Spectral Content of a Single Non-Brownian Trajectory

Time-dependent processes are often analyzed using the power spectral density (PSD) calculated by taking an appropriate Fourier transform of individual trajectories and finding the associated ensemble average. Frequently, the available experimental datasets are too small for such ensemble averages, and hence, it is of a great conceptual and practical importance to understand to which extent relevant information can be gained from S(f,T), the PSD of a single trajectory. Here we focus on the behavior of this random, realization-dependent variable parametrized by frequency f and observation time T, for a broad family of anomalous diffusions - fractional Brownian motion with Hurst index H - and derive exactly its probability density function. We show that S(f,T) is proportional - up to a random numerical factor whose universal distribution we determine - to the ensemble-averaged PSD. For subdiffusion (H<1/2), we find that S(f,T)∼A/f2H+1 with random amplitude A. In sharp contrast, for superdiffusion (H>1/2) S(f,T)∼BT2H-1/f2 with random amplitude B. Remarkably, for H>1/2 the PSD exhibits the same frequency dependence as Brownian motion, a deceptive property that may lead to false conclusions when interpreting experimental data. Notably, for H>1/2 the PSD is ageing and is dependent on T. Our predictions for both sub- and superdiffusion are confirmed by experiments in live cells and in agarose hydrogels and by extensive simulations.