The Hartree approximation in dynamics of polymeric manifolds in the melt

TL;DR

This paper applies the Hartree approximation within the Martin-Siggia-Rose framework to derive a generalized Rouse equation for polymeric manifolds in a melt, revealing critical dimensions and subdiffusive behavior consistent with simulations.

Contribution

It introduces a selfconsistent Hartree approximation to analyze the dynamics of polymeric manifolds, deriving a generalized Rouse equation and identifying critical dimensions for different regimes.

Findings

Static upper critical dimension: 2D/(2-D).

Dynamical critical dimension: twice the static one.

Rouse mode correlation function exhibits stretched exponential decay.

Abstract

The Martin-Siggia-Rose (MSR) functional integral technique is applied to the dynamics of a D - dimensional manifold in a melt of similar manifolds. The integration over the collective variables of the melt can be simply implemented in the framework of the dynamical random phase approximation (RPA). The resulting effective action functional of the test manifold is treated by making use of the selfconsistent Hartree approximation. As an outcome the generalized Rouse equation (GRE) of the test manifold is derived and its static and dynamic properties are studied. It was found that the static upper critical dimension, $d_{\rm uc}=2D/(2-D)$, discriminates between Gaussian (or screened) and non-Gaussian regimes, whereas its dynamical counterpart, ${\tilde d}_{uc}=2d_{\rm uc}$, distinguishes between the simple Rouse and the renormalized Rouse behavior. We have argued that the Rouse mode correlation function has a stretched exponential form. The subdiffusional exponents for this regime are calculated explicitly. The special case of linear chains, D=1, shows good agreement with MD- and MC-simulations.