Tutorial on stochastic systems

TL;DR

This tutorial explains the behavior of different stochastic oscillators, analyzing their correlation functions and spectra, and discusses the mathematical methods used to understand their steady states and dynamics.

Contribution

It provides a comprehensive explanation of stochastic oscillator models, including damped and undamped cases, with insights applicable to various physical systems.

Findings

Damped oscillators reach steady stochastic states with characteristic spectra.

Undamped oscillators' energy increases over time with spectra peaked at low frequencies.

Mathematical methods and physical concepts are explained with practical insights.

Abstract

In this tutorial, three examples of stochastic systems are considered: A strongly-damped oscillator, a weakly-damped oscillator and an undamped oscillator (integrator) driven by noise. The evolution of these systems is characterized by the temporal correlation functions and spectral densities of their displacements, which are determined and discussed. Damped oscillators reach steady stochastic states. Their correlations are decreasing functions of the difference between the sample times and their spectra have peaks near their resonance frequencies. An undamped oscillator never reaches a steady state. Its energy increases with time and its spectrum is sharply peaked at low frequencies. The required mathematical methods and physical concepts are explained on a just-in-time basis, and some theoretical pitfalls are mentioned. The insights one gains from studies of oscillators can be applied to a wide variety of physical systems, such as atom and semiconductor lasers, which will be discussed in a subsequent tutorial.