Harmonic Oscillator in Heat Bath: Exact simulation of time-lapse-recorded data, exact analytical benchmark statistics

TL;DR

This paper presents an exact simulation algorithm for the damped harmonic oscillator in a heat bath, providing analytical benchmark statistics for experimental data and exploring implications for various physical systems and sampling effects.

Contribution

It introduces an exact simulation method for the stochastic damped harmonic oscillator and derives analytical benchmark statistics applicable to experimental and theoretical systems.

Findings

Exact simulation algorithm for arbitrary time steps.

Analytical correlation functions and power spectra for experimental data.

Bridges different models of harmonic oscillators and discusses sampling effects.

Abstract

The stochastic dynamics of the damped harmonic oscillator in a heat bath is simulated with an algorithm that is exact for time steps of arbitrary size. Exact analytical results are given for correlation functions and power spectra in the form they acquire when computed from experimental time-lapse recordings. Three applications are discussed: (i) Effects of finite sampling-rate and -time, described exactly here, are similar for other stochastic dynamical systems-e.g. motile micro-organisms and their time-lapse recorded trajectories. (ii) The same statistics is satisfied by any experimental system to the extent it is interpreted as a damped harmonic oscillator at finite temperature-such as an AFM cantilever. (iii) Three other models of fundamental interest are limiting cases of the damped harmonic oscillator at finite temperature; it consequently bridges their differences and describes effects of finite sampling rate and sampling time for these models as well. Finally, we give a brief discussion of nondimensionalization.