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This study explores how stochastic forcing with memory shapes shear flows, leading to bimodal probability density functions (PDFs). Information length analysis reveals distinct relaxation and build-up dynamics in shear gradient formation.

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Area of Science:

  • Statistical Mechanics
  • Fluid Dynamics
  • Nonlinear Dynamics

Background:

  • Self-organized shear flows are crucial in various physical systems.
  • Stochastic forcing with finite memory time can induce complex dynamics.
  • Probability Density Functions (PDFs) offer insights into system state evolution.

Purpose of the Study:

  • To investigate the time-evolution of PDFs in a model of self-organized shear flows.
  • To understand how finite memory time in stochastic forcing influences shear flow formation.
  • To analyze the information geometry associated with PDF evolution using information length.

Main Methods:

  • Theoretical analysis of limiting cases.
  • Numerical solutions of the Fokker-Planck equation.
  • Calculation of information length (L) from time-dependent PDFs.

Main Results:

  • Demonstrated the emergence of bimodal PDFs indicating non-zero mean shear flows.
  • Conducted a parameter study of PDFs varying correlation time and forcing amplitude.
  • Identified differences in relaxation and build-up of shear gradients based on information change.

Conclusions:

  • The study elucidates the role of memory in stochastic forcing for shear flow emergence.
  • Information length effectively characterizes the information geometry of PDF dynamics.
  • Total information length (L∞) reveals attractor structures and is history-dependent.