space setting. We demonstrate our findings for simple state networks, a
time-dependent pump and for spin flips in the Ising model.

Statistical Mechanics