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Single-Molecule Tracking Microscopy - A Tool for Determining the Diffusive States of Cytosolic Molecules

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Testing of Multifractional Brownian Motion.

Michał Balcerek¹, Krzysztof Burnecki¹

¹Faculty of Pure and Applied Mathematics, Hugo Steinhaus Center, Wroclaw University of Science and Technology, Wyspianskiego 27, 50-370 Wroclaw, Poland.

Entropy (Basel, Switzerland)

|December 16, 2020

Summary

This study introduces a statistical test for multifractional Brownian motion (MFBM), a more flexible model for systems with changing dynamics. The test, validated by simulations, aids in analyzing anomalous diffusion.

Keywords:

Monte Carlo simulations autocovariance function multifractional Brownian motion power of the statistical test

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Area of Science:

Physics
Statistical Mechanics
Stochastic Processes

Background:

Fractional Brownian motion (FBM) models systems with self-similarity, characterized by the Hurst index (H).
Standard FBM's stationary increments limit its use for modeling systems with time-varying dynamics, common in experiments like single-particle tracking.
Multifractional Brownian motion (MFBM) generalizes FBM by allowing the Hurst index to vary with time, offering greater flexibility.

Purpose of the Study:

To introduce and rigorously test a statistical method for analyzing multifractional Brownian motion (MFBM).
To evaluate the performance of this test using Monte Carlo simulations for various time-dependent Hurst exponent functions (linear, logistic, periodic).
To analyze the mean-squared displacement (MSD) in MFBM and investigate ergodicity breaking.

Main Methods:

Development of a statistical test for MFBM based on its covariance function.
Utilization of Monte Carlo simulations to assess the power of the test against different MFBM alternatives.
Analysis of mean-squared displacement (MSD) by comparing ensemble averages.

Main Results:

The study presents a rigorous statistical test applicable to MFBM.
Monte Carlo simulations demonstrate the test's power for linear, logistic, and periodic Hurst exponent functions.
Analysis of MSD reveals insights into ergodicity breaking in different MFBM scenarios.

Conclusions:

The developed statistical test provides a valuable tool for analyzing MFBM.
The findings are expected to aid in the study of anomalous diffusion phenomena observed in various scientific experiments.
Understanding MFBM dynamics is crucial for accurately modeling complex systems with time-varying behaviors.