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

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.

accepted for publication in JSTAT

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• Effective Entropy Production and Thermodynamic Uncertainty Relation of Active Brownian Particles
• Information-driven transitions in projections of underdamped dynamics
• Variance sum rule: proofs and solvable models
• State-space renormalization group theory of nonequilibrium reaction networks: Exact solutions for hypercubic lattices in arbitrary dimensions
• Dissipation in non-steady state regulatory circuits
• Excess and loss of entropy production for different levels of coarse-graining