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The generation of electrical current in semiconductors is fundamentally driven by two mechanisms: drift and diffusion. These processes are essential for the functionality and performance of semiconductor-based devices.
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Ergodic Measure and Potential Control of Anomalous Diffusion.

Bao Wen1,2, Ming-Gen Li3, Jian Liu4

  • 1Institutes of Science and Development, Chinese Academy of Sciences, Beijing 100190, China.

Entropy (Basel, Switzerland)
|July 29, 2023
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This study explores how finite measurement times affect the ergodic hypothesis in anomalous diffusion. We found that non-ergodicity mimics medium sparseness and logarithmic potentials control diffusion behavior.

Keywords:
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Area of Science:

  • Statistical Mechanics
  • Complex Systems

Background:

  • The ergodic hypothesis, equating long-time and ensemble averages, is crucial for understanding anomalous diffusion, irreversibility, and entropy.
  • Finite measurement times pose challenges to establishing ergodicity, as time averages fluctuate and their convergence rate requires investigation.

Purpose of the Study:

  • To investigate the time-dependent fluctuation width of time averages for velocity and kinetic energy in a force-free particle system.
  • To determine the time scale for a system transitioning from a stationary state to an effective ergodic state.
  • To analyze the influence of a logarithmic spatial potential on diffusion processes.

Main Methods:

  • Utilizing the generalized Langevin equation to model particle dynamics.
  • Analyzing the stationary velocity autocorrelation function.
  • Introducing a logarithmic spatial potential to modulate diffusion.

Main Results:

  • The study quantifies the time-dependent fluctuation width for velocity and kinetic energy time averages.
  • An estimation of the shortest time scale for ergodic establishment is provided.
  • Logarithmic potential was shown to modulate free ballistic diffusion and control diffusion processes, realizing power-law regimes.

Conclusions:

  • Non-ergodicity in diffusion processes can effectively mimic the sparseness of a medium.
  • Logarithmic potentials play a unique role in modulating diffusion behavior and achieving specific power-law regimes.