Internal noise-driven generalized Langevin equation from a nonlocal continuum model

Abstract

Starting with a micropolar formulation, known to account for nonlocal microstructural effects at the continuum level, a generalized Langevin equation (GLE) for a particle, describing the predominant motion of a localized region through a single displacement degree-of-freedom (DOF), is derived. The GLE features a memory dependent multiplicative or internal noise, which appears upon recognising that the micro-rotation variables possess randomness owing to an uncertainty principle. Unlike its classical version, the new GLE qualitatively reproduces the experimentally measured fluctuations in the steady-state mean square displacement of scattering centers in a polyvinyl alcohol slab. The origin of the fluctuations is traced to nonlocal spatial interactions within the continuum. A constraint equation, similar to a fluctuation dissipation theorem (FDT), is shown to statistically relate the internal noise to the other parameters in the GLE.PACS numbers: 46.05.+b, 05.40.Ca, 78.35.+c As the space/time length scale in the forcing or the triggered deformation mechanism becomes comparable with the internal length scale of the material, the system response at the macro-continuum scale is significantly influenced by the material microstructure. This renders the classical continuum hypothesis of strictly local interactions, or its possible extension using adiabatic continuation arguments, untenable in the modeling of such response. Non-local modeling techniques such as micropolar [1], micromorphic [1] or gradient [2] theories, which aim at incorporating long-range inter-particle interactions, bring forth microstructural effects by introducing material length scales in the constitutive formulation. In materials like polymers, granular solids etc. where the length scale is of the macroscopic order, long-range effects could predominate and, in such cases, predictions through nonlocal models, unlike the the classical continuum model, are in closer conformity with experimental observations. If a continuum model can be replaced by a collection of harmonic oscillators and the focus is on the motion of a small region represented by a particle (an oscillator) with a single predominant translational degree-of-freedom (DOF), one arrives at a generalized Langevin equation (GLE) for the DOF after including the coupling effects from the neighboring oscillators [3]. The GLE, an expedient modeling tool that replaces the infinite dimensional continuum, is widely used in areas such as soft condensed matter physics and cell biology. Despite the correspondence as above between the GLE and the continuum, standard forms of the GLE do not include length scale information characteristic of nonlocal continuum theories and are therefore not equipped to describe the physically relevant microstructural effects.This work is partly motivated by the experimental data shown in Fig.1 (adapted from Fig. 6 of [4]). These correspond to the mean square displacement (MSD) plots of temperature-induced Brownian particles in a polyvinyl alcohol (PVA) slab, extracte...

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