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A hydrodynamic exercise in free probability: setting up free Euler equations. (English) Zbl 1461.46066

Summary: For the free probability analogue of Euclidean space endowed with the Gaussian measure we apply the approach of V. I. Arnol’d [Ann. Inst. Fourier 16, No. 1, 319–361 (1966; Zbl 0148.45301)] to derive Euler equations for a Lie algebra of non-commutative vector fields which preserve a certain trace. We extend the equations to vector fields satisfying non-commutative smoothness requirements. We introduce a cyclic vorticity and show that it satisfies vorticity equations and that it produces a family of conserved quantities.

MSC:

46L54 Free probability and free operator algebras
76B99 Incompressible inviscid fluids

Citations:

Zbl 0148.45301
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References:

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