Minimum entropy production, detailed balance and Wasserstein distance for continuous-time Markov processes

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