17 - Coarse-Graining

Summary

For many systems, the full information of an underlying Markovian decription is not accessible due to limited spatial or temporal resolution. We first show that such an often inevitable coarse-graining implies that, rather than the full entropy production, only a lower bound can be retrieved from coarse-grained data. As a technical tool, it is derived that the Kullback–Leibler divergence decreases under coarse-graining. For a discrete time-series obtained from an underlying time-continuous Markov dynamics, it is shown how the analysis of n-tuples leads to a better estimate with increasing length of the tuples. Finally, state-lumping as one strategy for coarse-graining an underlying Markov model is shown explicitly to yield a lower bound for the entropy production. However, in general, it does not yield a consistent interpretation of the first law along coarse-grained trajectories as exemplified with a simple model.