Zero modes, instantons and fluctuations in the Kraichnan model

Self-similar instanton solutions of the stochastic dynamics of the full passive scalar field in the Kraichnan model of turbulent advection are good candidates for describing the coherent motion of fluid particles leading to the production of strong gradients of the scalar and intermittency properties. By introducing a description of fluid particles in terms of quantum bosons, we show that instantons can be set in correspondence by a Legendre transformation with the so-called zero modes which are known, since the late 90s, to constitute the good mathematical framework for understanding and quantifying anomalous scaling in the Kraichnan model. Sticking to the classical level of approximation in the instanton approach leads clearly to an overestimation of scaling exponents of the structure functions of the scalar. To get further and incorporate the effect of fluctuations, we use the semiclassical instanton solutions as building blocks of more refined trial many-body wave functions for zero modes. One of the proposed approximation schemes gives surprisingly good estimates of scaling exponents even at low statistical orders and points towards a saturation of exponents at high statistical orders, as predicted by many authors either from numerics or analytic investigations in the large d limit. (paper: classical statistical mechanics, equilibrium and non-equilibrium)