Towards transition-based thermodynamics: Stopping criteria and Fluctuation Relations in partially accessible networks

Discrete state space stochastic processes are typically described as successions of states and permanence times. In the markovian case the process is fully described by transition rates between pairs of states and by exponentially distributed residence times. In this case knowledge of the rates of the systems allows estimation of the thermodynamic forces driving the non-equilibrium dynamics, and we can study the statistics of extensive current-like quantities which satisfy fluctuation relations, establishing symmetries for the probability distributions. The description in terms of states, reminiscent of equilibrium thermodynamics, may not to be the best way to describe systems where only a few degrees of freedom are accessible to observation, since it requires coarse-graining procedures that break markovianity, which is only restored in certain limits. In particular, a novel approach based on the occurrence of transitions has proven to successfully generate the discrete-time sequences of few transitions that are accessible to observation. In this thesis I will review the above-mentioned transition-based coarse-graining, to which I contributed, and study the properties of the discrete-time process which generates sequences of visible transitions. The most important result is that the statistics of the currents along the observed edges satisfy a fluctuation relation when observed up to some intrinsic time (paced by the occurrence of a fixed number of visible transitions), thus recovering thermodynamic consistency that is lost in general when observing a few transitions up to the elapsing of a fixed external clock time. It will be also shown that the same coarse-graining, originally derived from first-exit time problems, can be obtained in an alternative way by working directly in the space of visible transitions with a procedure known as stochastic complementation. Finally, a discussion on some attempted research is presented, some of it offering ideas for followups on the work discussed in most of this thesis.