Dynamic Linear Responses and Time Correlation Functions

Abstract

Although seemingly stationary, matter in equilibrium spontaneously fluctuates due to microscopic degrees of freedom thermally excited therein. Even the macroscopic properties, for example, the length of a rod or the polarization of a dielectric fluctuate, although imperceptibly; on a finer time scales the time series of these properties looks stochastic, with the variances reflecting the intrinsic response of the matter to a small external influence, as we studied in Chap. 9. Although apparently random, the time-series signals at different times are correlated at a close look. In this chapter we will find that the time correlation is directly related to the response of the system to a time-dependent perturbation, namely, the fluctuation-dissipation theorem. In particular, how the time correlation decays is same as how the non-equilibrium state relaxes after removal of the perturbation. From the knowledge of the time correlations, a variety of the associated dynamic response functions and transport coefficients can be obtained. In earlier chapters, we learned that these fluctuations become relatively larger for smaller systems as manifested in the Brownian motion. As an example for nanometer-sized systems, an RNA hairpin extended by an optical tweezer (Fig. 17.1) depicts temporal fluctuation (time-series signal) in extension of the RNA depending on the stretching force. A fundamental question is: what can we learn from the signal and its temporal correlation, for a small system in particular? What does a signal pattern imply with regards to the living nature of the biopolymers if it is in active state?