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An Analog Macroscopic Technique for Studying Molecular Hydrodynamic Processes in Dense Gases and Liquids

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Stochastic dynamics in systems with unidirectional delay coupling: two-state description.

Makoto Kimizuka¹, Toyonori Munakata

¹Department of Applied Mathematics and Physics, Graduate School of Informatics, Kyoto University, Kyoto 606-8501, Japan.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics

|October 2, 2009

Summary

This summary is machine-generated.

We analyzed delayed stochastic dynamics in coupled two-state particle systems. Exact solutions for stationary distribution and time correlation functions reveal a mapping to the Ising spin model, depending on feedback delay.

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Area of Science:

Statistical physics
Nonlinear dynamics
Complex systems

Background:

Stochastic processes are fundamental to many physical phenomena.
Understanding particle interactions with delayed coupling is crucial for complex systems.
Previous work by Tsimring and Pikovsky (2001) studied the N=1 case.

Purpose of the Study:

To investigate the stochastic dynamics of unidirectionally coupled two-state particles with delay.
To derive exact analytical solutions for the stationary distribution and time correlation functions.
To generalize findings from small systems (N=2, N=3) to arbitrary N-particle systems.

Main Methods:

Exact analytical derivations for stationary distribution (p(st)).
Exact analytical derivations for time correlation functions (TCF).
Extrapolation from N=2 and N=3 cases to the general N-particle system.

Main Results:

Exact expressions for p(st) and TCF derived for N=2 and N=3.
General exact expressions for p(st) and TCF for N particles obtained.
Demonstrated that the stationary state maps to the Ising spin model based on delay feedback (positive/negative).

Conclusions:

The study provides exact solutions for delayed stochastic particle systems.
Delay feedback significantly influences the system's stationary state.
A clear connection is established between delayed particle dynamics and the Ising spin model.