# Spectral content of a single non-Brownian trajectory

## TL;DR

This paper analyzes the spectral content of single trajectories in anomalous diffusion, deriving exact distributions for the PSD and revealing how subdiffusive and superdiffusive behaviors manifest in spectral properties, with implications for experimental data interpretation.

## Contribution

It provides an exact analytical characterization of the PSD for single trajectories of fractional Brownian motion, highlighting differences between subdiffusion and superdiffusion and their impact on spectral analysis.

## Key findings

- PSD scales as 1/f^{2H+1} for subdiffusion
- PSD scales as T^{2H-1}/f^2 for superdiffusion
- PSD exhibits aging and T-dependence for superdiffusion

## Abstract

Time-dependent processes are often analysed 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 data sets 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 (fBm) 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 $S(f,T)\sim A/f^{2H+1}$ with random-amplitude $A$. In sharp contrast, for superdiffusion $(H>1/2)$ $S(f,T)\sim BT^{2H-1}/f^2$ 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.