Instantons and Intermittency

Abstract

We propose the new method for finding the non-Gaussian tails of probability distribution function (PDF) for solutions of a stochastic differential equation, such as convection equation for a passive scalar, random driven Navier-Stokes equation etc. Existence of such tails is generally regarded as a manifestation of intermittency phenomenon. Our formalism is based on the WKB approximation in the functional integral for the conditional probability of large fluctuation. We argue that the main contribution to the functional integral is given by a coupled field-force configuration -- {\em instanton}. As an example, we examine the correlation functions of the passive scalar $u$ advected by a large-scale velocity field $δ$-correlated in time. We find the instanton determining the tails of the generating functional and show that it is different from the instanton that determines the probability distribution function of high powers of $u$. We discuss the simplest instantons for the Navier-Stokes equation.