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Generalized Einstein relations and conditions for anomalous relaxation.

Jing-Dong Bao1

  • 1Department of Physics, Beijing Normal University, Beijing 100875, People's Republic of China.

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|December 25, 2019
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Summary

This study explores the generalized Einstein relation (GER) for nonergodic processes, identifying conditions for anomalous relaxation and deviations from equilibrium. Findings clarify the relationship between GER, system equilibrium, and particle velocity dynamics.

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Area of Science:

  • Statistical Mechanics
  • Non-equilibrium Physics
  • Theoretical Physics

Background:

  • The generalized Einstein relation (GER) is crucial for understanding particle dynamics in complex systems.
  • Nonergodic processes exhibit anomalous relaxation behaviors, deviating from standard equilibrium assumptions.
  • Generalized Langevin equations provide a framework for analyzing these complex dynamics.

Purpose of the Study:

  • To investigate the generalized Einstein relation (GER) for nonergodic processes.
  • To identify and distinguish conditions leading to anomalous relaxation, such as long-tail decay and non-vanishing velocity autocorrelation function (VAF).
  • To analyze deviations from equilibrium in nonergodic systems.

Main Methods:

  • Utilizing the framework of the generalized Langevin equation.
  • Analyzing the velocity autocorrelation function (VAF) for different nonergodic scenarios.
  • Investigating the conditions for asymptotic GER and system equilibrium.

Main Results:

  • Conditions for anomalous relaxation (long-tail decay, non-vanishing VAF) are proposed and differentiated.
  • An asymptotic GER is observed for stationary nonergodic processes with non-thermal initial velocity preparation.
  • GER holding is shown to be a necessary but not sufficient condition for near-equilibrium systems.
  • For high-frequency cutoff-induced nonergodic processes, VAF oscillates, GER holds, but equilibrium fails in the long-time limit.

Conclusions:

  • The study provides a nuanced understanding of the generalized Einstein relation in nonergodic systems.
  • Theoretical findings are confirmed through applications to practical examples.
  • The research clarifies the conditions under which systems approach or deviate from equilibrium.