An equivalence of moment closure and nonlinear variational approximation of the Fokker-Planck equation for dilute polymeric flow

TL;DR

This paper rigorously proves the equivalence between classical moment closure and a nonlinear variational approximation of the Fokker-Planck equation in dilute polymeric flow, providing a foundation for systematic reduced models.

Contribution

It establishes the equivalence in the linearized setting and introduces a variational framework for developing reduced models for nonlinear polymeric flows.

Findings

Equivalence between moment closure and variational approximation in linearized flow

Exact evolution of the conformation tensor via invariance of the Gaussian manifold

Framework for error analysis and systematic model reduction

Abstract

We establish rigorously the equivalence between classical moment closure and a nonlinear variational approximation of the Fokker-Planck equation for dilute polymeric flow in the linearized Hookean spring chain setting. The variational formulation is based on the Dirac-Frankel principle applied to a Gaussian approximation manifold endowed with the Fisher-Rao information metric. We show that the invariance of this manifold under the linear configurational dynamics yields an exact evolution for the macroscopic conformation tensor, recovering the classical diffusive Oldroyd-B closure. While the equivalence only holds in the linearized setting, the associated variational framework provides an abstract error representation and a starting point for the systematic construction of reduced approximation schemes for polymeric flows with nonlinear forcing laws.