  In this work, a physical system described by Hamiltonian $\mathbf{H}_\omega =
\mathbf{H}_0 + \mathbf{V}_\omega(\mathbf{x},t)$ consisted of a solvable model
$\mathbf{H}$ and external random and time-dependent potential
$\mathbf{V}_\omega(\mathbf{x},t)$ is investigated. Under the conditions that
the average external potential with respect to the configuration $\omega$ is
constant in time, and, for each configuration, the potential changes smoothly
that the evolution of the system follows Schr\"odinger dynamics, the
mean-dynamics can be derived from taking average of the equation with respect
to configuration parameter $\omega$. It provides extra contributions from the
deviations of the Hamiltonian and evolved state along the time to the
Heisenberg and Liouville-von Neumann equations. Consequently, the Kubo's
formula and the fluctuation-dissipation relation obtained from the construction
is modified in the sense that the contribution from the information of
randomness and memory effect from time-dependence are present.
