  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, of which relation to the microscopic dynamics is
well-established. In this work, we show that the dynamic equations with 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 equations with 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 equations for the transient
potential can be also 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 equations for
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 equation for the transient potential is the
generalized Langevin equation with the memory effect. Our result justifies the
use of the transient potential to describe the coarse-grained dynamics. We
propose several approximations to obtain simplified dynamics model. We show
that, under several approximations, the dynamic equation for the transient
potential reduces to the relatively simple Markovian dynamic equation for the
potential parameters.
