# Conditional $1/f^{\alpha}$ noise: from single molecules to macroscopic measurements

## TL;DR

This paper reveals that $1/f^{\alpha}$ noise exhibits fundamentally different behaviors at the single molecule level compared to macroscopic measurements, highlighting the nonstationary nature of the noise even when it appears stationary in large samples.

## Contribution

It introduces the concept of conditional spectra in single particle measurements and demonstrates the nonstationary nature of $1/f^{\alpha}$ noise through various models.

## Key findings

- Single molecule measurements show nonstationary $1/f^{\\alpha}$ noise.
- Macroscopic measurements appear stationary due to increasing units.
- $1/f^{\\alpha}$ spectrum is inherently nonstationary despite macroscopic obscuration.

## Abstract

We demonstrate that the measurement of $1/f^{\alpha}$ noise at the single molecule or nano-object limit is remarkably distinct from the macroscopic measurement over a large sample. The single particle measurements yield a conditional time-dependent spectrum. However, the number of units fluctuating on the time scale of the experiment is increasing in such a way that the macroscopic measurements appear perfectly stationary. The single particle power spectrum is a conditional spectrum, in the sense that we must make a distinction between idler and non-idler units on the time scale of the experiment. We demonstrate our results based on stochastic and deterministic models, in particular the well known superposition of Lorentzians approach, the blinking quantum dot model, and deterministic dynamics generated by non-linear mapping. Our results show that the $1/f^\alpha$ spectrum is inherently nonstationary even if the macroscopic measurement completely obscures the underlying time dependence of the phenomena.