Abstract
The recent proposal by D. Ruelle for a theory of the corrections to the OK theory (“intermittency corrections”) is to take into account that the Kolmogorov scale itsef should be regarded as a fluctuating variable. Some quantitative aspects of the theory can be quite easily studied also via computer and will be presented.
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Notes
- 1.
If the box \(\varDelta = (n,j) \subset \varDelta ' = (n - 1,j')\) then the distribution Π(W | W Δ′) of \(W_{\varDelta } \equiv \mathbf{v}_{\varDelta }^{3}\) is conditioned to be such that \(\boldsymbol{\langle }\,W\,\boldsymbol{\rangle } =\kappa ^{-1}W_{\varDelta '}\); therefore the maximum entropy condition is that \(-\int \varPi (W\vert W')\log \varPi (W\vert W')dW -\lambda _{\varDelta }\int W\varPi (W\vert W')dW\), where \(\lambda _{\varDelta }\) is a Lagrange multiplier, is maximal under the constraint that \(\boldsymbol{\langle }\,W\,\boldsymbol{\rangle } = W'\kappa ^{-1}\): this gives the expression, called Boltzmannian in [7], for \(\varPi (W\vert W')\).
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Acknowledgements
The above comments are based on numerical calculations first done by P. Garrido and confirmed by G. Gallavotti. This is the text of our comments (requested by the organizers) to the talk by D. Ruelle at the CHAOS15 conference, Institut Henri Poincaré, Paris, May 26–29, 2015.
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Gallavotti, G., Garrido, P. (2016). Non-equilibrium Statistical Mechanics of Turbulence. In: Skiadas, C. (eds) The Foundations of Chaos Revisited: From Poincaré to Recent Advancements. Understanding Complex Systems. Springer, Cham. https://doi.org/10.1007/978-3-319-29701-9_4
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