The Kraichnan Model and Non-Equilibrium Statistical Physics of Diffusive Mixing

TL;DR

This paper applies the Kraichnan model to analyze non-equilibrium concentration fluctuations during diffusion in liquids, revealing that nonlinear advection can be treated exactly and reproduces linear theory predictions for structure functions.

Contribution

It introduces a method that treats nonlinear advection exactly, showing that structure functions match linearized fluctuating hydrodynamics predictions, and discusses implications for higher-order correlations.

Findings

Structure functions match linear theory predictions.

Nonlinear advection can be treated exactly.

Higher-order cumulants are predicted to be non-zero.

Abstract

We discuss application of methods from the Kraichnan model of turbulent advection to the study of non-equilibrium concentration fluctuations arising during diffusion in liquid mixtures at high Schmidt numbers. This approach treats nonlinear advection of concentration fluctuations exactly, without linearization. Remarkably, we find that static and dynamic structure functions obtained by this method reproduce precisely the predictions of linearized fluctuating hydrodynamics. It is argued that this agreement is an analogue of anomaly non-renormalization which does not, however, protect higher-order multi-point correlations. The latter should thus yield non-vanishing cumulants, unlike those for the Gaussian concentration fluctuations predicted by linearized theory.