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

## TL;DR

This paper reviews how coarse-graining affects entropy production estimates in non-equilibrium systems, discusses existing corrections, and introduces a new approach to accurately recover entropy production by explicitly including microscopic variables.

## Contribution

It generalizes the correction for entropy production under different FPE derivations and proposes a novel method to recover true entropy production by incorporating microscopic currents.

## Key findings

- Exact correction for entropy production in FPE derived from Langevin equations.
- New approach to include microscopic variables for accurate entropy estimation.
- Discussion of open problems in Gaussian transition rates and underdamped limits.

## Abstract

Systems operating out of equilibrium exchange energy and matter with the environment, thus producing entropy in their surroundings. Since the entropy production depends on the current flowing throughout the system, its quantification is affected by the level of coarse-graining we adopt. In particular, it has been shown that the description of a system via a Fokker-Planck equation (FPE) lead to an underestimation of the entropy production with respect to the corresponding one in terms of microscopic transition rates. Moreover, such a correction can be derived exactly. Here we review this derivation, generalizing it when different prescriptions to derive the FPE from a Langevin equation are adopted. Then, some open problems about Gaussian transition rates and underdamped limit are discussed. In the second part of the manuscript we present a new approach to dealing with the discrepancy in entropy production due to the coarse graining by introducing enough microscopic variables to correctly estimate the entropy production within the FPE description. We show that any discrete state system can be described by making explicit the contribution of each microscopic current.