Stochastic dynamics of time correlation in complex systems with discrete time

Area of Science:

• Complex Systems Science
• Statistical Physics
• Information Theory

Background:

• Describing random processes in complex systems with discrete time is challenging.
• Existing methods often rely on Markovian assumptions, which may not capture intricate dynamics.
• Quantifying time correlation and memory in stochastic processes is crucial for understanding system behavior.

Purpose of the Study:

• To present a novel theoretical framework for describing discrete-time random processes in complex systems.
• To introduce dynamic information Shannon entropy as a measure of stochastic dynamics and disorder.
• To develop a method for analyzing non-Markovian phenomena and their implications in real-world data.

Main Methods:

• Developed a chain of finite-difference non-Markov equations for time correlation functions (TCFs).
• Introduced dynamic information Shannon entropy S(i)(t) to quantify time correlation and memory disorder.
• Employed Gram-Schmidt orthogonalization to generate an infinite chain of kinetic equations for TCFs and memory functions (MFs).

Main Results:

• Established recurrence relations between TCFs and MFs of different orders.
• Enabled the detection of frequency spectra for the entropy function S(i)(t).
• Demonstrated non-Markovian phenomena in human ECG RR interval dynamics, showing distinct short- and long-range scaling.

Conclusions:

• The developed theory provides a quantitative measure of disorder in time correlation and memory.
• The method offers new opportunities for analyzing stochastic dynamics in discrete random processes.
• The findings suggest potential applications in distinguishing healthy from pathological physiological data based on non-Markovian properties.