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Fluctuation relation for a Lévy particle.

H Touchette1, E G D Cohen

  • 1School of Mathematical Sciences, Queen Mary, University of London, London E1 4NS, United Kingdom. ht@maths.qmul.ac.uk

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|October 13, 2007
PubMed
Summary

This study explores particle work fluctuations under Lévy-type random forces. Results show unusual "fat" power-law tails in work probability, violating standard fluctuation theorems.

Area of Science:

  • Statistical mechanics
  • Non-equilibrium physics
  • Stochastic processes

Background:

  • Understanding particle dynamics under external forces is crucial in statistical mechanics.
  • Investigating systems driven by non-Gaussian random processes reveals complex behaviors beyond standard models.
  • Work fluctuation theorems provide fundamental insights into thermodynamics of small systems.

Purpose of the Study:

  • To analyze the work fluctuations of a particle subjected to deterministic drag and Lévy-type random forcing.
  • To investigate the stationary probability density of work and its statistical properties.
  • To examine the implications of Lévy statistics on existing fluctuation theorems.

Main Methods:

  • Analytical treatment of a particle's Langevin equation with Lévy noise.

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  • Derivation of the stationary probability density function for work.
  • Analysis of the behavior of work fluctuations and their ratio for positive and negative values.

    • Main Results:

    • The probability density of work exhibits "fat" power-law tails, indicating a higher likelihood of large fluctuations compared to Gaussian forcing.
    • These power-law tails lead to a significant violation of established fluctuation theorems.
    • The ratio of probabilities for positive and negative work fluctuations of equal magnitude displays non-monotonic behavior.

    Conclusions:

    • Lévy-type random forcing fundamentally alters work fluctuation statistics in driven systems.
    • Existing fluctuation theorems, typically derived for Gaussian processes, are inadequate for systems with Lévy noise.
    • The predicted non-monotonic behavior and fat tails offer testable predictions for future experiments.