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The gain of free energy upon unmixing is determined via application of Markovstate modeling (MSM), using an Ising model with a fixed number of up- anddown-spins. MSM yields reasonable estimates of the free energies. However, acloser look reveals significant differences which point to residualnon-Markovian effects. These non-Markovian effects are rather unexpected sincethe typical criteria to study the quality of Markovianity indicate completeMarkovian behavior. We identify the sparse connectivity between differentMarkov states as a likely reason for the observed bias. By studying a simplefive state model system we can analytically elucidate different sources of thebias and thus explain the different deviations that were observed for the Isingmodel. Based on this insight we can modify the determination of the countmatrix in the MSM approach. In this way, the estimation of the free energy issignificantly improved.

Exploring the free energy gain of phase separation via Markov State Modeling