Abstract
The Kraichnan flow provides an example of a random dynamical system accessible to an exact analysis. We study the evolution of the infinitesimal separation between two Lagrangian trajectories of the flow. Its long-time asymptotics is reflected in the large deviation regime of the statistics of stretching exponents. Whereas in the flow that is isotropic at small scales the distribution of such multiplicative large deviations is Gaussian, this does not have to be the case in the presence of an anisotropy. We analyze in detail the flow in a two-dimensional periodic square where the anisotropy generally persists at small scales. The calculation of the large deviation rate function of the stretching exponents reduces in this case to the study of the ground state energy of an integrable periodic Schrödinger operator of the Lamé type. The underlying integrability permits to explicitly exhibit the non-Gaussianity of the multiplicative large deviations and to analyze the time-scales at which the large deviation regime sets in. In particular, we indicate how the divergence of some of those time scales when the two Lyapunov exponents become close allows a discontinuity of the large deviation rate function in the parameters of the flow. The analysis of the two-dimensional anisotropic flow permits to identify the general scenario for the appearance of multiplicative large deviations together with the restrictions on its applicability.
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References
L. Arnold, Random Dynamical Systems (Springer, Berlin 2003).
E. Balkovsky, G. Falkovich, and A. Fouxon, Intermittent distribution of inertial particles in turbulent flows. Phys. Rev. Lett. 86:2790–2793 (2001).
E. Balkovsky and A. Fouxon, Universal long-time properties of Lagrangian statistics in the Batchelor regime and their application to the passive scalar problem. Phys. Rev. E 60:4164–4174 (1999).
P. H. Baxendale, The Lyapunov spectrum of a stochastic flow of diffeomorphisms, In L. Arnold and V. Wihstutz, Eds., Lyapunov Exponents, Bremen 1984, Lecture Notes in Math. vol. 1186 (Springer 1986), pp. 322–337.
P. H. Baxendale and D. W. Stroock, Large deviations and stochastic flows of diffeomorphisms. Prob. Theor.& Rel. Fields 80:169–215 (1988).
J. Bec, K. Gawędzki, and P. Horvai, Multifractal clustering in compressible flows. Phys. Rev. Lett. 92:224501–2240504 (2004).
D. Bernard, K. Gawędzki, and A. Kupiainen, Slow modes in passive advection, J. Stat. Phys. 90:519–569 (1998).
F. Bonetto, G. Gallavotti, and G. Gentile, A fluctuation theorem in a random environment. mp_arc/06-139
M. Chertkov, Polymer stretching by turbulence. Phys. Rev. Lett. 84:4761–4764 (2000).
M. Chertkov, G. Falkovich, I. Kolokolov, and V. Lebedev, Statistics of a passive scalar advected by a large-scale 2D velocity field: analytic solution. Phys. Rev. E 51:5609–5627 (1995).
M. Chertkov, G. Falkovich, I. Kolokolov, and M. Vergassola, Small-scale turbulent dynamo. Phys. Rev. Lett. 83:4065–4068 (1999).
M. Chertkov, I. Kolokolov, and M. Vergassola, Inverse versus direct cascades in turbulent advection. Phys. Rev. Lett. 80:512–515 (1998).
P. Etingof and A. Kirillov, Jr., Representations of affine Lie algebras, parabolic differential equations and Lamé Functions. Duke Math. J. 74:585–614 (1994).
A. Erdélyi, W. Magnus, F. Oberhettinger, and F. Tricomi, Higher Transcendental Functions III (McGraw-Hill, New York, 1955).
D. J. Evans, E. G. D. Cohen, and G. P. Morriss, Probability of second law violations in shearing steady states. Phys. Rev. Lett. 71:2401–2404 and 3616 (1993).
G. Falkovich, K. Gawędzki, and M. Vergassola, Particles and fields in fluid turbulence, Rev. Mod. Phys. 73: 913–975 (2001)
G. Gallavotti, Fluctuation patterns and conditional reversibility in nonequilibrium systems. Ann. Inst. H. Poincaré 70:429–443 (1999).
G. Gallavotti and E. D. G. Cohen, Dynamical ensembles in non-equilibrium statistical mechanics. Phys. Rev. Lett. 74:2694–2697 (1995).
K. Gawędzki, On multiplicative large deviations (unpublished).
K. Gawędzki and F. Falceto, Elliptic Wess-Zumino-Witten model from elliptic Chern-Simons theory. Lett. Math. Phys. 38:155–175 (1996).
I. S. Gradstein and I. M. Rhyzik, Table of Integrals, Series, and Products, V th edition (Academic Press, New York, 1994).
P. Grassberger, R. Baddi, and A. Politi, Scaling laws for invariant measures on hyperbolic and nonhyperbolic attractors. J. Stat. Phys. 51:135–178 (1988).
R. H. Kraichnan, Small scale structure of a scalar field convected by turbulence, Phys. Fluids 11:945–953 (1968).
H. Kunita, Stochastic Flows and Stochastic Differential Equations (Cambridge University Press, London, 1990).
Y. Le Jan, On isotropic Brownian motions. Zeit. Wahrschein. verw. Gebite 70:609–620 (1985).
V. I. Oseledec, Multiplicative ergodic theorem: Characteristic Lyapunov exponents of dynamical systems. Trudy Moskov. Mat. Obšč 19:179–210 (1968).
D. Ruelle, Ergodic theory of differentiable dynamical systems. Publications Mathématiques de l'IHéS 50:275–320 (1979).
B. Shraiman and E. Siggia, Symmetry and scaling of turbulent mixing. Phys. Rev. Lett. 77:2463–2466 (1996).
H. Volkmer, Lamé functions. To appear in Digital Library of Mathematical Functions, National Institute of Standards and Technology, Gaithersburg, MD, http://dlmf.nist.gov.
E. T. Whittaker and G. N. Watson, A Course of Modern Analysis (Cambridge University Press, London, 1927).
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Krzysztof Gawedzki is Member of C.N.R.S.
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Chetrite, R., Delannoy, JY. & Gawedzki, K. Kraichnan Flow in a Square: An Example of Integrable Chaos. J Stat Phys 126, 1165–1200 (2007). https://doi.org/10.1007/s10955-006-9225-5
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DOI: https://doi.org/10.1007/s10955-006-9225-5