Linear response and modified fluctuation-dissipation relation in random potential

In this work, a physical system described by the Hamiltonian ${\mathbf{H}}_{\omega }={\mathbf{H}}_0+{\mathbf{V}}_{\omega }\left(t\right)$ consisting of a solvable model ${\mathbf{H}}_0$ and external random and time-dependent potential ${\mathbf{V}}_{\omega }\left(t\right)$ is investigated. Under the conditions in which, for each realization, the potential changes smoothly so that the evolution of the system follows the Schrödinger dynamics, and that the average external potential with respect to all realizations is constant in time, an adjusted equilibrium state can be defined as a reference state and the mean dynamics can be derived from taking the average of the equation with respect to the configuration parameter $\omega$. It provides extra contributions from the deviations of the Hamiltonian and evolves the state along the time by the Heisenberg and Liouville–von Neumann equations. Consequently, the Kubo formula and the fluctuation-dissipation relation (FDR) are modified in the sense that the contribution from the information of randomness and memory effects from the time dependence is also present. The modified Kubo formula now has a contribution from two terms. The first term is an antisymmetric cross correlation between two observables measured by a probe as expected, and the latter term is an accumulation of the propagation of the effects from the randomness. When the considered system is in the adjusted equilibrium state at the time the measurement probe interacts, the latter contribution vanishes, and the standard FDR is recovered.

Figure 1

Schematic illustration of the system's time evolution. The system is in the adjusted equilibrium ${\sigma }_{\beta }^{\prime }$ at time $t=0$, and it begins to contact with the bath at time $t=t^{\prime }.$ During time $0\le s\le t^{\prime }$, the system experiences random effects from the nonhomogeneous term that accumulate until time $t^{\prime }$.