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Systems operating out of equilibrium exchange energy and matter with theenvironment, thus producing entropy in their surroundings. Since the entropyproduction depends on the current flowing throughout the system, itsquantification is affected by the level of coarse-graining we adopt. Inparticular, it has been shown that the description of a system via aFokker-Planck equation (FPE) lead to an underestimation of the entropyproduction with respect to the corresponding one in terms of microscopictransition rates. Moreover, such a correction can be derived exactly. Here wereview this derivation, generalizing it when different prescriptions to derivethe FPE from a Langevin equation are adopted. Then, some open problems aboutGaussian transition rates and underdamped limit are discussed. In the secondpart of the manuscript we present a new approach to dealing with thediscrepancy in entropy production due to the coarse graining by introducingenough microscopic variables to correctly estimate the entropy productionwithin the FPE description. We show that any discrete state system can bedescribed by making explicit the contribution of each microscopic current.

Entropy production in Master Equations and Fokker-Planck Equations: facing the coarse-graining and recovering the information loss