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No-go theorem for ergodicity and an Einstein relation.

D Froemberg1, E Barkai

  • 1Department of Physics, Institute of Nanotechnology and Advanced Materials, Bar Ilan University, Ramat-Gan 52900, Israel.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|September 17, 2013
PubMed
Summary
This summary is machine-generated.

A new theorem shows that for anomalous diffusion, ergodicity and the generalized Einstein relation are typically broken, unlike normal diffusion. This impacts understanding of particle movement and transport properties.

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Area of Science:

  • Statistical Physics
  • Physical Chemistry
  • Complex Systems

Background:

  • Anomalous diffusion processes deviate from standard Brownian motion.
  • Ergodicity and the Einstein relation are fundamental concepts in diffusion theory.

Purpose of the Study:

  • To establish a no-go theorem for ergodicity and the generalized Einstein relation in anomalous diffusion.
  • To provide a general relation for time-averaged quantities in ergodic anomalous diffusion.

Main Methods:

  • Theoretical derivation of a no-go theorem.
  • Analysis of mean squared displacements (MSD) and time-averaged diffusivity.
  • Exemplification using the Lévy walk model.

Main Results:

  • A theorem demonstrating that ergodicity (equal time and ensemble averaged MSD) is broken for anomalous diffusion.
  • The generalized Einstein relation for time-averaged diffusivity and mobility is invalid.
  • A general relation for time averages of drift and MSD, dependent on measurement time, is derived for ergodic anomalous diffusion.

Conclusions:

  • Anomalous diffusion fundamentally differs from normal diffusion regarding ergodicity and the Einstein relation.
  • The findings necessitate revised theoretical frameworks for understanding anomalous transport phenomena.
  • The Lévy walk model serves as a concrete example illustrating these theoretical limitations.