Three Routes to the Friction Matrix and Their Application to the Coarse‐Graining of Atomic Lattices

Kauzlarić, David; Español, Pep; Greiner, Andreas; Succi, Sauro

doi:10.1002/mats.201100014

Cited by 24 publications

(15 citation statements)

References 35 publications

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“…We immediately see that the last term gives rise to an approximation to the memory term, with memory kernel exactly given by (D.4), which as explained at the beginning of this section, correspond to the rational approximation of the Laplace transform (30). In addition, the first two terms form a stationary Gaussian process, denoted by g (t ), since the Lyapunov condition has been imposed.…”

Section: Appendix C: Properties Of the Memory Kernelmentioning

confidence: 82%

“…This approach is directly based on the dynamics of the full system, rather than the equilibrium statistical properties. Such formalism (or similar reduction procedure) has recently been widely used to derive CG models based on the deterministic Newton's equations of motion 8,12,28,30,31,34,39,43,48,55 , known as molecular dynamics (MD) models. The typical result is a generalized Langevin equation (GLE), with a memory (or frictional) kernel implicitly incorporating the influences of the degrees of freedom that have been projected out.…”

Section: Introductionmentioning

confidence: 99%

“…(t )d t = A 12 , Γ 12 G −1Using the form of the rational function R 2,2(30) and the properties of Laplace transform, we can write down a differential equation for the approximate memory kernel,θ ′′ = B 0 θ ′ + B 1 θ, (D.1)together with the initial conditions,θ(0) = M 0 , θ ′ (0) = M 1 (D.2)…”

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confidence: 99%

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The derivation and approximation of coarse-grained dynamics from Langevin dynamics

2016

The Journal of Chemical Physics

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We present a derivation of a coarse-grained description, in the form of a generalized Langevin equation, from the Langevin dynamics model that describes the dynamics of bio-molecules. The focus is placed on the form of the memory kernel function, the colored noise, and the second fluctuation-dissipation theorem that connects them. Also presented is a hierarchy of approximations for the memory and random noise terms, using rational approximations in the Laplace domain. These approximations offer increasing accuracy. More importantly, they eliminate the need to evaluate the integral associated with the memory term at each time step. Direct sampling of the colored noise can also be avoided within this framework. Therefore, the numerical implementation of the generalized Langevin equation is much more efficient.

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Section: Appendix C: Properties Of the Memory Kernelmentioning

confidence: 82%

Section: Introductionmentioning

confidence: 99%

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The derivation and approximation of coarse-grained dynamics from Langevin dynamics

2016

The Journal of Chemical Physics

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“…It is at this point that we recognize the similarity to an order reduction problem: The large-dimensional dynamics (13) contains an input variable u(t ), which is low-dimensional. Moreover, of direct importance to the coarse-grained dynamics (19) is Ly, which again is low-dimensional. Also observed, however, is that the dimensions of L and R are different.…”

Section: A Reformulation Of the Orthogonal Dynamicsmentioning

confidence: 99%

Coarse-graining Langevin dynamics using reduced-order techniques

2019

Journal of Computational Physics

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This paper considers the reduction of the Langevin equation arising from bio-molecular models. To facilitate the construction and implementation of the reduced models, the problem is formulated as a reduced-order modeling problem. The reduced models can then be directly obtained from a Galerkin projection to appropriately defined Krylov subspaces. The equivalence to a moment-matching procedure, previously implemented in , 2), is proved. A particular emphasis is placed on the reduction of the stochastic noise, which is absent in many order-reduction problems. In particular, for order less than six we can show the reduced model obtained from the subspace projection automatically satisfies the fluctuation-dissipation theorem. Details for the implementations, including a bi-orthogonalization procedure and the minimization of the number of matrix multiplications, will be discussed as well.There are multiple benefits from such an approach. For example, reduced models can capture directly the dynamics of certain quantities of interest. Secondly, with the reduction of the dimension, the computational cost can be reduced dramatically. In addition, the quantities of interest often correspond to slow variables. By eliminating fast variables, the time step can also be increased considerably. This allows one to access longer time scales ?.There has been tremendous recent progress in the development of coarse-grained models , 7, 9, 1, 3, 7,?, 5); , 0). Most effort, however, is thermodynamics based. Namely, one aims to construct the free energy associated with the reduced variables, which then yields the driving force for the reduced dynamics, known as the potential of mean forces (PMF) , 3, 4). As pointed out in , 3, 4), the damping mechanics, which also plays an important role in the reduced dynamics, is not part of the construction.In this work, we are interested in an equation-based derivation, where the reduced model can be derived directly from the Langevin dynamics. Deriving reduced models from a stochastic dynamical system has been a subject of extensive studies, the most well known of which is the homogenization approach , 9). Another important approach is to employ a coordinate transformation using normal forms to separate out the degrees of freedom that are less relevant , 3). More recently, Legoll and Lelievre proposed to use conditional expectations to derived reduced models , 5). Overall, these methods require either significant scale separation assumption, or simple functions forms in the stochastic differential equations, which for bi-molecular models, does not apply. For example, the force-field for biomolecular models typically involves complicated function forms.Meanwhile, in the field of molecular modeling there are also many methods that were proposed to coarse-grain a molecular dynamics model. Most of these methods are derived from a Hamiltonian system of ODEs ,9,1,7,8,4); ; ; , either motivated by or directly obtained, from the Mori-Zwanzig projection formalism , 5, 9). Strictly speaking, such a procedure...

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“…Early work was based on top-down CG CNT-models assuming a priori the DPD-form of the equations of motion for centre of mass (COM) variables of groups of carbon atoms [23,24]. In a systematic bottom-up approach for two-dimensional (2D) motion of graphene, a Markovian approximation was applied to GLEs obtained from Mori's or Zwanzig's projection operators [25][26][27]. COM-based relevant variables seemed to show not only less non-Markovian features than finite element based variables but also showed weaknesses in the reproduction of cross-correlation functions of the relevant variables or long-wavelength normal mode decay, while the reproduction of autocorrelation functions is nearly perfect.…”

Section: Introductionmentioning

confidence: 99%

Markovian equations of motion for non-Markovian coarse-graining and properties for graphene blobs

et al. 2013

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We obtain Markovian equations of motion for a many body system of interacting coarse-grained (CG) variables and additional fluxes. The investigated CG variables belong to the special family of linear combinations of atomistic degrees of freedom. The system of Markovian equations of motion approximates Mori's exact non-Markovian generalized Langevin equation and is easy to solve by computer simulation. All parameters of the equations can be obtained from equilibrium molecular dynamics simulations of the investigated microscopic system. These parameters are either equal to the famous static covariances from Mori's continued fraction or they represent generalized constant friction matrices. We propose two different ways to compute these friction matrices based on Mori's continued fraction. Finally, some of the parameters are computed numerically for the special case of centre of mass 6

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Three Routes to the Friction Matrix and Their Application to the Coarse‐Graining of Atomic Lattices

Cited by 24 publications

References 35 publications

The derivation and approximation of coarse-grained dynamics from Langevin dynamics

The derivation and approximation of coarse-grained dynamics from Langevin dynamics

Coarse-graining Langevin dynamics using reduced-order techniques

Markovian equations of motion for non-Markovian coarse-graining and properties for graphene blobs

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