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.
Similar content being viewed by others
Notes
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].
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 \(p^*\ge 1\).
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.
References
Abramov, R.V., Majda, A.: New approximations and tests of linear fluctuation-response for chaotic nonlinear forced-dissipative dynamical systems. J. Nonlinear Sci. 18, 303–341 (2008). doi:10.1007/s00332-007-9011-9
Benettin, G., Galgani, L., Giorgilli, A., Strelcyn, J.M.: Lyapunov characteristic exponents for smooth dynamical systems and for Hamiltonian systems; a method for computing all of them. Part 1: Theory. Meccanica 15(1), 9–20 (1980)
Blender, R., Wouters, J., Lucarini, V.: Avalanches, breathers, and flow reversal in a continuous Lorenz-96 model. Phys. Rev. E 88, 013201 (Jul 2013)
Bowman, A.W., Azzalini, A.: Applied Smoothing Techniques for Data Analysis. Oxford University Press, Oxford (1997)
Dettman, C., Morriss, G.: Proof of conjugate pairing for an isokinetic thermostat. Phys. Rev. E 53, 5545–5549 (1996)
Dressler, U.: Symmetry property of the lyapunov exponents of a class of dissipative dynamical systems with viscous damping. Phys. Rev. A 38, 2103–2109 (1988)
Eckmann, J.-P., Ruelle, D.: Ergodic theory of chaos and strange attractors. Rev. Mod. Phys. 57, 617655 (1985)
Gallavotti, G.: Reversible Anosov diffeomorphisms and large deviations. Math. Phys. Electron. J. 1, 1–12 (1995)
Gallavotti, G.: Extension of Onsager’s reciprocity to large fields and the chaotic hypothesis. Phys. Rev. Lett. 77, 4334–4337 (1996)
Gallavotti, G.: Dynamical ensembles equivalence in fluid mechanics. Phys. D 105, 163–184 (1997)
Gallavotti, G.: Breakdown and regeneration of time reversal symmetry in nonequilibrium statistical mechanics. Phys. D 112, 250–257 (1998)
Gallavotti, G.: Fluctuations and entropy driven space-time intermittency in Navier–Stokes fluids. In: Fokas, E., Grigoryan, A., Kibble, T., Zegarlinski, B. (eds.) Mathematical Physics 2000. World Scientific, London (2000)
Gallavotti, G.: Non equilibrium in statistical and fluid mechanics. ensembles and their equivalence. Entropy driven intermittency. J. Math. Phys. 41, 4061–4081 (2000)
Gallavotti, G.: Foundations of Fluid Dynamics, vol. 2. Springer, Berlin (2005)
Gallavotti, G.: Microscopic chaos and macroscopic entropy in fluids. J. Stat. Mech. 2006:P10011 (+9) (2006).
Gallavotti, G.: Aspects of Lagrange’s Mechanics and their Legacy. arXiv:1305.3438, pp. 1–23 (2013).
Gallavotti, G., Bonetto, F., Gentile, G.: Aspects of the Ergodic, Qualitative and Statistical Theory of Motion. Springer, Berlin (2004)
Gallavotti, G., Cohen, E.G.D.: Dynamical ensembles in nonequilibrium statistical mechanics. Phys. Rev. Lett. 74, 2694–2697 (1995)
Gallavotti, G., Cohen, E.G.D.: Dynamical ensembles in stationary states. J. Stat. Phys. 80, 931–970 (1995)
Gallavotti, G., Rondoni, L., Segre, E.: Lyapunov spectra and nonequilibrium ensembles equivalence in 2D fluid. Phys. D 187, 358–369 (2004)
Hallerberg, Sarah, Pazo, Diego, Lopez, Juan M., Rodriguez, Miguel A.: Logarithmic bred vectors in spatiotemporal chaos: structure and growth. Phys. Rev. E 81, 066204 (Jun 2010)
Karimi, A., Paul, M.R.: Extensive chaos in the Lorenz-96 model. Chaos: an Interdisciplinary. J. Nonlinear Sci. 20(4), 043105 (2010)
Livi, R., Politi, A., Ruffo, S.: Distribution of characteristic exponents in the thermodynamic limit. J. Phys. A 19, 2033–2040 (1986)
Lorenz, E.: Deterministic non periodic flow. J. Atmos. Sci. 20, 130–141 (1963)
Lorenz, E.: Designing chaotic models. J. Atmos. Sci. 62, 1574–1587 (2005)
Lorenz, E., Emanuel, K.: Optimal sites for supplementary weather observations: simulation with a small model. J. Atmos. Sci. 55, 399–414 (1998)
Lucarini, V.: Response theory for equilibrium and non-equilibrium statistical mechanics: causality and generalized Kramers–Kronig relations. J. Stat. Phys. 131(3), 543–558 (2008)
Lucarini, V.: Evidence of dispersion relations for the nonlinear response of Lorenz 63 system. J. Stat. Phys. 134, 38140 (2009)
Lucarini, V., Blender, R., Herbert, C., Pascale, S., Ragone, F., Wouters, J. Mathematical and physical ideas for climate science. ArXiv e-prints, Nov 2013.
Lucarini, V., Sarno, S.: A statistical mechanical approach for the computation of the climatic response to general forcings. Nonlinear Process. Geophys. 18, 7–28 (2011)
Orrell, D. Model error and predictability over different timescales in the Lorenz ’96 systems. J. Atmos. Sci. 60(17), 2219–2228 (2003). 24 Mar 2014.
Pope, Stephen: Turbulent Flows. Cambridge University Press, Cambrdge (2000)
Ragone, F., Lucarini, V., Lunkeit, F. A new framework for climate sensitivity and prediction. ArXiv e-prints, March 2014.
Ruelle, D.: Chaotic Evolution and Strange Attractors. Cambridge University Press, Cambridge (1989)
Ruelle, D.: General linear response formula in statistical mechanics, and the fluctuation–dissipation theorem far from equilibrium. Phys. Lett. A 245, 220–224 (1998)
Ruelle, D.: A review of linear response theory for general differentiable dynamical systems. Nonlinearity 22, 855–870 (2009)
Ruelle, D., Takens, F.: On the nature of turbulence. Commun. Math. Phys. 20, 167–192 (1971)
Sagaut, Pierre: Large Eddy Simulation for Incompressible Flows. Springer, New York (2006)
Sagaut, P., Garnier, E., Adams, N.: Large Eddy Simulation for Compressible Flows. Springer, New York (2009)
She, Z.S., Jackson, E.: Constrained Euler system for Navier Stokes turbulence. Phys. Rev. Lett. 70(9), 1255–1258 (1993)
Sinai, Y.G.: Lectures in Ergodic Theory. Lecture Notes in Mathematics. Princeton University Press, Princeton (1977)
Smagorinsky, J.: Large eddy simulation of complex engineering and geophysical flows. In: Galperin, B., Orszag, S.A. (eds.) Evolution of Physical Oceanography, pp. 3–36. Cambridge University Press, Cambridge (1993)
Trevisan, Anna, D’Isidoro, Massimo, Talagrand, Olivier: Four-dimensional variational assimilation in the unstable subspace and the optimal subspace dimension. Q. J. R. Meteorol. Soc. 136(647), 487–496 (2010)
Trevisan, A., Uboldi, F. Assimilation of standard and targeted observations within the unstable subspace of the observation-analysis-forecast cycle system. J. Atmos. Sci. 61(1), 103–113 (2004). 24 Mar 2014.
Wilks, D.S.: Effects of stochastic parametrizations in the Lorenz ’96 system. Q. J. R. Meteorol. Soc. 131(606), 389–407 (2005)
Author information
Authors and Affiliations
Corresponding author
Appendices
Appendix 1
Appendix 2
Rights and permissions
About this article
Cite this article
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
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10955-014-1051-6