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 inves
tigated. 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 Schrodinger 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 Kubos 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.
