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A path integral for turbulence in incompressible fluids. (English) Zbl 0868.76041
Summary: A classical path integral is formulated for incompressible fluids that evolve according to the Navier-Stokes equation. The path integral propagates probability distributions deterministically on the space \({\mathcal G}_{\text{vol}}\) of solenoidal velocity fields. We construct a set of \(ISp(2)\) charges associated with the geometry of \({\mathcal G}_{\text{vol}}\) and its Poisson structure, and a pair of supersymmetry charges connected with the Hamiltonian. These charges generate exact symmetries of the classical path integral when the viscosity is set equal to zero. When the effect of dissipation is included, the charges associated with the Poisson structure and the Hamiltonian are no longer conserved. Charges that generate Kolmogorov scaling and Galilean transformations are also constructed. The classical path integral is formulated in terms of vorticity as well.

MSC:
76F99 Turbulence
76D05 Navier-Stokes equations for incompressible viscous fluids
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