Abstract
Systems out of equilibrium exhibit a net production of entropy. We study the dynamics of a stochastic system represented by a Master equation (ME) that can be modeled by a Fokker–Planck equation in a coarse-grained, mesoscopic description. We show that the corresponding coarse-grained entropy production contains information on microscopic currents that are not captured by the Fokker–Planck equation and thus cannot be deduced from it. We study a discrete-state and a continuous-state system, deriving in both the cases an analytical expression for the coarse-graining corrections to the entropy production. This result elucidates the limits in which there is no loss of information in passing from a ME to a Fokker–Planck equation describing the same system. Our results are amenable of experimental verification, which could help to infer some information about the underlying microscopic processes.
Highlights
Any physical system, and its characterizing processes, can be depicted by making use of different levels of description
6 The symmetry condition on f ensures that ázñ = 0. This condition is necessary to have the drift coefficient of order 1. This result agrees with the fact that, in order to consistently describe a microscopic dynamics as a Fokker–Planck equation (FPE), one needs to assume Gaussian transition rates, otherwise inconsistencies in non-equilibrium quantities may arise [42, 43]
We have proven that the same applies when a mesoscopic description of the dynamics is adopted, i.e. when a coarse-graining is performed on the dynamics
Summary
Its characterizing processes, can be depicted by making use of different levels of description. We address the basic question of how equations (2) and (4) are related The former is derived within a framework considering discrete states systems, whereas the latter arises directly in the continuum limit, where many microscopic details are ignored, i.e. after a suitable coarse-graining on the dynamics. Since both formulas refer to the same quantity at two different levels of description, we naively expect that one can be obtained from the other. In general, equation (4) does not fully capture the contribution to the entropy production stemming from the microscopic currents, which do not enter explicitly in the FPE
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