A nonequilibrium statistical model of spectrally truncated Burgers-Hopf dynamics

TL;DR

This paper introduces a novel statistical model reduction technique for spectral truncations of the Burgers-Hopf equation, deriving a closed system for low modes that accurately captures the dynamics through a fractional diffusion and modified nonlinear interactions.

Contribution

A new nonequilibrium statistical closure method for spectral truncations of the Burgers-Hopf equation using Hamilton-Jacobi theory and trial densities.

Findings

Reduced equations include fractional diffusion and modified nonlinear terms.

The model accurately predicts dynamics compared to direct numerical simulations.

The closure parameter effectively captures the system's stochastic behavior.

Abstract

Exact spectral truncations of the unforced, inviscid Burgers-Hopf equation are Hamiltonian systems with many degrees of freedom which exhibit intrinsic stochasticity and coherent scaling behavior. For this reason recent studies have employed these systems as prototypes to test stochastic mode reduction strategies. In the present paper the Burgers-Hopf dynamics truncated to n Fourier modes is treated by a new statistical model reduction technique, and a closed system of evolution equations for the mean values of the m lowest modes is derived for m << n. In the reduced model the m-mode macrostates are associated with trial probability densities on the phase space of the n-mode microstates, and a cost functional is introduced to quantify the lack of fit of paths of these densities to the Liouville equation. The best-fit macrodynamics is obtained by minimizing the cost functional over paths, and the equations governing the closure are then derived from Hamilton-Jacobi theory. The resulting reduced equations have a fractional diffusion and modified nonlinear interactions, and the explicit form of both are determined up to a single closure parameter. The accuracy and range of validity of this nonequilibrium closure is assessed by comparison against direct numerical simulations of statistical ensembles, and the predicted behaviour is found to be well represented by the reduced equations.