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Equivalence of Non-equilibrium Ensembles and Representation of Friction in Turbulent Flows: The Lorenz 96 Model

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Abstract

We construct different equivalent non-equilibrium statistical ensembles in a simple yet instructive \(N\)-degrees of freedom model of atmospheric turbulence, introduced by Lorenz in 1996. The vector field can be decomposed into an energy-conserving, time-reversible part, plus a non-time reversible part, including forcing and dissipation. We construct a modified version of the model where viscosity varies with time, in such a way that energy is conserved, and the resulting dynamics is fully time-reversible. For each value of the forcing, the statistical properties of the irreversible and reversible model are in excellent agreement, if in the latter the energy is kept constant at a value equal to the time-average realized with the irreversible model. In particular, the average contraction rate of the phase space of the time-reversible model agrees with that of the irreversible model, where instead it is constant by construction. We also show that the phase space contraction rate obeys the fluctuation relation, and we relate its finite time corrections to the characteristic time scales of the system. A local version of the fluctuation relation is explored and successfully checked. The equivalence between the two non-equilibrium ensembles extends to dynamical properties such as the Lyapunov exponents, which are shown to obey to a good degree of approximation a pairing rule. These results have relevance in motivating the importance of the chaotic hypothesis. in explaining that we have the freedom to model non-equilibrium systems using different but equivalent approaches, and, in particular, that using a model of a fluid where viscosity is kept constant is just one option, and not necessarily the only option, for describing accurately its statistical and dynamical properties.

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Notes

  1. In \(3\) dimensions the natural cut-off would be the Kolmogorov scale; in \(2\) dimensions the cascade is inverse and the interpretation is more subtle, see [10].

  2. This is a function \(\gamma (p)\) such that the probability the \(p\in \Delta \) is asymtpotic as \(\tau \rightarrow \infty \) to \(const\,e^{\tau \max _{p\in \Delta }\gamma (p) }\): in Anosov systems it is analytic, [17, 41], in \(p\) for \(|p|<p^*\) for some \(p^*\); in time reversal symmetric Anosov systems \(p^*\ge 1\).

  3. Also featuring modest deviations , compared to the size of the strongly \(R\)-dependent Lyapunov exponents, from what we would have obtained (a constant -1 value) had the inviscid, unforced dynamics been Hamiltonian.

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Author information

Authors and Affiliations

  1. Department of Physics, University of Rome La Sapienza, Rome, Italy

    Giovanni Gallavotti

  2. Department of Mathematics, Rutgers University, Piscataway, NJ, USA

    Giovanni Gallavotti

  3. Institute of Meteorology, University of Hamburg, Klimacampus, Hamburg, Germany

    Valerio Lucarini

  4. Department of Mathematics and Statistics, University of Reading, Reading, UK

    Valerio Lucarini

  5. Walker Institute for Climate Change Research, University of Reading, Reading, UK

    Valerio Lucarini

Authors
  1. Giovanni Gallavotti
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  2. Valerio Lucarini
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Correspondence to Valerio Lucarini.

Appendices

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Gallavotti, G., Lucarini, V. Equivalence of Non-equilibrium Ensembles and Representation of Friction in Turbulent Flows: The Lorenz 96 Model. J Stat Phys 156, 1027–1065 (2014). https://doi.org/10.1007/s10955-014-1051-6

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  • DOI: https://doi.org/10.1007/s10955-014-1051-6

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