Anomalous Polymer Dynamics Is Non-Markovian: Memory Effects and The Generalized Langevin Equation Formulation

TL;DR

This paper demonstrates that anomalous polymer dynamics are inherently non-Markovian and can be effectively modeled using a unified Generalized Langevin Equation with power-law memory kernels, linking microscopic polymer physics to macroscopic dynamics.

Contribution

It introduces a unified GLE framework with power-law memory kernels to describe and analyze anomalous polymer dynamics, emphasizing their non-Markovian nature.

Findings

All studied polymer systems exhibit non-Markovian dynamics characterized by power-law memory kernels.

The GLE with these kernels accurately reproduces the subdiffusive behavior of tagged monomers.

Memory kernels derived from polymer physics explain the response to external fields.

Abstract

Any first course on polymer physics teaches that the dynamics of a tagged monomer of a polymer is anomalously subdiffusive, i.e., the mean-square displacement of a tagged monomer increases as $t^\alpha$ for some $\alpha<1$ until the terminal relaxation time $\tau$ of the polymer. Beyond time $\tau$ the motion of the tagged monomer becomes diffusive. Classical examples of anomalous dynamics in polymer physics are single polymeric systems, such as phantom Rouse, self-avoiding Rouse, self-avoiding Zimm, reptation, translocation through a narrow pore in a membrane, and many-polymeric systems such as polymer melts. In this pedagogical paper I report that all these instances of anomalous dynamics in polymeric systems are robustly characterized by power-law memory kernels within a {\it unified} Generalized Langevin Equation (GLE) scheme, and therefore, are non-Markovian. The exponents of the power-law memory kernels are related to the relaxation response of the polymers to local strains, and are derived from the equilibrium statistical physics of polymers. The anomalous dynamics of a tagged monomer of a polymer in these systems is then reproduced from the power-law memory kernels of the GLE via the fluctuation-dissipation theorem (FDT). Using this GLE formulation I further show that the characteristics of the drifts caused by a (weak) applied field on these polymeric systems are also obtained from the corresponding memory kernels.