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 -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.PACS numbers 47.10.+g, 47.27.-i, 05.40.+j

Short-time height distribution in the one-dimensional Kardar-Parisi-Zhang equation: Starting from a parabola

“…[31]. Similar approaches have been applied, under different names, to turbulence [32][33][34], stochastic reactions [35,36], diffusive lattice gases [37], and non-equilibrium surface growth [21-23, 25, 38-41] including the KPZ equation itself. The WNT equations can be formulated as a classical Hamiltonian field theory.…”

Statistical analysis and modeling of intermittent transport events in the tokamak scrape-off layer

“…The instanton method is a non-perturbative way of calculating PDF tails, which was adopted from quantum field theory and then modified to classical statistical physics for Burgers turbulence and a passive scalar model. 9,10 For instance, using the instanton method, it has been shown in Ref. 11 that the PDF tails of momentum flux Q are significantly enhanced over the Gaussian prediction.…”

Исследование Асимптотик Высоких Порядков Квантово-Полевых Разложений В Теории Двумерной Развитой Турбулентности

“…В работе [10] была предпринята попытка решения аналогич-ного уравнения (динамического) в области малых волновых чисел p. Отметим, что такое решение в нашем случае не позволяет решить поставленную задачу, поскольку разложение реально ведется по параметру p/Λ, все члены этого разложения дают вклады одного порядка на верхнем пределе интегралов по волновому числу как в интегральном уравнении (12), так и во входящие в S(v st ) интегралы. Рассмотрим, тем не менее, асимптотики возможного решения данного уравнения.…”