Exploring the free energy gain of phase separation via Markov state modeling.

The gain of free energy upon unmixing is determined via application of Markov state modeling (MSM), using an Ising model with a fixed number of up- and down-spins. MSM yields reasonable estimates of the free energies. However, a closer look reveals significant differences that point to residual non-Markovian effects. These non-Markovian effects are rather unexpected since the typical criteria to study the quality of Markovianity indicate complete Markovian behavior. We identify the sparse connectivity between different Markov states as a likely reason for the observed bias. By studying a simple five state model system, we can analytically elucidate different sources of the bias and thus explain the different deviations that were observed for the Ising model. Based on this insight, we can modify the determination of the count matrix in the MSM approach. In this way, the estimation of the free energy is significantly improved.