An equivalence of moment closure and nonlinear variational approximation of the Fokker-Planck equation for dilute polymeric flow
Caroline Lasser, Stephan B. Lunowa, Barbara Wohlmuth
TL;DR
This work shows that, in the Hookean linear-chain regime, classical moment closure for dilute polymeric flows is mathematically equivalent to a nonlinear variational approximation based on a Gaussian manifold and Dirac–Frenkel projection with the Fisher–Rao metric. The invariance of the Gaussian manifold under linear configurational dynamics yields an exact evolution for the macroscopic conformation tensor, reproducing the diffusive Oldroyd–B closure and enabling an a posteriori error representation. The framework provides a systematic route to construct reduced, variationally consistent models for nonlinear polymeric forces and offers a foundation for algorithmic reduced-order schemes when exact closures are unavailable. It also clarifies the limitations of the approach for nonlinear springs and sketches directions for stable numerical schemes and deeper micro–macro coupling analyses.
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.