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The most commonly used measure of variation is the standard deviation. It is a numerical value measuring how far data values are from their mean. The standard deviation value is small when the data are concentrated close to the mean, exhibiting slight variation or spread. The standard deviation value is never negative, it is either positive or zero. The standard deviation is larger when the data values are more spread out from the mean, which means the data values are exhibiting more variation.
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Anomalous Scaling of Dynamical Large Deviations.

Daniel Nickelsen1, Hugo Touchette1

  • 1National Institute for Theoretical Physics (NITheP), Stellenbosch 7600, South Africa and Institute of Theoretical Physics, Department of Physics, University of Stellenbosch, Stellenbosch 7600, South Africa.

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Large deviation functions characterize nonequilibrium processes. This study reveals anomalous time scaling in these functions can occur even without long-range correlations in simple Markovian systems like the Langevin equation.

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Area of Science:

  • Statistical mechanics
  • Non-equilibrium thermodynamics
  • Complex systems

Background:

  • Large deviation functions generalize thermodynamic potentials to systems out of equilibrium.
  • Typically, these functions exhibit linear scaling with integration time (τ) in the steady state.
  • Anomalous scaling (τ^ξ, ξ≠1) is usually associated with long-range correlations.

Purpose of the Study:

  • To investigate the origin of anomalous time scaling in large deviation functions.
  • To demonstrate that anomalous scaling can arise in Markovian processes without long-range correlations.
  • To elucidate the underlying mechanism and physical consequences of this phenomenon.

Main Methods:

  • Analysis of time-integrated observables in nonequilibrium steady states.
  • Application of path integral formalism.
  • Investigation of Markovian processes, including the Langevin equation.

Main Results:

  • Demonstrated anomalous power-law scaling (τ^ξ) in large deviation functions for Markovian processes without long-range correlations.
  • Identified the underlying mechanism responsible for this non-standard scaling behavior.
  • Showcased that simple systems like the Langevin equation can exhibit such anomalous scaling.

Conclusions:

  • Anomalous time scaling in large deviations is not exclusively linked to long-range correlations.
  • The findings extend the understanding of large deviation theory in nonequilibrium statistical mechanics.
  • The results have implications for analyzing a broader range of physical processes and complex systems.