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Multivariate Ornstein-Uhlenbeck processes with mean-field dependent coefficients: application to postural sway
T D Frank1, A Daffertshofer, P J Beek
1Faculty of Human Movement Sciences, Vrije Universiteit, Van der Boechorststraat 9, 1081 BT Amsterdam, The Netherlands.
Summary
This study explores many-particle systems using Ornstein-Uhlenbeck processes, revealing complex behaviors like multiple solutions and bifurcations. Findings describe erratic motion in quiet standing using unique autocorrelation functions.
Area of Science:
- Statistical physics
- Nonlinear dynamics
- Stochastic processes
Background:
- Many-particle systems exhibit complex transient and stationary behaviors.
- Mean-field approximations are crucial for analyzing these systems.
- Ornstein-Uhlenbeck processes are a common tool for modeling stochastic systems.
Purpose of the Study:
- To investigate the transient and stationary dynamics of many-particle systems.
- To analyze systems with nonlinear friction and diffusion coefficients dependent on mean fields.
- To characterize the behavior of autocorrelation functions in such systems.
Main Methods:
- Utilizing multivariate Ornstein-Uhlenbeck processes.
- Deriving mean-field approximations via Fokker-Planck equations.
- Analyzing the properties of stationary solutions and bifurcations.
Main Results:
- Identified the occurrence of multiple stationary solutions and bifurcations.
- Discovered strictly monotonically decreasing steady-state autocorrelation functions.
- Demonstrated faster-than-exponential decay in these autocorrelation functions.
Conclusions:
- The study provides a theoretical framework for understanding complex behaviors in many-particle systems.
- The derived Fokker-Planck equations offer insights into nonlinear mean-field interactions.
- The unique autocorrelation functions successfully model erratic center-of-pressure motion during quiet standing.