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We show that a mesoscopic coarse-grained dynamics model which incorporatesthe transient potential can be formally derived from an underlying microscopicdynamics model. As a microscopic dynamics model, we employ the overdampedLangevin equation. By utilizing the path probability and the Onsager-Machluptype action, we calculate the path probability for the coarse-grainedmesoscopic degrees of freedom. The action for the mesoscopic degrees of freedomcan be simplified by incorporating the transient potential. Then the dynamicequation for the mesoscopic degrees of freedom can be simply described by theLangevin equation with the transient potential (LETP). As a simple andanalytically tractable approximation, we introduce additional degrees offreedom which express the state of the transient potential. Then weapproximately express the dynamics of the system as the the combination of theLETP and the dynamics model for the transient potential. The resulting dynamicsmodel has the same dynamical structure as the responsive particle dynamics(RaPiD) type models [W. J. Briels, Soft Matter 5, 4401 (2009)] and themulti-chain slip-spring type models [T. Uneyama and Y. Masubuchi, J. Chem.Phys. 137, 154902 (2012)]. As a demonstration, we apply our coarse-grainingmethod with the LETP to a single particle dynamics in a supercooled liquid, andcompare the results of the LETP with the molecular dynamics simulations andother coarse-graining models.

Coarse-Graining of Microscopic Dynamics into Mesoscopic Transient Potential Model