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Synthesis of Cyclic Polymers and Characterization of Their Diffusive Motion in the Melt State at the Single Molecule Level
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Published on: September 26, 2016

Generalization of the Einstein relation for single trajectories in deterministic subdiffusion.

Takuma Akimoto1

  • 1Department of Mechanical Engineering, Keio University, Yokohama 223-8522, Japan. akimoto@z8.keio.jp

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|April 3, 2012
PubMed
Summary
This summary is machine-generated.

This study explores random diffusion coefficients in subdiffusion, revealing a universal constant relating velocity and Lyapunov exponents under bias. The findings generalize the Einstein relation for single particle trajectories.

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Area of Science:

  • Physics
  • Statistical Mechanics
  • Physical Chemistry

Background:

  • Subdiffusion is characterized by random diffusion coefficients due to power-law trapping.
  • The Einstein relation is a fundamental concept in diffusion studies.

Purpose of the Study:

  • To investigate the Einstein relation for single trajectories in subdiffusion using deterministic models.
  • To establish a relationship between bias, velocity, and Lyapunov exponents in subdiffusion.

Main Methods:

  • Utilized deterministic biased and unbiased diffusion models.
  • Applied Hopf's ergodic theorem to analyze single trajectories.
  • Calculated generalized Lyapunov exponents for biased and unbiased diffusion.

Main Results:

  • The difference in generalized Lyapunov exponents between biased and unbiased diffusion correlates with the applied bias velocity.
  • Ratios of velocities to Lyapunov exponents for single trajectories converge to a universal constant proportional to bias strength.
  • A generalized Einstein relation for single trajectories was derived based on a single-trajectory transport coefficient.

Conclusions:

  • The study provides a generalized Einstein relation applicable to single trajectories in subdiffusion.
  • Demonstrates a universal relationship between transport properties and bias in subdiffusion systems.
  • Offers new insights into the statistical mechanics of anomalous diffusion.