Phase Space Reduction and the Instanton Crossover in (1+1)-Dimensional Turbulence

Abstract

We study (1+1)-dimensional turbulence in the framework of the Martin-Siggia-Rose field theory formalism. The analysis is focused on the asymptotic behaviour at the right tail of the probability distribution function (pdf) of velocity differences, where shock waves do not contribute. A BRS-preserving scheme of phase space reduction, based on the smoothness of the relevant velocity fields, leads to an effective theory for a few degrees of freedom. The sum over fluctuations around the instanton solution is written as the expectation value of a functional of the time-dependent physical fields, which evolve according to a set of Langevin equations. A natural regularization of the fluctuation determinant is provided from the fact that the instanton dominates the action for a finite time interval. The transition from the turbulent to the instanton dominated regime is related to logarithmic corrections to the saddle-point action, manifested on their turn as multiplicative power law corrections to the velocity differences pdf.