Linear systems with adiabatic fluctuations

TL;DR

This paper develops a theoretical approach to analyze linear systems influenced by slow, weak external fluctuations, deriving effective parameters like diffusion and damping constants, with applications to oscillators and turbulent fluids.

Contribution

It introduces an adiabatic expansion method to derive effective linear equations for systems under slow external fluctuations, providing a new analytical tool.

Findings

Derived a linear differential equation for the average solution under adiabatic fluctuations.

Applied the theory to damped harmonic oscillators and turbulent diffusion.

Obtained renormalized diffusion and damping constants for the studied systems.

Abstract

We consider a dynamical system subjected to weak but adiabatically slow fluctuations of external origin. Based on the ``adiabatic following'' approximation we carry out an expansion in \alpha/|\mu|, where \alpha is the strength of fluctuations and 1/|\mu| refers to the time scale of evolution of the unperturbed system to obtain a linear differential equation for the average solution. The theory is applied to the problems of a damped harmonic oscillator and diffusion in a turbulent fluid. The result is the realization of `renormalized' diffusion constant or damping constant for the respective problems. The applicability of the method has been critically analyzed.