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Effects from Additional Random Configuration to Linear Response and Modified Fluctuation-Dissipation Relation

In this work, a physical system described by Hamiltonian $\mathbf{H}_ω= \mathbf{H}_0 + \mathbf{V}_ω(\mathbf{x},t)$ consisted of a solvable model $\mathbf{H}$ and external random and time-dependent potential $\mathbf{V}_ω(\mathbf{x},t)$ is investigated. Under the conditions that the average external potential with respect to the configuration $ω$ is constant in time, and, for each configuration, the potential changes smoothly that the evolution of the system follows Schrödinger dynamics, the mean-dynamics can be derived from taking average of the equation with respect to configuration parameter $ω$. 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.