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113 | 2014
Annuaire du Collège de France 2012-2013
Résumé des cours et travaux 113e année
Conférenciers invités

Gravity and a Geometrization of Turbulence

Yaron Oz
p. 923

Notes de la rédaction

M. Yaron Oz a été invité par l’Assemblée des professeurs sur proposition du Pr Gabriele Veneziano pour deux conférences données les 24 et 28 juin 2013.

Texte intégral

1The dynamics of fluids is a long standing challenge that remained as an unsolved problem for centuries. Understanding its main features, chaos and turbulence, is likely to provide an understanding of the principles and non-linear dynamics of a large class of systems far from equilibrium. In the lectures we will present a conceptually new viewpoint to study these features using black hole dynamics. Since the gravitational field is characterized by a curved geometry, the gravity variables provide a geometrical framework for studying the dynamics of fluids: A geometrization of turbulence.

2In the first part of the lectures we will consider the dynamics of both non-relativistic and relativistic fluids from the field theory viewpoint. We will discuss the phenomena of turbulence and its universal statistical characteristics in the inertial range of scale. We will present the anomalous scaling exponents of turbulence as seen by experimental and numerical studies, and discuss the major open problems in the field. We will outline Kolmogorov theory of turbulence in non-relativistic incompressible fluids, and the deviation of its predictions from the experimental and numerical data. We will derive new scaling relations, both for the non-relativistic and relativistic theories in the inertial range. We will briefly discuss the relativistic fluid produced by heavy ion collisions, and its notably low shear viscosity to entropy density ratio. We will discuss the Millenium problem in mathematics concerning the smoothness of the space of solutions of the Navier-Stokes equations.

3In the second part of the lectures we will introduce the concept of Holography relating quantum field theories in (+ 1) space-time dimensions to quantum gravity in one higher space dimension. We will outline the basic features of the AdS/CFT correspondence and the interpretation of the extra spatial dimension as the renormalization group (energy) scale. We will show that both the relativistic and non-relativistic Navier-Stokes equations describe deformations of black hole geometries in one higher space dimension and discuss the geometrical interpretation of the fluid variables. We will consider the analogy between cosmic censorship and the absence of naked singularities in gravity, and the smoothness of solutions to the Navier-Stokes equations. We will discuss the Penrose inequality as an example of a global method in geometry that can be used to study smoothness of fluid flows.

4We will discuss a recent development in the field of hydrodynamics and its gravitational description: the role of quantum field theory anomalies in fluid dynamics. We will consider the chiral, mixed chiral-gravitational and the conformal anomalies. We will show how these anomalies are manifested in hydrodynamics and outline several experimental consequences.

5Finally, we will present an outlook and discuss numerous open problems.

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Référence papier

Cours et travaux du Collège de France. Annuaire 113e année, Collège de France, Paris, avril 2014, p. 923. ISBN 978-2-7226-0329-5

Référence électronique

Yaron Oz, « Gravity and a Geometrization of Turbulence », L’annuaire du Collège de France [En ligne], 113 | 2014, mis en ligne le 15 août 2014, consulté le 28 mai 2018. URL :

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Yaron Oz

Professeur, université de Tel-Aviv (Israël)

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Droits d’auteur

Collège de France

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