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Simple models for strictly non-ergodic stochastic processes of macroscopic systems.

G George1, L Klochko1, A N Semenov1

  • 1Institut Charles Sadron, Université de Strasbourg & CNRS, 23 rue du Loess, 67034, Strasbourg Cedex, France.

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This study explores non-ergodic stochastic processes using simple models. Researchers found the non-ergodicity parameter stabilizes over time and linked these models to amorphous glass behavior.

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Area of Science:

  • Statistical Physics
  • Condensed Matter Physics

Background:

  • Stochastic processes are fundamental in modeling complex systems.
  • Non-ergodicity presents unique challenges in statistical analysis.
  • Understanding empirical variance is crucial for time series analysis.

Purpose of the Study:

  • To investigate simple models for strictly non-ergodic stochastic processes.
  • To analyze the expectation value and standard deviation of empirical variance.
  • To explore the volume dependence of the non-ergodicity parameter.

Main Methods:

  • Focusing on expectation value (v) and standard deviation (σ) of empirical variance (χ²).
  • Averaging over a fluctuating field (ξ) characterized by a quenched spatially correlated Gaussian field (η).
  • Investigating the volume dependence for different spatial correlations.

Main Results:

  • The empirical variance (χ²) converges to a finite constant (χ²∞) for large sampling times (t).
  • The non-ergodicity parameter (Δ) shows volume dependence influenced by spatial correlations.
  • Models with marginally long-ranged correlations successfully map onto simulated amorphous glass shear stress data.

Conclusions:

  • Simple models can capture essential features of non-ergodic processes.
  • The non-ergodicity parameter's behavior is sensitive to field correlations and system size.
  • These findings provide insights into the physics of amorphous materials.