Extended Kramers-Moyal analysis applied to optical trapping

TL;DR

This paper enhances Kramers-Moyal analysis by addressing finite time effects with error estimates, applying it to optical trapping data to reveal memory effects and validate Markovian models of Brownian motion.

Contribution

It introduces an extended Kramers-Moyal method with error estimation, improving analysis reliability in finite sampling scenarios and applied to optical trapping data.

Findings

Finite time effects cause systematic errors in analysis.

Memory effects extend the Markov-Einstein time scale.

Classical Markov models are valid beyond this time scale.

Abstract

The Kramers-Moyal analysis is a well established approach to analyze stochastic time series from complex systems. If the sampling interval of a measured time series is too low, systematic errors occur in the analysis results. These errors are labeled as finite time effects in the literature. In the present article, we present some new insights about these effects and discuss the limitations of a previously published method to estimate Kramers-Moyal coefficients at the presence of finite time effects. To increase the reliability of this method and to avoid misinterpretations, we extend it by the computation of error estimates for estimated parameters using a Monte Carlo error propagation technique. Finally, the extended method is applied to a data set of an optical trapping experiment yielding estimations of the forces acting on a Brownian particle trapped by optical tweezers. We find an increased Markov-Einstein time scale of the order of the relaxation time of the process which can be traced back to memory effects caused by the interaction of the particle and the fluid. Above the Markov-Einstein time scale, the process can be very well described by the classical overdamped Markov model for Brownian motion.