Frequency-frequency correlations of single-trajectory spectral densities of Gaussian processes

Abstract

We investigate the stochastic behavior of the single-trajectory spectral density S(ω, T ) of several Gaussian stochastic processes, i.e., Brownian motion, the Ornstein-Uhlenbeck process, the Brownian gyrator model and fractional Brownian motion, as a function of the frequency ω and the observation time T . We evaluate in particular the variance and the frequency-frequency correlation of S(ω, T ) for different values of ω. We show that these properties exhibit different behaviors for different physical cases and can therefore be used as a sensitive probe discriminating between different kinds of random motion. These results may prove quite useful in the analysis of experimental data.

Spectral fingerprints of nonequilibrium dynamics: The case of a Brownian gyrator

“…As shown in Ref. [74], variables corresponding to different frequencies are independent. Therefore, the form of the ensemble-averaged power spectrum density can in principle be inferred from a single, albeit long, realization of V 1 , without any reference to ensemble averaging.…”