The Geometry of Non-Ideal Fluids

Details The Geometry of Non-Ideal Fluids The Geometry of Non-Ideal Fluids

2010-04-14

arxiv.org/abs/1004.2462v1

Physics

Mathematical Physics

Talk at the Quantum Theory and Symmetries 6 Conference at the University of Kentucky

Scientific paper

Arnold showed that the Euler equations of an ideal fluid describe geodesics on the Lie algebra of incompressible vector fields. We generalize this to fluids with dissipation and Gaussian random forcing. The dynamics is determined by the structure constants of a Lie algebra, along with inner products defining kinetic energy, Ohmic dissipation and the covariance of the forces. This allows us to construct tractable toy models for fluid mechanics with a finite number of degrees of freedom. We solve one of them to show how symmetries can be broken spontaneously.In another direction, we derive a deterministic equation that describes the most likely path connecting two points in the phase space of a randomly forced system: this is a WKB approximation to the Fokker-Plank-Kramer equation, analogous to the instantons of quantum theory. Applied to hydrodynamics, we derive a PDE system for Navier-Stokes instantons.

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