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A system is linear if it displays the characteristics of homogeneity and additivity, together termed the superposition property. This principle is fundamental in all linear systems. Linear time-invariant (LTI) systems include systems with linear elements and constant parameters.
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Scale-invariant Green-Kubo relation for time-averaged diffusivity.

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Summary
This summary is machine-generated.

Anomalous diffusion studies reveal distinct time- and ensemble-averaged mean-squared displacements. This research connects time-averaged diffusivity to velocity correlations, crucial for single-particle tracking experiments.

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Area of Science:

  • Physics
  • Physical Chemistry
  • Statistical Mechanics

Background:

  • Anomalous diffusion exhibits differing time- and ensemble-averaged mean-squared displacements.
  • Ensemble-averaged diffusivity links to scale-invariant velocity correlations via Green-Kubo relations.

Purpose of the Study:

  • Establish the relationship between time-averaged diffusivity and scale-invariant velocity correlation functions.
  • Clarify differences in time-averaged behavior compared to ensemble-averaged behavior in anomalous diffusion systems.

Main Methods:

  • Derivation of the relation for time-averaged mean-squared displacement: 〈δ^{2}[over ¯]〉∼2D_{ν}t^{β}Δ^{ν-β}.
  • Analysis of anomalous diffusion exponent (ν) and kinetic energy scaling (β).
  • Demonstration using nonstationary scale-invariant stochastic and deterministic models.

Main Results:

  • A direct connection is established between exponents derived from velocity correlation functions and the transport constant D_{ν}.
  • Systems with identical ensemble-averaged behavior can show significant differences in time-averaged properties.
  • When averaged kinetic energy is finite (β=0), time scaling of displacements matches, but transport coefficients differ.

Conclusions:

  • The study provides a framework for understanding time-averaged diffusivity in anomalous diffusion.
  • Highlights the importance of considering both time and ensemble averages for a complete picture of transport phenomena.
  • Offers insights applicable to interpreting single-particle tracking experiments in complex systems.