Deceleration of one-dimensional mixing by discontinuous mappings

Area of Science:

• Computational physics
• Dynamical systems theory
• Statistical mechanics

Background:

• Understanding mixing processes in dynamical systems is crucial for various scientific fields.
• Previous studies often focused on individual components like stretching or diffusion separately.

Purpose of the Study:

• To computationally investigate the combined effects of stretching, permutation, and diffusion on a one-dimensional map.
• To analyze the resulting eigenmode decay rates and their dependence on system parameters.

Main Methods:

• Developed a computational model for a one-dimensional map incorporating stretching, cell permutations, and diffusion.
• Analyzed the long-time behavior of the system's eigenmodes.
• Examined the relationship between diffusion coefficient, permutation strategy, and decay rates.

Main Results:

• The combined dynamics lead to eigenmodes exhibiting exponential decay.
• Decay rates depend non-monotonically on the diffusion coefficient and the chosen permutation.
• Global mixing rates approximate eigenmode decay for low diffusivity but fail to bound them for higher values.

Conclusions:

• The interplay of stretching, permutation, and diffusion creates complex mixing behaviors.
• An increasing diffusion rate can counterintuitively decelerate the asymptotic mixing rate.
• Results have implications for understanding and predicting mixing in finite-time scenarios.