Systems out of equilibrium are characterized by a continuous exchange of energy and matter with the environment through different mechanisms, thus producing entropy at the macroscopic level. We study the entropy production of a discrete-state system with random transition rates and the injection of an external probability current. At stationarity, the entropy production is shown to be composed bytwo contributions whose exact distributions are evaluated in the large system size limit and close to equilibrium. The first one is related to a generalized Joule's law, while the second contribution has a Gaussian universal distribution which depends just on two topological parameters. In the second part of the talk we compare the entropy production of a Master Equation system with the one of the corresponding continuum limit, i.e. a Fokker-Planck equation. We demonstrate that the Seifert's formula for the entropy production provides just a lower bound for the exact entropy production. In fact, this formula ought to be corrected keeping into account information about the microscopic transition rates, and this correction survives in the continuum limit. This effect of the coarse-graining has shown to be glaring in the simple example of a n-step random walk. The difference between discrete-state systems and continuous systems in evaluating the entropy production has been addressed also in comparing non-equilibrium steady states, NESS, and stochastic pumping, SP. We show that the entropy production of a system with SP is greater than the one produced by a system in a NESS, generating the same (time-averaged) probability current and exhibiting the same (time-averaged) probability distribution.
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