Numerical integration of the extended variable generalized Langevin equation with a positive Prony representable memory kernel

Baczewski, Andrew David; Bond, Stephen D.

doi:10.1063/1.4815917

Cited by 75 publications

(116 citation statements)

References 47 publications

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“…In both the simulations Prony series approximations are adopted for characterizing the memory kernels and the non-Markovian SDEs are converted to a set of Markovian SDEs [12]. From the nonlocal interaction effects, as anticipated, the fluctuations in the steady-state regime are indeed captured in the simulation via the proposed GLE (Fig.2), which parallels the experimental observation (Fig.1).…”

supporting

confidence: 58%

“…Neglecting contributions from tr ∇uR T ∇uR T and tr(RR) T , the discrete Hamiltonian finally takes the following form. The governing dynamics for the system and bath variables, described through Hamilton's equations in terms of the displacement DOFs and the momenta, are given by the equation pairs (11,12) and (13,14) …”

Section: Formulation Of Discrete Hamiltonianmentioning

confidence: 99%

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

et al. 2015

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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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confidence: 58%

Section: Formulation Of Discrete Hamiltonianmentioning

confidence: 99%

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

et al. 2015

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“…The general advantage of the stochastic GLE is that dissipation and the statistical properties of the noise are entirely described by the so-called memory kernel being simply a function of time. If such a memory kernel can be obtained for a real system then the full quantum-mechanical treatment of the bath can be performed analytically, leading to a quantum version of the GLE [28][29][30][31][32] [39,40]. Therefore the GLE formalism has become a popular tool for assigning system properties in macroscopic environments.…”

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confidence: 99%

Parametrizing linear generalized Langevin dynamics from explicit molecular dynamics simulations

Gottwald

Karsten

Ivanov

et al. 2015

The Journal of Chemical Physics

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Fundamental understanding of complex dynamics in many-particle systems on the atomistic level is of utmost importance. Often the systems of interest are of macroscopic size but can be partitioned into few important degrees of freedom which are treated most accurately and others which constitute a thermal bath. Particular attention in this respect attracts the linear generalized Langevin equation (GLE), which can be rigorously derived by means of a linear projection (LP) technique. Within this framework a complicated interaction with the bath can be reduced to a single memory kernel. This memory kernel in turn is parametrized for a particular system studied, usually by means of time-domain methods based on explicit molecular dynamics data. Here we discuss that this task is most naturally achieved in frequency domain and develop a Fourier-based parametrization method that outperforms its time-domain analogues. Very surprisingly, the widely used rigid bond method turns out to be inappropriate in general. Importantly, we show that the rigid bond approach leads to a systematic underestimation of relaxation times, unless the system under study consists of a harmonic bath bi-linearly coupled to the relevant degrees of freedom. INTRODUCTIONStudying complex dynamics of many-particle systems has become one of the main goals in modern molecular physics. The fundamental understanding of the underlying microscopical processes requires the interplay of elaborate experimental techniques and sophisticated theoretical approaches. Experimentally, (non-)linear spectroscopy revealed itself as a powerful tool for probing the dynamics and for determining the characteristic timescales, such as dephasing/relaxation times and reaction rates to name but two. For interpreting the experimental spectra theoretical models are needed which can give insight into the atomistic dynamics. Often, a reduction of the description to few variables is convenient in many cases since this can not only ease the interpretation, but enable the identification of key properties [1]. Such a reduced description can formally be obtained from the so-called system-bath partitioning, where only a small subset of degrees of freedom (DOFs), referred to as system, is considered as important for describing a physical process under study. All the other DOFs, referred to as bath, are regarded as irrelevant in the sense that they might influence the time evolution of the system but do not explicitly enter any dynamical variable of interest. Practically, such a separation is often natural, for instance, when studying a reaction with a clearly defined reaction center or solute dynamics in a solvent environment. Further, reduced equations of motion (EOMs) for the system DOFs can be derived in which the influence of the bath is limited to dissipation and fluctuations.The most simple formulation of this idea is provided by the Markovian Langevin equation, where dissipation and fluctuations take the form of static friction and stochastic white noise, respectively [2][3][4]. Sit...

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“…As we will show, these two approximations are often insufficient to predict dynamics properties; however, our hierarchy can be used to construct arbitrarily high-order models to characterize long-time behaviors and complex transition dynamics. The technique of using auxiliary variables has been demonstrated by others (27)(28)(29) to treat the memory term in the GLE. However, the current work makes a direct connection between the parameters in the extended system and the statistics of the CG variables.…”

Section: Significancementioning

confidence: 99%

Data-driven parameterization of the generalized Langevin equation

Lei

Baker

2016

Proc. Natl. Acad. Sci. U.S.A.

148

178

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We present a data-driven approach to determine the memory kernel and random noise in generalized Langevin equations. To facilitate practical implementations, we parameterize the kernel function in the Laplace domain by a rational function, with coefficients directly linked to the equilibrium statistics of the coarse-grain variables. We show that such an approximation can be constructed to arbitrarily high order and the resulting generalized Langevin dynamics can be embedded in an extended stochastic model without explicit memory. We demonstrate how to introduce the stochastic noise so that the second fluctuationdissipation theorem is exactly satisfied. Results from several numerical tests are presented to demonstrate the effectiveness of the proposed method.generalized Langevin dynamics | data-driven parameterization | coarse-grained molecular models | reaction rate | model reduction G eneralized Langevin equations (GLEs) have recently reemerged in the area of molecular modeling as a promising description of reduced-dimension coarse-grained variables. In principle, GLEs can be derived using the Mori-Zwanzig projection formalism (1, 2). Examples of such derivations can be found for a variety of applications (3-10), for example, climate modeling (11, 12). The GLE approach eliminates a large number of irrelevant degrees of freedom, reducing system dimensions to make direct computation feasible. Because this elimination often projects out high-frequency modes, the GLE can also extend the time scale of simulations. The GLE does this by describing the dynamics for explicit quantities of interest and implicitly describing the remaining degrees of freedom through a memory term and a random noise term. The random noise term is often strongly correlated in time.However, practical implementations of GLEs require specification of the memory function, which can be difficult to obtain, even when the full dynamics of the system is known. For example, the memory functions obtained in past studies (8,9,13,14) have involved functions of high-dimensional matrices. Darve et al. (14) proposed a more efficient algorithm to compute the memory kernel by solving an equation for the orthogonal dynamics derived from the Mori-Zwanzig formalism. However, the orthogonal dynamics equation can be expensive to solve when the original system is large. Furthermore, even when the memory kernel function is available, direct evaluation of the memory term can be costly because it requires the history of the coarse-grained (CG) variables at every time step and the associated numerical integration. Sampling of the random noise is also a challenging component of GLEs: To generate the correct equilibrium statistics for the CG model, the random noise has to obey the second fluctuation-dissipation theorem (FDT) (15). The theory of stationary processes (16) states that the random process is uniquely determined by the correlation function, which is proportional to the memory kernel; however, sampling the random noise is nontrivial in practice. Methods based ...

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Numerical integration of the extended variable generalized Langevin equation with a positive Prony representable memory kernel

Cited by 75 publications

References 47 publications

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

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

Parametrizing linear generalized Langevin dynamics from explicit molecular dynamics simulations

Data-driven parameterization of the generalized Langevin equation

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