Susceptibility and transport coefficients in a transient state on a one-dimensional lattice I. Extended linear response and diffusion

In this paper (Part I) we extend the linear response analysis to calculate the complex dynamic susceptibility and the complex dynamic mobility/conductivity for a system in a transient state relaxing to equilibrium. This analysis has a meaning in the intermediate time and frequency region; for example, for solvation dynamics and nonlinear relaxation in electrolyte systems. The discussed situation differs from the one considered by the usual linear response theory since our system during its relaxation is still under the action of an external force. It relaxes and simultaneously somehow responds to the applied force. As a test model we assume a single-particle one-dimensional random walk on a lattice in an inhomogeneous periodic potential where a transient state is created by a nonequilibrium initial probability distribution. Using spectral analysis we derived spectral and summed formulas for the above mentioned dynamic quantities. The feature which distinguishes the present result is a force-dependence, since summed formulas depend, in general, on the external force but do not depend on its amplitude. The spectral analog of the dissipation-fluctuation theorem of the first kind was derived. In addition, a time- and frequency-dependent diffusion coefficient was studied. As a striking effect, we found a nonmonotonic frequency and time dependence of transport coefficients. The reason for this effect is a competition between the terms belonging to the different modes and contributing to the spectral analog of the current-current correlation functions with opposite signs, in contrast to the situation in equilibrium. The external force can additionally increase this effect. The dc values are well described by the usual linear response expressions for the thermalized syste. The relation between the dynamic mobility and the frequency-dependent diffusion coefficient is still an open question for the system in a transient state. The systematic numerical studies of the results were performed in Part II by the Exact Enumeration method and by Monte Carlo simulation.