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To maximize velocity correlation in persistent random walks, the optimal number of direction changes scales with the square root of time. This finding reveals an invariant relationship between reversals and persistence length across different durations.

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Area of Science:

  • Physics
  • Statistical Mechanics
  • Complex Systems

Background:

  • Persistent random walks model systems with memory.
  • Understanding velocity fluctuations is key in diffusion processes.

Purpose of the Study:

  • Determine the optimal frequency of direction changes for a random walker.
  • Maximize velocity correlation in a 1D persistent random walk over a finite time.

Main Methods:

  • Analysis of correlation and mutual information in velocity trajectories.
  • Varying persistence level and observation time as parameters.
  • Mathematical modeling of the persistent random walk.

Main Results:

  • Optimal persistence level found.
  • Average direction reversals scale as the square root of observation time.
  • Identified a square-root scaling law for optimal diffusion.

Conclusions:

  • The optimal number of direction reversals is proportional to the square root of observation time.
  • This scaling law ensures invariance in the growth rate of reversals versus persistence length.