A general approach for analyzing baseline power spectral densities: Zwanzig-Mori projection operators and the generalized Langevin equation

TL;DR

This paper develops a theoretical framework using Zwanzig-Mori projection operators and the generalized Langevin equation to analyze baseline power spectral densities, revealing how different bath models influence observed PSD behaviors like 1/f divergence.

Contribution

It introduces a rigorous approach to connect system and bath PSDs, providing insights into the origins of 1/f behavior without relying on self-organized criticality.

Findings

Debye model leads to 1/f divergence at low frequencies

Different bath models produce various power law behaviors

Analyzing baseline PSDs can reveal bath characteristics

Abstract

There continues to be widespread interest in 1/f^(alpha) behavior in baseline power spectral densities (PSD's) but its origins remain controversial. Zwanzig-Mori projection operators provide a rigorous, common starting place for building a theory of PSD's from the bottom up. In this approach, one separates out explicit "system" degrees of freedom (which are experimentally monitored) from all other implicit or "bath" degrees of freedom, and then one "projects" or integrates out all the implicit degrees of freedom. The result is the generalized Langevin equation. Within this formalism, the system PSD has a simple relation to the bath PSD. We explore how several models of the bath PSD affect the system PSD. We suggest that analyzing the baseline can yield valuable information on the bath. The Debye model of acoustic bath oscillations in particular gives rise to a low frequency 1/f divergence. Other power law behaviors are possible with other models. None of these models require self-organized criticality.