Bottom-up coarse-graining of a simple graphene model: The blob picture

Kauzlarić, David; Meier, Julia; Español, Pep; Succi, Sauro; Greiner, Andreas; Korvink, Jan G.

doi:10.1063/1.3554395

Cited by 38 publications

(48 citation statements)

References 80 publications

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“…To this end, we apply the three forms to the case of a blob-based CG procedure, as applied to a simple microscopic model of graphene. [15] After a short summary of Mori-Zwanzig theory, where we work out the connection between Mori's and Zwanzig's SDE, we use Mori's theory to derive the Green-Kubo, Onsager and Einstein-Helfand routes to compute the friction matrix. Subsequently, we show that, for our specific example of coarse-grained (CG) graphene blobs, a combination of the Onsager and Einstein-Helfand routes is the best choice.…”

Section: Introductionmentioning

confidence: 99%

“…Shell's method, [14] is a powerful and elegant approach for the parameter estimation of the effective potential [14,15] and it has been recently generalized to dynamic situations. [16] Concerning the friction matrix in Zwanzig's theory, in the cases where the fluctuations of the CG variables are sufficiently small, and the equilibrium distribution function of CG variables is highly peaked, we are allowed to approximate its expression with Mori's expression for the friction matrix.…”

Section: Introductionmentioning

confidence: 99%

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

Kauzlarić

Español

Greiner

et al. 2011

Macro Theory & Simulations

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The calculation of the friction matrix in the coarse‐grained (CG) description of an atomistic system is a crucial issue, in order to properly account for the dissipative effects inherent to any reduced representation of the atomistic dynamics. Within the Mori‐Zwanzig projection operator approach to CG, there are several possibilities for the definition of the friction matrix, depending on the projector that is being used. In this paper, the connection of two of these projectors (Mori's and Zwanzig's) is discussed and the corresponding merits and limitations are analysed. Moreover, three different ways of computing the friction matrix in the Mori's framework are presented, along with a discussion of their mutual connections. By the example of CG centre of mass blob variables in the graphene lattice, it is shown that, even though the three approaches are equivalent from a theoretical viewpoint, they may differ considerably in terms of practical implementation for computer simulation purposes. In the given example, which is representative for atomic lattices, it turns out that a linear regression of the velocity–velocity correlation function, inspired by the Einstein–Helfand approach, is the least error‐prone against disturbances from optical modes. magnified image

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Kauzlarić

Español

Greiner

et al. 2011

Macro Theory & Simulations

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“…As a result, no memory term needs to be evaluated, and no colored noise needs to be sampled. The first two approximations in the rational approximation hierarchy correspond to the Markovian approximation (6,25,26) and approximation of noise by a Ornstein-Uhlenbeck process, which is the ansatz used in some previous work. 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.…”

Section: Significancementioning

confidence: 99%

Data-driven parameterization of the generalized Langevin equation

Lei

Baker

2016

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

147

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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“…The memory term often comes from a coarse-graining step, e.g., 2,[8][9][10][11]15,16,19,22,[25][26][27]29,30,32,34,40,43,46 , which has also been an recent emerging area of interest in molecular modeling. Among the many approximation schemes [6][7][8]12,20,23,32,34 for the GLE model, the BD model (3) is clearly the simplest. To understand how the BD approximation can naturally come about, we first present a derivation which eliminates momentum variables from the GLE, so that the resulting model only involves the coordinates of the molecular variables.…”

Section: Introductionmentioning

confidence: 99%

From generalized Langevin equations to Brownian dynamics and embedded Brownian dynamics

2016

The Journal of Chemical Physics

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We present the reduction of generalized Langevin equations to a coordinateonly stochastic model, which in its exact form, involves a forcing term with memory and a general Gaussian noise. It will be shown that a similar fluctuation-dissipation theorem still holds at this level. We study the approximation by the typical Brownian dynamics as a first approximation. Our numerical test indicates how the intrinsic frequency of the kernel function influences the accuracy of this approximation. In the case when such an approximate is inadequate, further approximations can be derived by embedding the nonlocal model into an extended dynamics without memory. By imposing noises in the auxiliary variables, we show how the second fluctuation-dissipation theorem is still exactly satisfied.

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Bottom-up coarse-graining of a simple graphene model: The blob picture

Cited by 38 publications

References 80 publications

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

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

Data-driven parameterization of the generalized Langevin equation

From generalized Langevin equations to Brownian dynamics and embedded Brownian dynamics

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