Application of projection operator method to coarse-grained dynamics with transient potential.

Abstract

We show that the coarse-grained dynamics model with the time-dependent and fluctuating potential (transient potential) can be derived from the microscopic Hamiltonian dynamics. The concept of the transient potential was first introduced rather phenomenologically, and its relation to the underlying microscopic dynamics has not been clarified yet. This is in contrast to the generalized Langevin equation, the relation of which to the microscopic dynamics is well-established. In this work, we show that the dynamic equationswith the transient potential can be derived for the coupled oscillator model, without any approximations. It is known that the dynamics of the coupled oscillator model can be exactly described by the generalized Langevin-type equations. This fact implies that the dynamic equationswith the transient potential can be utilized as a coarse-grained dynamics model in a similar way to the generalized Langevin equation. Then we show that the dynamic equationsfor the transient potential can also be formally derived for the microscopic Hamiltonian dynamics, without any approximations. We use the projection operator method for the coarse-grained variables and transient potential. The dynamic equationsfor the coarse-grained positions and momenta are similar to those in the Hamiltonian dynamics, but the interaction potential is replaced by the transient potential. The dynamic equationfor the transient potential is the generalized Langevin equationwith the memory effect. Our result justifies the use of the transient potential to describe the coarse-grained dynamics. We propose several approximations to obtain the simplified dynamics model. We show that, under several approximations, the dynamic equationfor the transient potential reduces to the relatively simple Markovian dynamic equationfor the potential parameters. We also show that with several additional approximations, the approximate dynamics model further reduces to the Markovian Langevin-type equationswith the transient potential.