Abstract
Some models for developed turbulence are considered; they are shown to obey a large-fluctuations theorem, and one of them also obeys a response reciprocity relation of Onsager type. This illustrates and extends ideas and techniques developed in earlier works mainly for nonequilibrium problems in statistical mechanics.
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References
V. Arnold and A. Avez,Ergodic Problems of Classical Mechanics (Benjamin, New York, 1966).
L. Boltzmann, Über die Eigenschaften monzyklischer und anderer damit verwandter Systeme, inWissenschaftliche Abhandlungen, Vol. III, F. P. Hasenöhrl, ed. (reprint, Chelsea, New York, 1968).
R. Bowen,Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms (Springer-Verlag, Berlin, 1975).
S. de Groot and P. Mazur,Non Equilibrium Thermodynamics (Dover, New York, 1984).
D. J. Evans, E. G. D. Cohen, and G. P. Morriss, Viscosity of a simple fluid from its maximal Lyapunov exponents,Phys. Rev. 42A:5990–5997 (1990).
D. J. Evans, E. G. D. Cohen, and G. P. Morriss, Probability of second law violations in shearing steady flows,Phys. Rev. Lett. 71:2401–2404 (1993).
G. L. Eyink, J. L. Lebowitz, and H. Spohn, Hydrodynamics and fluctuations outside of local equilibrium: driven diffusive systems [mp_arc@math.utexas.edu#95-168].
D. J. Evans and G. P. Morriss,Statistical Mechanics of Nonequilibrium Fluids (Academic Press, New York, 1990).
R. Esposito, R. Marra, and H. T. Yau, Diffusive limit of asymmetric simple exclusion, inThe State of Matter, M. Aizenmann and H. Araki, eds. (World Scientific, Singapore, 1994).
J. P. Eckmann and D. Ruelle, Ergodic theory of strange attractors,Rev. Mod. Phys. 57:617–656 (1985).
D. J. Evans and D. J. Searles, Equilibrium microstates which generate second law violating steady states, preprint, Research School of Chemistry, Canberra, ACT, Australia (1993).
V. Franceschini and F. Zironi, On constructing Markov partitions by computer,J. Stat. Phys. 40:69–91 (1985).
G. Gallavotti, InTopics in Chaotic Dynamics, P. L. Garrido and J. Marro, eds. (Springer-Verlag, Berlin, 1995).
G. Gallavotti, Reversible Anosov diffeomorphisms and large deviations,Math. Phys. Electron. J. 1:1–12 (1995).
G. Gallavotti, Ergodicity, ensembles, irreversibility in Boltzmann and beyond,J. Stat. Phys. 78:1571–1579 (1995).
G. Gallavotti, Equivalence of dynamical ensembles and Navier-Stokes equations [mp_arc 96-131; chao-dyn 9604006]; Dynamical ensembles equivalence in statistical mechanics [mp_arc@math.utexas.edu #96-182; chao-dyn@xyz.lanl.gov. #9605006].
G. Gallavotti, Chaotic hypothesis: Onsager reciprocity and fluctuation-dissipation theorem,J. Stat. Phys. 84:899–925.
G. Gallavotti, Extension of Onsager's reciprocity to large fields and the chaotic hypothesis [mp_arc 96-109; chao-dyn 9603003].
G. Gallavotti and E. G. D. Cohen, Dynamical ensembles in nonequilibrium statistical mechanics,Phys. Rev. Lett. 74:2694–2697 (1995).
G. Gallavotti and E. G. D. Cohen, Dynamical ensembles in stationary states,J. Stat. Phys. 80:931–970 (1995).
R. H. Kraichnan, Classical fluctuation-relaxation theorem,Phys. Rev. 113:1181–1182 (1959).
R. H. Kraichnan, Remarks on turbulence theory,Adv. Math. 16:305–331 (1975).
C. Kipnis, S. Olla, and S. R. S. Varadhan,Commun. Pure Appl. Math. XLII:243 (1989).
E. Lorenz, Deterministic non periodic flow,J. Atmos. Sci. 20:130–141 (1963).
T. Levi-Civita and U. Amaldi,Lezioni di Meccanica Razionale (Zanichelli, Bologna, 1927; reprinted, 1974).
L. D. Landau and E. M. Lifshitz,Fluid Mechanics (Pergamon Press, Oxford, 1959).
C. Landim, S. Olla, and H. T. Yau, First-order correction for the hydrodynamic limit of asymmetric simple exclusion processes in dimensiond≥3, preprint, Ecole Polytechnique, R.I. No. 307 (November 1994).
R. Livi, A. Politi, and S. Ruffo, Distribution of characteristic exponents in the thermodynamic limit,J. Phys. 19A:2033–2040 (1986).
C. Marchioro, An example of absence of turbulence for any Reynold number,Commun. Math. Phys. 105:99–106 (1986);108:647–651 (1987).
H. A. Posch and W. G. Hoover, Non equilibrium molecular dynamics of a classical fluid, inMolecular Liquids: New Perspectives in Physics and Chemistry, J. Teixeira-Dias, ed. (Kluwer, Dordrecht, 1992), pp. 527–547.
D. Ruelle, Measures describing a turbulent flow,Ann. N. Y. Acad. Sci. 357:1–9 (1980); see also D. Ruelle,Chaotic Motions and Strange Attractors, notes by S. Isola (Accademia Nazionale dei Lincei, and Cambridge University Press, 1989).
D. Ruelle, A measure associated with axiom A attractors,Am. J. Math. 98:619–654 (1976).
Y. G. Sinai, Gibbs measures in ergodic theory,Russ. Math. Surv. 27:21–69 (1972);Lectures in Ergodic Theory (Princeton University Press, Princeton, New Jersey, 1977).
Z. S. She and E. Jackson, Constrained Euler system for Navier-Stokes turbulence,Phys. Rev. Lett. 70:1255–1258 (1993).
G. E. Uhlenbeck and G. W. Ford,Lectures in Statistical Mechanics (American Mathematical Society, Providence, Rhode Island, 1963), pp. 5, 16, 30.
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Gallavotti, G. Chaotic principle: Some applications to developed turbulence. J Stat Phys 86, 907–934 (1997). https://doi.org/10.1007/BF02183608
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DOI: https://doi.org/10.1007/BF02183608