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On Functional and Holographic Renormalization Group Methods in Stochastic Theory of Turbulence

A nonlocal quantum-field model is constructed for the system of hydrodynamic equations for incompressible viscous fluid (the stochastic Navier--Stokes (NS) equation and the continuity equation). This model is studied by the following two mutually parallel methods: the Wilson--Polchinski functional renormalization group method (FRG), which is based on the exact functional equation for the generating functional of amputated connected Green's functions (ACGF), and the Heemskerk--Polchinski holographic renormalization group method (HRG), which is based on the functional Hamilton--Jacobi (HJ) equation for the holographic boundary action. Both functional equations are equivalent to infinite hierarchies of integro-differential equations (coupled in the FRG case) for the corresponding families of Green's functions (GF). The RG-flow equations can be derived explicitly for two-particle functions. Because the HRG-flow equation is closed (contains only a two-particle GF), the explicit analytic solutions are obtained for the two-particle GF (in terms of the modified Bessel functions $I$ and $K$) in the framework of the minimal holographic model and its simple generalization, and these solutions have a remarkable property of minimal dependence on the details of the random force correlator (the function of the energy pumping into the system). The restrictions due to the time-gauged Galilean symmetry present in this theory, the problem of choosing the pumping function, and some generalizations of the standard RG-flow procedures are discussed in detail. Finally, the question of whether the HRG-solutions can be used to solve the FRG-flow equation for the two-particle GF (in particular, the relationship between the regulators in the two methods) is studied.