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Relaxation functions of the Ornstein-Uhlenbeck process with fluctuating diffusivity.

Takashi Uneyama1, Tomoshige Miyaguchi2, Takuma Akimoto3

  • 1Center for Computational Science, Graduate School of Engineering, Nagoya University, Furo-cho, Chikusa, Nagoya 464-8603, Japan.

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We investigated the Ornstein-Uhlenbeck process with fluctuating diffusivity (FD). Our findings reveal unique relaxation behaviors distinct from conventional models, offering new insights into complex dynamics.

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Area of Science:

  • Statistical Physics
  • Complex Systems Dynamics

Background:

  • The Ornstein-Uhlenbeck process models particle dynamics with a restoring force.
  • Fluctuating diffusivity (FD) is crucial for modeling heterogeneous environments or internal degrees of freedom.

Purpose of the Study:

  • To analyze the relaxation behavior of the Ornstein-Uhlenbeck process under fluctuating diffusivity.
  • To develop a general theoretical framework for systems with FD.

Main Methods:

  • Utilized functional integral expressions and the transfer matrix method.
  • Derived relaxation functions based on eigenvalues and eigenfunctions of the transfer matrix.

Main Results:

  • Developed a general theory for relaxation functions in FD processes.
  • Obtained analytic expressions for relaxation functions in two-state and OU-type FD models.
  • Identified distinct relaxation behaviors compared to generalized Langevin equations.

Conclusions:

  • The Ornstein-Uhlenbeck process with FD exhibits unique relaxation dynamics.
  • The developed theory provides a framework for analyzing complex relaxational systems.
  • This work advances the understanding of dynamics in heterogeneous or internally complex systems.