Exact Computations in the Burgers Problem

TL;DR

This paper performs exact calculations of fluctuations around the instanton solution in the Burgers equation with quadratic forcing, confirming a differential equation for the probability distribution without anomaly contributions.

Contribution

It provides an exact summation of fluctuations around the instanton in the Burgers problem with quadratic forcing, clarifying the role of anomaly terms.

Findings

Probability distribution satisfies Polyakov's differential equation without anomaly.

Fluctuations do not generate anomaly terms in the quadratic forcing case.

Supports that anomaly terms, if present, originate from other instanton solutions.

Abstract

We complete the program outlined in the paper of the author with A. Migdal and sum up exactly all the fluctuations around the instanton solution of the randomly large scale driven Burgers equation. We choose the force correlation function $\kappa$ to be exactly quadratic function of the coordinate difference. The resulting probability distribution satisfy the differential equation proposed by Polyakov without an anomaly term. The result shows that unless the anomaly term is indeed absent it must come from other possible instanton solutions, and not from the fluctuations.