Parametric Hamilton's equations for Lagrangian model for passive scalar gradients

TL;DR

This paper introduces a modified parametrization of Hamilton's equations for instanton methods, enabling arbitrary speeds, correcting action calculations, and improving probability density function approximations in turbulence models.

Contribution

It generalizes Hamilton's equations parametrization, corrects the action for finite Hamiltonians, and enhances instanton-based PDF predictions in turbulence modeling.

Findings

Generalized parametrization to any instanton speed.

Corrected the parametric action for finite Hamiltonians.

Improved instanton approximation accuracy for turbulence statistics.

Abstract

In the context of instanton method for stochastic system this paper purposes a modification of the arclength parametrization of the Hamilton's equations allowing for an arbitrary instanton speed. The main results of the paper are: (i) it generalizes the parametrized Hamilton's equations to any speed required. (ii) corrects the parametric action on the occasion that the Hamiltonian is small but finite and how it adjusts to the probability density function (pdf). (iii) Improves instanton approximation to pdf by noise and propagator renormalization. As an application of the above set up we predict the statistics of passive scalar gradients in a Lagrangian model for turbulence, namely the scalar gradient Recent Fluid Deformation Closure (sgRFD).