Burgers Turbulence and Interface Growth III. Similarity Functional Solution of Hopf Equation. the Case of Random Gaussian Forcing

TL;DR

This paper presents a similarity functional solution to the Hopf equation for Burgers turbulence with Gaussian forcing, analyzing its properties across dimensions and comparing with other theoretical approaches.

Contribution

It introduces a similarity functional solution for the Hopf equation in Burgers turbulence with Gaussian forcing and explores its implications across different dimensions.

Findings

Self-similar behavior of local fluctuations in one dimension

Gaussian steady-state distribution within correlation length

Higher-dimensional cumulants deviate from Gaussian, showing logarithmic dependence

Abstract

For the problem of Burgers turbulence with random gaussian forcing a similarity functional solution of Hopf equation is presented and compared with scaling arguments and replica Bethe-anzatz treatments. The corresponding field theory is almost non-anomalous. In one dimension the local fluctuations develop self-similar time-dependent behavior, while relative fluctuations within the correlation length form a steady-state with gaussian distribution. This is the precise meaning of the so-called fluctuation-dissipation theorem. The one-dimensional properties are also studied numerically. It is shown that the fluctuation-dissipation theorem is invalid above one dimension and higher-order cumulants are non-zero. In two dimensions the cumulants exhibit logarithmic spatial dependence which is close to but different from that in the Edwards-Wilkinson case. No other similarity functional solution is found which may indicate that the ``strong-coupling'' results are not described by Burgers equation with gaussian noise.