Coarse graining and hidden entropy production*

There are many components of living systems that seem well described as two state systems, including allosteric molecules such as hemoglobin, chemotaxis receptors, and the bursting behavior of gene transcription. In two state systems information about the arrow of time and energy dissipation is obscured since thermodynamic cycles and fluxes can’t be observed.  Nonetheless, these systems could be out of equilibrium.  Where is the associated entropy production “hiding”?  We study the simplest example, a three-state system that we approximate as a two-state system; this approximation becomes exact when there is a separation of time scales.  In this limit, the true entropy production rate approaches a finite value.  If the relevant rates are finite but very large, then the approximate two state system is not quite Markovian, and this is visible in a non-exponential distribution of waiting times between transitions. We quantify entropy production by the Kullback-Leibler divergence between the distributions of forward and backward trajectories, and show how to estimate this divergence directly from simulations.  We combine these tools to understand whether we can uncover the arrow of time in effectively two state systems, provided we measure with sufficient time resolution.