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Summary

Information length, a metric for dynamical systems, preserves the linear geometry of Ornstein-Uhlenbeck processes. It effectively detects changes in coupled systems where other metrics fail, highlighting subsystem evolution.

Keywords:
Fisher informationFokker–Planck equationLangevin equationO-U processinformation lengthmetricsprobability density functionstochastic processes

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Area of Science:

  • Statistical physics
  • Dynamical systems theory
  • Information geometry

Background:

  • Statistical formulations offer key insights into complex dynamical systems.
  • Metrics quantify state changes in time-evolving systems.
  • Relative entropy is a common measure of information change.

Purpose of the Study:

  • Investigate the relaxation problem in single and coupled Ornstein-Uhlenbeck (O-U) processes.
  • Compare information length with entropy-based metrics (relative entropy, Jensen divergence).
  • Evaluate metrics' ability to capture system dynamics and subsystem evolution.

Main Methods:

  • Analysis of single and coupled Ornstein-Uhlenbeck processes.
  • Calculation and comparison of information length against relative entropy and Jensen divergence.
  • Examination of metrics in the long-time limit and for subsystem dynamics.

Main Results:

  • Information length uniquely preserves the linear geometry of the O-U process.
  • In coupled O-U processes, information length detects changes in both components.
  • Other metrics show limited sensitivity in one component of coupled O-U processes.

Conclusions:

  • Information length is a superior metric for analyzing O-U processes, especially coupled ones.
  • Information length is sensitive to the evolution of subsystems within complex dynamical systems.
  • This metric offers a more comprehensive understanding of information flow and system changes.