Markovian dissipative coarse grained molecular dynamics for a simple 2D graphene model David Kauzlari, Pep Español, Andreas Greiner, and Sauro Succi Citation: The Journal of Chemical Physics 137, 234103 (2012); doi: 10.1063/1.4771656 View online: http://dx.doi.org/10.1063/1.4771656 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/137/23?ver=pdfcov Published by the AIP Publishing This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 207.237.201.241 On: Thu, 13 Mar 2014 19:02:56
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Markovian dissipative coarse grained molecular dynamics for a simple 2D graphenemodelDavid Kauzlari, Pep Español, Andreas Greiner, and Sauro Succi Citation: The Journal of Chemical Physics 137, 234103 (2012); doi: 10.1063/1.4771656 View online: http://dx.doi.org/10.1063/1.4771656 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/137/23?ver=pdfcov Published by the AIP Publishing
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THE JOURNAL OF CHEMICAL PHYSICS 137, 234103 (2012)
Markovian dissipative coarse grained molecular dynamicsfor a simple 2D graphene model
David Kauzlaric,1,2,a) Pep Español,2 Andreas Greiner,3 and Sauro Succi1,4
1School of Soft Matter Research, Freiburg Institute for Advanced Studies, University of Freiburg, Albertstr. 19,79104 Freiburg, Germany2Departamento de Física Fundamental, Universidad Nacional de Educación a Distancia (UNED),Aptdo. 60141 E-28080, Madrid, Spain3Laboratory for Simulation, Department of Microsystems Engineering (IMTEK), University of Freiburg,Georges-Köhler-Allee 103, 79110 Freiburg, Germany4Istituto Applicazioni Calcolo, CNR, via dei Taurini 9, 00185, Roma, Italy
(Received 10 October 2012; accepted 28 November 2012; published online 19 December 2012)
Graphene1 and related macromolecules represent a newclass of fascinating materials not only from a quantum butalso from a classical mechanical point of view due to, e.g.,an extraordinarily high elastic modulus or thermal conductiv-ity. The next generation of micro devices might be composedof just one molecule and there are already attempts to designor investigate high frequency resonators from graphene basedmaterials such as carbon nanotubes (CNTs)2–4 or graphenenanoscrolls.5 Besides their resonance frequency, the Q-factoris a crucial characteristic quantity,6, 7 which directly translatesinto the sensitivity of sensor devices. It is finite for any non-ideal resonator and describes its quality in terms of the char-acteristic time it takes for the energy of an excited mode todecay significantly, i.e., to get redistributed into other modes.
When tracking excited normal modes in crystalline struc-tures (such as graphene) in a molecular dynamics (MD) sim-ulation a general observation is that their decay is extremelyslow as compared to the MD time scale.8, 9 This is to be ex-pected since elastic conservative forces dominate lattice dy-namics. Even though the dissipation rate (in units of a fre-quency) usually seems to be negligible as compared to the
a)Author to whom correspondence should be addressed. Electronic mail:[email protected].
oscillation frequencies, the dissipation rate is nonetheless cru-cial to describe the quality factor of a resonator quantitatively.The time scale of the dissipation makes its study on, e.g.,resonator devices with thousands to millions of atoms espe-cially difficult with MD. This situation immediately calls fora coarse-grained (CG) description, both in space and time.Previous works include a mechanical model for single-walledCNTs based on the dissipative particle dynamics (DPD)method which was developed by Liba et al.10 with device sim-ulation in mind and was used to analyze the resonance behav-ior of CNT-resonators.11 In these works, the parameters of theDPD model have been determined top-down by a calibrationto macroscopically known properties, such as the elastic mod-ulus. Buehler12 used a bottom-up approach to construct a CGmodel in the sense of fitting observables, such as fracture be-havior, that were obtained previously from MD simulations.We will use an approach similar in spirit to Buehler’s in thiswork to obtain the dissipation parameters of our CG model.
The first important step in any CG procedure is the selec-tion of the CG variables as functions of the molecular degreesof freedom. A criterion for this choice is the rather genericproperty of being “slow” as compared to the “rest” of elim-inated degrees of freedom. In previous work, we describedgraphene at a CG level in terms of center of mass (COM)variables of lumps of carbon atoms or “blobs.”13, 14 A poorreproduction of the relevant low wave number part of the
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234103-2 Kauzlaric et al. J. Chem. Phys. 137, 234103 (2012)
dispersion relation of the microscopic system by the COMvariables was observed. In addition, the Markovian DPD-model developed there shows a very strong over-estimationof the system’s dissipation which expresses itself in a decayof sound waves, which is orders of magnitude too fast. In or-der to elucidate these difficulties we use in this work a new setof CG variables. One alternative to a simple local grouping ofatoms into relevant COM variables is a more non-local finite-element (FE) based approach called coarse-grained molecu-lar dynamics (CGMD).15 By a least squares approach, onederives a scheme that in fact uses a sort of inverse of the FE-weighting functions to transfer atomic information to CGMD-nodes. It was shown that CGMD reproduces dispersion rela-tions very well.15 In its original form, CGMD only constructsan effective harmonic potential and does not consider dissipa-tive effects. Since the effective potential is harmonic, the lin-ear Mori theory16 fits well if one wishes to introduce dissipa-tive effects. While doing this, one of the questions we addressin this work is whether a dissipative CGMD (D-CGMD) alsoshows advantages with respect to the correct representation ofdissipative behavior. We present an attempt to approach thisquestion by constructing a CGMD-model for a simple micro-scopic model of graphene which is comparable with the pre-viously developed COM description.13
Kobayashi et al.17 also applied CGMD to the graphenesheet and reproduced successfully the static deflection whencompared to a full MD-simulation. Inoue et al.18 combinedthe CGMD approach with the Lattice Boltzmann and an im-mersed boundary method to investigate graphene in a flowfield. Before coupling the molecule to a dissipative environ-ment, we wish to understand in this work how it is possibleto describe on a CG level the dissipation emerging from theinternal dynamics of the molecule, in particular, the phonon-mediated dissipation detectable by MD-simulation, which isof major importance for nanomechanical devices and greatlyinfluences the Q-factor of molecular resonators.6, 7 With a dif-ferent motivation, which is the elimination of wave reflectionsat boundaries between different levels of coarse-graining,there have already been attempts to introduce non-Markoviandissipative effects into CGMD.19, 20
First, we present the underlying microscopic graphenemodel in Sec. II which we use for the computation of theCGMD stiffness and friction matrices. In Sec. III, we thenfirst define our CGMD variables in terms of the mesh andthe coarse-graining number in such a way that the CGMD-nodes correspond to the COMs used in Ref. 13. Afterwards,the equations of motion of the CGMD-variables including dis-sipative effects are derived from Mori-theory, in their non-Markovian exact and Markovian approximate forms.
In Sec. IV, we find that the Markovian approximationwhich we were able to use for COM-variables13 is not justi-fied for CGMD-variables. This can be traced back to the factthat the CGMD-weighting function suppresses more stronglyhigh-frequency modes than a COM based CG procedure. Re-garding our original question, this result gives the partial an-swer that the choice of CGMD variables makes the construc-tion of their dissipative dynamics harder, because in a strictsense, the simpler Markovian models are excluded and non-Markovian models must be used instead.
From the point of view of a practical tool for device sim-ulation a non-Markovian model is cumbersome and we askinstead whether we can at least construct a Markovian D-CGMD model that correctly reproduces the effective soundattenuation of low wave number modes of the microscopicMD-system. The requirement of the term “effective” will be-come clear later where it becomes obvious that the true dis-sipative dynamics of the modes is more complex than just anexponential decay as enforced by a linear Markovian model.Hence, we attempt to construct a model that predicts onlythe effective decay of the modes neglecting their additionalmore detailed dynamics requiring the consideration of mem-ory terms.
In Sec. V, we show how to extract from MD simula-tions the parameters of the D-CGMD model. The requiredinverse of the singular displacement covariance matrix occur-ring in a system with periodic boundary conditions is shownto be computable as a pseudoinverse. Then we describe an ap-proach that, by construction, reproduces correctly the dissipa-tive decay of excited normal modes of the system as computedfrom MD within an exponential approximation by construct-ing a friction matrix in the eigenmode space and a compatiblenoise term.
Simulation results are presented in Sec. VI. First, correla-tion functions of the relevant variables as computed from MDand D-CGMD are compared. It turns out that CGMD is lessdissipative than COM-variables, which is a desirable feature.This is reflected by a smaller deviation of CGMD’s conser-vative part from the microscopic dynamics of the relevant CGvariables in terms of the variables’ correlation functions. Thenit is verified that D-CGMD reproduces within an exponentialapproximation, the decay of important modes of the investi-gated system.
II. MICROSCOPIC MODEL
We consider here the in-plane 2D motion of the graphenehoneycomb lattice (cf. Fig. 1). The dynamics of the graphenemodel is described by Hamilton’s equations of motion for theset of microscopic variables {ri , pi}, where mi is the mass,ri is the position, and pi is the momentum of C-atom i. Forthe sake of simplicity, instead of more realistic potentials suchas the Brenner bond order potential,21 we use a simple two-dimensional potential of the same form as already used inRef. 13, but with slightly modified parameters to match twospeeds of sound instead of only one as shown below. The po-tential is based on pairwise and angular springs. Specifically,the force Fi on the ith C-atom is defined by
Fi =∑Pij
Fij +∑Tijk
Fijk, (1)
where Pij denotes all pairs of C-atoms i and j which arebonded in the graphene sheet and Tijk denotes all triplets ofparticles {i, j, k} which are connected by two bonds. The pairforce is defined as
Fij (rij ) = −ks(rij − r0)eij , (2)
where ks > 0 is a stiffness constant, r0 is the equilibriumbond-length, rij = |rij |, eij = rij /rij , and rij = ri − rj . The
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234103-3 Kauzlaric et al. J. Chem. Phys. 137, 234103 (2012)
x
y
FIG. 1. Usage of triangular elements for the CG of the 2D graphene lat-tice. The elements of the regular grid coincide with the COM-coordinatesof hexagonal blobs from Ref. 13 which are also shown. The figure showsnodes/bobs with a coarse-graining number Ncg = 24 as example. The rectan-gles containing two nodes show the primitive unit-cell when using rectangu-lar periodic boundary conditions.
angular force is Fijk(α) = −∇riU (αijk) where the potential
energy is defined as
U (αijk) = 1
2ka(cos αijk − cos α0)2 (3)
with the equilibrium angle α0 = 2π /3 and
cos(π − αijk) = rij · rjk
rij rjk
. (4)
Here, ka is a second stiffness constant for angular changes.Since we are mainly interested in the CG of finite-sized
and relatively small systems, all MD-simulations will be per-formed on a periodic graphene sheet of NC = 10 368 atoms.
The unit of length l* is fixed by setting l∗ ≡ r0
≈ 0.142 nm in graphene. The unit of mass is m* ≡ mC wheremC ≈ 2 × 10−26 kg is the mass of one C-atom. We choosethe unit of time t* ≡ (mC/ks)1/2, which amounts to set thevalue of ks = 1 in the selected units. The angular spring con-stant ka = 0.12 is found by requiring the model to reproducethe ratio cT/cL of the transversal speed of sound cT and lon-gitudinal speed of sound cL in real graphene. The in-planedensity ρ ≈ 7.61 × 10−7 kg/m2, the in-plane Young modulusY = 342 N/m measured in Ref. 22, and the Poisson ratio ν
= 0.165 known from Ref. 23 allow to obtain this ratio throughthe relations
cL =√
Y (1 − ν)
ρ(1 + ν)(1 − 2ν), (5)
cT =√
Y
2ρ(1 + ν), (6)
which gives cT/cL ≈ 0.633. The measured Young modulus isan average and isotropic quantity.22 As shown below, also oursimple graphene model produces speeds of sound which are
TABLE I. Computed frequencies ω and sound speeds c of longitudinal (L)and transverse (T) waves for the given wavelengths λx and λy, respectively.The given wavelengths are equal to the dimensions of the simulation domaincontaining NC = 10 368 atoms. The numbers in brackets denote the error ofthe last decimal place.
Wavelength [l*] Frequency [1/t*] Speed of sound [l*/t*]
isotropic and hence, for our model, the isotropic informationis sufficient.
We compute the frequencies ωL(λ) and ωT(λ) and the cor-responding speeds of sound cL(λ) and cT(λ) of a longitudinalwave and a transverse wave, respectively. For the computa-tion we choose the largest wavelength fitting into the periodicsimulation domain, i.e., λx = 2π /Lx and λy = 2π /Ly whereLx and Ly are the two domain sizes in x-direction and in y-direction, respectively (cf. Fig. 1 for the orientation of theaxes). We have Lx �= Ly because the honeycomb lattice doesnot allow for square shaped periodic simulation domains. Theresulting frequencies and speeds of sound are given in Table I.Within our numerical accuracy of around 1%, we do not ob-serve any anisotropy for the two speeds of sound, at least forthe ratio of force constants ka/ks = 0.12 reproducing the ratiocT/cL ≈ 0.633 of real graphene.
From the longitudinal speed of sound cL ≈ 0.80l*/t* (cf.Table I), we can estimate the unit of time t* by setting thenumerical sound speed equal to its experimental value. Anin-plane sound speed of cL ≈ 21 930 m/s can be determinedfrom the in-plane elastic constants for graphene, as experi-mentally obtained by Lee et al.22 For our selected unit oftime, this yields t∗ ≈ 5.18 fs. We measure entropy (or heatcapacity) in units of the Boltzmann constant, so that the unitof temperature is T ∗ ≡ mCr2
0 /kBt∗2 ∼ 1.09 × 106 K. MD-simulations are performed in the (N,V,E)-ensemble at a tem-perature of T = 2.74 × 10−4 (corresponding to T ≈ 298 K).Typically, the system is equilibrated for a sufficiently longtime, of the order of 10t* and data are collected subsequently.The used time step in all MD-simulations is �t = 0.005t*.
III. CGMD MODEL FOR 2D-GRAPHENE
A. Mesh and shape functions
The first step is to choose a suitable set of elements andcorresponding shape functions. The NC = 10 368 atoms willbe coarse-grained to Nn = 108 nodes, i.e., we can define acoarse-graining number Ncg ≡ Nc/Nn = 96. Figure 1 showsa triangular grid, where, for a regular mesh, the nodes co-incide (in equilibrium) with the COMs of the blobs fromRef. 13. The shape functions of the triangles are linear func-tions of the form
Nμ(r) = Aμ + Bμx + Cμy (7)
defined on each node μ where r ≡ {x, y} and {Aμ, Bμ, Cμ}are coefficients depending on the specific triangle. The major
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234103-4 Kauzlaric et al. J. Chem. Phys. 137, 234103 (2012)
advantage of the triangular elements is that they allow for adirect comparison of the results with those from the blob-CGperformed in Ref. 13.
For general triangles we obtain the coefficients {Aμ, Bμ,Cμ} of the shape functions of nodes i from⎛
⎝A1 B1 C1
A2 B2 C2
A3 B3 C3
⎞⎠ = A−1 ≡
⎛⎝ 1 1 1
x1 x2 x3
y1 y2 y3
⎞⎠
−1
= 1
2A
⎛⎜⎝
x2y3 − x3y2 y2 − y3 x3 − x2
x3y1 − x1y3 y3 − y1 x1 − x3
x1y2 − x2y1 y1 − y2 x2 − x1
⎞⎟⎠ , (8)
where A ≡ (det A)/2 is the area of the triangle.Using periodic boundary conditions, we require rectan-
gular primitive unit cells containing two nodes as indicated inFig. 1. This leads to optical modes on the CG-level, which,microscopically, are in fact associated with acoustic modes.
B. Relevant CGMD variables
The coarse-graining approach considered here is basedon the displacement variables Xμ of the CGMD-nodes whereeach node will be identified by a Greek index μ. In termsof the microscopic variables, the nodal displacement and thevelocity are defined as follows:15
Xμ ≡∑
i
fμixi , (9)
Vμ ≡ Xμ =∑
i
fμi xi =∑
i
fμivi , (10)
fμi =∑
ν
N−1μν Nνi, (11)
Nμν ≡∑
i
NμiNiν, (12)
Nμi ≡ Nμ(ri0) (13)
with the atomic equilibrium positions ri0 and displacementsxi ≡ ri − ri0 of atom i. The inverse in Eq. (11) denotes amatrix inverse and the shape function Nμ(r) was defined inEq. (7). It can be shown that∑
i
fμi = 1,∑
μ
fμi = 1/Ncg (14)
for any μ or i, respectively, and for any Ncg. The second equa-tion shows that fμi does not have a partition of unity property.For Eq. (9) this is of no concern, because the displacementand the velocity are intensive variables.
We will also require the definition of a CG momentumPμ and its associated CG force Fμ. These quantities will berequired within the Mori-formalism but not in the final CGmodel. As will be seen later, even though the standard formof the CGMD-equations that we will finally use only requiresthe definitions (9) and (10), the CGMD-equations can be ex-pressed in terms of Pμ and Fμ, demonstrating the compatibil-ity of CGMD with the predictions of the Mori-formalism. To
achieve this we define the momentum consistently within theCGMD-approach as
Pμ ≡∑
i
f ′νipi =
∑i
Ncgfνimi xi ≡ MXμ, (15)
Fμ ≡ Pμ =∑
i
f ′νi pi =
∑i
f ′νiFi , (16)
where the last equality in Eq. (15) holds for our monoatomiccase with atomic masses m and defines the mesoscopic mo-mentum as the mesoscopic velocity times a mesoscopic massM ≡ mNcg. For this definition we require a scaled weightingfunction f ′
μi = Ncgfμi (which amounts to new shape func-tions N ′
μi = Nμi/Ncg). Now, f ′μi is a partition of unity, but
N ′μi is not in order to fulfill∑
i
pi =∑
μ
Pμ (17)
for the momentum, which is an extensive variable.
C. Mori theory for CGMD
The decay of an excited normal mode of a resonator thatwe intuitively describe with the term dissipation is in fact,from a classical microscopic point of view, the consequenceof conservative atomic interactions through a nonlinear poten-tial energy function. For the decay of excited normal modestowards a thermal equilibrium, nonlinearity is thereby neces-sary but not sufficient.24 Dissipation is just one way to de-scribe the nonlinear effect on a CG level of description withrelevant variables. This is, in particular, the route of choicewhen using harmonic near-equilibrium descriptions for ef-fective CG potentials such as in the Mori projection opera-tor theory16 which enforces us to describe all additional ef-fects by a dissipative term, possibly containing a memory.Alternatively we might transfer some of the nonlinear effectsinto nonlinearities of a CG model potential.25 Replacing Moriprojection operators by those of Zwanzig26 leads to rigorousequations for CG variables allowing for nonlinear terms in theCG potential (and also in the dissipative terms). This consid-eration of nonlinear terms in principle allows the applicationof Zwanzig theory for problems far from equilibrium. Here,we will assume near-equilibrium situations only and hence weuse Mori theory in the following.
For arbitrary relevant variables Aμ(t) obeying 〈Aμ(t)〉 = 0we obtain from Mori theory16 the exact equations of motion
Aμ(t) = μνAν(t) −∫ t
0Gμν(t − t ′)Aν(t ′)dt ′ + Fμ(t).
(18)
Here and in the following repeated indices are summed.The different elements of this generalized Langevin equation(GLE) require the definition of an inner product 〈φ, ψ〉. Weuse Mori’s definition as an equilibrium average16
〈φ,ψ〉 ≡∫
dzρeq(z)φ(z)ψ(z) (19)
with the equilibrium ensemble
ρeq = 1
Zexp[−βH (z)] (20)
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234103-5 Kauzlaric et al. J. Chem. Phys. 137, 234103 (2012)
microscopic variables z, the partition function Z, β = 1/kBT,and the Hamiltonian H(z).
In Eq. (18) the drift , the memory kernel G(t) and therandom force Fμ(t) are given by
μν = 〈Aν,Aν ′ 〉−1⟨Aν ′ , iLAμ
⟩, (21)
Gμν(t) = 〈Aν,Aν ′ 〉−1⟨Fν ′(0), Fμ(t)
⟩, (22)
Fμ(t) = Q exp(iLQt)iLAμ. (23)
L is the Liouville operator associated with the HamiltonianH(z). Q is a projection operator projecting onto the space or-thogonal to the space spanned by the relevant variables A(z).It is defined by Q ≡ 1 − P where P is a projection operatorprojecting any phase function ψ onto the space of relevantvariables according to
Pψ = Aμ〈Aμ,Aν〉−1〈Aν,ψ〉. (24)
The random force Fμ(t) satisfies
〈Fμ(t)〉 = 0, (25)
〈AνFμ(t)〉 = 0, (26)
〈Fν ′ Fμ(t)〉 = Kμν(t)(Aν,Aν ′ ) (27)
for t ≥ 0.Before we proceed a short remark on the notation: In all
what follows, we will avoid to denote the contraction of aCartesian index by a dot “ · ”, unless it is the only contraction,i.e., e.g.,
ri · rj ≡∑
α
riαrj
α ,
�μνXν ≡∑νβ
μανβ Xνβ,
Mμν ′Gν ′ν ≡∑ν ′β ′
Mμα
ν ′β ′Gνβ
ν ′β ′ .
Let us now particularize Mori’s GLE (18) for the CGMD-variables Aμ(z) ≡ {Xμ, Pμ} as defined in (9) and (15). By in-serting those definitions we formally get
Aμ(t) = �μνAν(t) −∫ t
0Gμν(t − t ′)Aν(t ′)dt ′ + Fμ(t) (28)
with
�μν ≡[
�XXμν �XP
μν
�PXμν �PP
μν
], (29)
Gμν(t) ≡[
GXXμν (t) GXP
μν (t)
GPXμν (t) GPP
μν(t)
], (30)
Fμ(t) ≡ [FX
μ(t) FPμ(t)
]T. (31)
From (15) we directly get Xμ = Pμ/M which means that thistime derivative is proportional to a relevant variable. Since FX
μ
contains only the contributions from the irrelevant variableswe have FX
μ = 0μ and as a consequence GXXμν (t) = GXP
μν (t)= GPX
μν (t) = 0μν . This also fixes �XXμν Xν = 0μ and �XP
μν Pν
= Pμ/M . The equation of motion for the CG momentum re-quires
for t ≥ t′ where Eq. (33) was used again. Note that the mem-ory kernel contains a coupling of projected force correlations〈FP
ν ′ FPμ(t − t ′)〉 of different pairs of CGMD-nodes ν ′, μ by
the CGMD mass matrix Mνν ′ . Since CGMD is based on localnearest-neighbor finite elements, the mass matrix is local andnearest-neighbor as well (also cf. Sec. V A).
In summary the exact CGMD equations of motion are
Xμ = Pμ/M
Pμ = �PXμν Xν
−∫ t
0GPP
μν(t − t ′)Pν(t ′)dt ′ + FPμ(t), (37)
where �PXμν is given by (32) and GPP
μν(t − t ′) is given by (36).
D. Markovian dissipative CGMD
The Markovian approximation of Eq. (37) is
Pμ(t) = − MM−1μν ′Kν ′νXν(t) − Gμν ′Pν(t) + FP
μ(t) (38)
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234103-6 Kauzlaric et al. J. Chem. Phys. 137, 234103 (2012)
containing the approximation GPPμν(t − t ′) ≈ Gμνδ
+(t − t ′)where the right hand Dirac delta function has the property27∫ ∞
0δ+(t)dt = 1 (39)
and where the friction matrix Gμν obeys
Gμν = 1
M2kBT
(∫ ∞
0
⟨FP
μ(t)FPν ′⟩dt
)Mν ′ν . (40)
Equation (38) may also be expressed in a form more commonfor CGMD,
MμνXν(t) = −KμνXν(t) − �μνXν(t) + Fμ(t), (41)
where Eq. (15) was used, where �μν is a still to be determinedfriction matrix with
�μν ≡ Mμν ′Gν ′ν (42)
and Fμ(t) ≡ MμνFPν (t)/M . The force Fμ(t) containing linear
combinations of entries of a stochastic force is hence again astochastic force and it follows from (36) and (42) that it obeysthe fluctuation-dissipation theorem⟨
Fμ(t)Fν(t ′)⟩ = 2kBT �μνδ(t ′ − t). (43)
Multiplying (41) from the left with M−1μν and introducing new
expressions for the effective matrices and the noise vector bypriming the old expressions we get
Xμ(t) = −K′μνXν(t) − �′
μνXν(t) + F′μ(t). (44)
Equations (38), (41), and (44) are an approximationto the exact Eq. (37). The main approximation leading toEq. (44) is the Markovian assumption.26, 28 This approxima-tion is based on the observation that, whenever the correla-tion time of the microscopically defined projected force Fμ(t),is much smaller than the correlation time of the momentumPμ(t) then the memory kernel can be approximated as in (43).In Sec. IV, we will show that, strictly speaking, this assump-tion is violated for our microscopic graphene model.
IV. GRAPHENE CGMD-VARIABLES ARENON-MARKOVIAN
We determine the separation of time scales at the levelof description of the CGMD-nodes that will indicate whetherthe time-evolution of the CG variables is Markovian or not.A good measure for the separation of time scales is the com-parison of the decay times of the momentum auto-correlationfunction (PACF),
CPμ(t) ≡ ⟨
Pμ(t) · Pμ(0)⟩
(45)
and its (negative) second derivative, the force auto-correlationfunction (FACF),
CFμ(t) ≡ − d2
dt2
⟨Pμ(t) · Pμ(0)
⟩= ⟨
iLPμ(t) · iLPμ(0)⟩ ≡ ⟨
Fμ(t) · Fμ(0)⟩. (46)
While, strictly speaking we should be computing the correla-tion function of the projected forces entering the fluctuation-dissipation theorem (43), both correlations are similar at short
0.5
-0.5
0
0
1
100
nor
mal
ised
AC
Fs
time [t∗]
CGMD PACF for Ncg = 96CGMD FACF for Ncg = 96COM FACF for Ncg = 96
FIG. 2. The autocorrelation functions 〈Pμ(t) · Pμ〉 (PACF) and 〈Fμ(t) ·Fμ〉 (FACF) for Ncg = 96. In addition, 〈Fμ(t) · Fμ〉 is plotted for COM-variables.13 For better visibility, the maxima of all functions have beennormalized.
times, allowing to use the real CG forces in order to estimatethe correlation time of the memory.29 We have also testedthat no qualitative difference in the time scales and the over-all behavior of the FACFs can be observed whether includingthe nearest-neighbor coupling by the mass matrix as given inEq. (36) or not.
A. Correlation functions
Figure 2 plots the autocorrelations 〈Pμ(t) · Pμ〉 (PACF),〈Fμ(t) · Fμ〉 (FACF), and the numerical time derivative ofthe former giving a numerical approximation for 〈Pμ(t) · Fμ〉(PFACF), for Ncg = 96. Figure 3 shows the same for Ncg
= 24. All correlation functions display peaks after t ≈ 100(not shown in the plots) due to sound waves traveling throughthe periodic domain which are in accordance with the speedsof sound found in Table I. We can observe that for the CGMDvariables the separation of time scales seems to be much lesspronounced than for the COM-variables as used in Ref. 13(cf. Fig. 2). This can also be seen in the behavior of the
0.5
-0.5
0
0
1
25 50
nor
mal
ised
AC
Fs
time [t∗]
CGMD PACF for Ncg = 24
−d/dt CGMD PACF for Ncg = 24CGMD FACF for Ncg = 24
FIG. 3. As Fig. 2 but for Ncg = 24. Additionally, the numerical time deriva-tive of the PACF giving a numerical approximation for 〈Pμ(t) · Fμ〉 is shown.
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234103-7 Kauzlaric et al. J. Chem. Phys. 137, 234103 (2012)
function 〈Pμ(t) · Fμ〉 (cf. Fig. 3) which does not show aplateau region after its maximum. This plateau is requiredto determine unambiguous dissipation coefficients within aMarkovian approximation.30, 31 Similar observations can bemade for cross-correlation functions. It seems that CGMDshows no separation of time scales because the CGMD-force became a variable as slow as the CGMD- or COM-momentum. This means that rather than some underlying uni-versal microscopic physics, it is the coarse-graining, i.e., thechoice of variables that determines whether there is a separa-tion of time scales or not in our problem. In the following wepresent the source of non-Markovianity in CGMD.
B. fμi in Fourier space
The time scales of the CG variables and their associ-ated forces must be a result of the definition of the CG vari-ables. The essential part of the definition (15) is the weightingfunction fμi which is plotted in Fig. 4 (for simplicity) for a1D-chain of 200 equidistant atoms and three different num-bers of nodes, i.e., levels of coarse-graining Ncg. The behav-ior in 2D is qualitatively the same: the function is oscilla-tory, piecewise linear due to the underlying linear FE basisfunctions, exponentially decaying, and increasingly non-localwith increasing Ncg.
In the following, we analyze the 2D-function fμi inFourier space in comparison to the weighting function usedfor COM-variables. We expect that the CGMD-weightingfunction suppresses large wave numbers (and hence large fre-quencies) more strongly than the COM-weighting function.
This standard analysis assumes plane wave solutions inour microscopic 2D system of the form32
xj
(r0j , t
) ∼ 1√Nk
exp[ − i
(k · r0
j + ωt)]
≡ T(r0j , k
)exp(−iωt), (47)
where the displacement x can readily be replaced by the ve-locity, momentum or force; k is the wave vector, and ω is
FIG. 4. The CGMD weighting function fμi for a 1D-chain of 200 equidistantatoms and three different levels of coarse-graining Ncg. The symbols repre-sent the discrete values and the lines serve as a guide to the eye. The orderingin the legend from top to bottom corresponds to the ordering of the maximafμ0 at the central atom i = 0.
the frequency of the wave. The discrete Fourier transformT (r0
j , k) was implicitly defined by this equation. In order tofind the admissible wave vectors k, we must take into accountthe periodic boundary conditions. Then the admissible wavevectors have the well known form
k =D∑α
kα
Nα
bα (48)
with D = 2 dimensions in our case and where the lattice vec-tors for graphene may be chosen, e.g., as a1 = r0(3,−√
3)/2and a2 = r0(3,
√3)/2, and the relation bα · aα = 2πδαβ then
leads to the reciprocal vectors
b1 = 2π
r0
(1
3,− 1√
3
)and b2 = 2π
r0
(1
3,
1√3
). (49)
The periodicity limits the number of orthogonal waves andcan be accessed by the range kα = −(Nα − 1)/2..(Nα − 1)/2for odd Nα and kα = −Nα/2 + 1..Nα/2 if Nα is even. Forgraphene this gives Nk ≡ N1 × N2 = N/2 different wavevectors (N: number of atoms), each associated with D =2 acoustical and D = 2 optical eigenmodes, i.e., a total of2DNk = 2DN eigenmodes.
We can define the Fourier transform of fμi by
fμ(k) ≡ fμiT (ri , k), (50)
where the function T (r, k) was defined in (47) and the k takesthe admissible values. For a rectangular periodic graphenesheet of 10 368 atoms Fig. 5 shows the Fourier space rep-resentations fμ(k) of the weighting functions fμi of CGMDand COM variables for Ncg = 96 in the first hexagonal Bril-louin zone (BZ) in the direction b′ ≡ b1 + b2 (k1 = k2). TheBZ has a length of 2π /3r0 in this direction. The definition of,e.g., the COM CG momentum is13
PCOMμ ≡ f COM
μi pi , (51)
where the symbol f COMμi takes the value 1 if atom i is in the
hexagonal region of blob μ (cf. Fig. 1) and zero otherwise. In
-π/3 π/3
0.5
1.5
00
1
CGMD (96)
COMs (96)
|k| [1/l∗]
f μ(k
)
FIG. 5. Fourier space representations fμ(k) of the weighting functionsfμi of CGMD and COM variables for Ncg = 96 for the rectangular peri-odic graphene sheet of 10 368 atoms in the first Brillouin zone. Symbolsdenote the actual values for the admissible wave-vectors in the directionb1 + b2 (k1 = k2). The lines only serve as a guide to the eye.
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234103-8 Kauzlaric et al. J. Chem. Phys. 137, 234103 (2012)
-π/3 π/30
0
1
1.5
0.5
CGMD (24)COMs (24)
|k| [1/l∗]
f μ(k
)
FIG. 6. Same as Fig. 5 but for Ncg = 24.
1D, this function would correspond to a rectangular windowfor which the Fourier transform (50) is analytically known andshows an identical behavior to the one seen in Figs. 5 and 6.We clearly see the expected behavior: The CGMD weightingfunction reduces the intensity of high-k modes much morestrongly than the weighting function corresponding to thechoice of COM variables because the former shows a muchsteeper decay and much smaller ripples for high k. Assumingphysical dispersion relations this picture can be translated tothe frequency domain. Hence, CGMD is damping high fre-quency modes more strongly than the COMs. But the highfrequency modes are exactly what would make the FACF de-cay fast. Therefore, the CGMD force correlation is sloweddown and decays on the same time scale as the momentumcorrelation. A Markovian approximation is then unjustified ina strict sense since no clear separation of time scales can beobserved.
Striking is the overshoot of the CGMD weighting func-tion before its first drop to zero. Its consequences have notbeen investigated in detail. It might be related to the im-proved reproduction of dispersion relations as compared tolocal FEM-interpolation on one hand.15 But it is also evi-dently the reason why the reproduction is not perfect as com-pared to a rectangular fμ(k) that would be used by choosingthe first few normal modes as CG-variables. The behavior ofCGMD in Fourier space reminds of elliptic filters, which areoptimized for a maximal slope at the transition at the expenseof introducing ripples.
Figure 6 shows the results for Ncg = 24. They showquantitative differences, but the qualitative interpretation re-mains the same. The number of maxima (for |k| ≥ 0) hashalved compared to Ncg = 96. This is consistent with thehalving of the radius of influence of the nodes from Ncg = 96to Ncg = 24. For Ncg = 6 and Ncg = 5413 we would have oneand three maxima, respectively. Still, the CGMD weightingfunction reduces high-k modes more strongly than the COMweighting function, due to the steeper slope, the amplificationof the low-k modes and the fact that the second ripple decaysat lower k than for the COM weighting function.
V. D-CGMD PARAMETERS FROM MD
In this section, we present methods for computing the pa-rameters required for the CGMD model which are containedin the matrices Mμν , Kμν , �μν . First, we briefly discuss howto deal with singularities in a periodic system when comput-ing matrix inverses. Finally, we present a top down method tocompute the dissipation matrix �μν from shear and pressurewave decay simulated with MD.
A. Computation of the mass and stiffness matrix
Equations (33) and (34) give the prescription for com-puting the CGMD mass and stiffness matrix in terms of thematrix inverses of the equilibrium averages of the covari-ances 〈VμVν〉MD and 〈XμXν〉MD. These matrices are straight-forwardly computed from MD. While we will perform thesecomputations for the stiffness matrix, we will use the simplerCGMD-relation
Mμν = m∑
i
NμiNiνδαβ = mNμνδαβ (52)
for the mass matrix,15 where m is the mass of a C-atom andNμν was defined in Eq. (12). The indices α, β stand for theCartesian components. Hence, the matrix is only diagonal inthe Cartesian indices and otherwise non-diagonal, i.e., dis-tributed. This is in accordance with Eq. (33) where we expectstatistical independence between different Cartesian compo-nents, but, due to the CGMD-weighting function fμi we expectcorrelations between Vμ and Vν , i.e., non-zero off-diagonalterms (μ �= ν) for α = β. We avoid finite temperature equi-librium averages for the mass matrix since we do not expectan influence of the temperature on inertia but there might bean influence on the effective stiffness coming from nonlin-ear effects. The stiffness matrix we get from Eq. (34) is usu-ally called the quasi-harmonic approximation in contrast tothe zero-temperature harmonic approximation.
For systems with fixed atoms as boundary conditions,computing the inverses is usually possible since the covari-ance matrices are non-singular. For the system with periodicboundary conditions investigated here, this is no longer true.In this case, the system is translationally invariant and the ma-trix 〈XμXν〉MD will be singular with D zero eigenvalues, if theaverage is understood in its exact sense and not as an approx-imate numerical result. This latter point is important since weshould enforce this singularity when computing the covari-ance numerically. But first of all, the singularity means thatthe inverse in Eq. (34) does not exist.
Going a step back from Eq. (34) we are in fact searchingfor a solution Kμν of the under-determined equation
〈XμXν〉MDKνσ ≡ CμνKνσ = kBT Iμσ , (53)
where Iμσ is the identity matrix. This is a linear systemof equations with kBT times the identity as the multipleright-hand-side. If there are solutions, then K∗
νσ = kBT C+μν
is a solution where C+μν is the Moore-Penrose pseudoinverse
of the matrix Cμν and ‖K∗νσ‖F ≤ ‖Kνσ‖F for all solutions
Kνσ .33 The norm ||x||F is the Frobenius norm, i.e., K∗νσ is the
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234103-9 Kauzlaric et al. J. Chem. Phys. 137, 234103 (2012)
solution in least-squares sense. Hence we have to computeminKνσ ∈RM′×M′ ‖CμνKνσ − kBT Iμσ‖F , where M′ ≡ DNn.
In theory, the matrix Cμν has the same properties as thezero-temperature form of Kνσ : it is square, symmetric, posi-tive semi-definite and circulant. This leads to the identity of itssingular values required to obtain the pseudoinverse with theeigenvalues. The matrix obtained from averages over a finitenumber of MD-timesteps does fulfill these properties only ap-proximately. But we can enforce the theoretical properties byadditional averaging. Ideally, we should, e.g., transform thematrix into its eigenspace and enforce the same degenera-cies of the eigenvalues (corresponding to symmetries in realspace) as for the zero temperature version of the stiffness ma-trix. Since it would require additional effort to compute thezero temperature-form,15 we instead take the following twoapproximate measures which do not enforce all, but sufficientsymmetries to enforce the singularities: (i) All matrix entriesbelonging to equivalent pairs of nodes in terms of the relativepositions of the nodes are averaged. (ii) We enforce
∑μ
〈XμXν〉MD = 0ν
by subtracting from each column the small deviation fromzero that we still get after step (i), divided by DNn. Thesetwo steps do not remove the statistical error completely, butthey singularize the matrix giving two zero eigenvalues withinfloating point precision.
B. Top-down derivation of a friction matrix
Due to the non-Markovianity of the CGMD-variablesshown in Sec. IV, a Markovian approximation will not re-produce correctly correlation functions of the CG variables,nor fine details of the dynamics of shear or pressure wave de-cay. But from a pragmatic engineering point of view we mayat least ask for a model reproducing the average decay of themodes correctly, and for this purpose, a Markovian dissipa-tive model should be adequate. This will be the attempt inthis section.
In particular, we measure a few eigenmode frequenciesand decay times from MD computer experiments and deter-mine a CGMD dissipation matrix �μν which reproduces thisbehavior together with the already known CGMD stiffnessand mass matrices. Since we assume a Markovian system, ourdissipation matrix will be time-independent and the associ-ated fluctuations will represent white noise. This also impliesthat we are assuming an exponential decay of the eigenmodes.As will be seen from the results, the true decay is more com-plex, and, in addition we can only observe its initial behaviorfrom the MD-simulations. This is another way of seeing thatour Markovian assumption is only an approximation.
We will assume mode amplitudes which are large com-pared to the noise and hence we neglect the latter in the fol-lowing. At the end we will reintroduce the noise and computeit from the dissipation matrix by the fluctuation-dissipationtheorem (43). Applying a discrete spatial Fourier transformon our CGMD node lattice, we can transform Eq. (44) with-
out noise term into
Xb(k, t) = −K′bb′ (k)Xb′ (k, t) − �′
bb′ (k)Xb′ (k, t) (54)
with
Xb(k, t) = Tbμ(k)Xb
μ(t) (55)
and
K′bb′ (k) = Tb
μ(k)K′bb′μν Tb′
ν (−k) (56)
and analogously for �′bb′ (k). Even though the wave vector k
can only take discrete values, we write it as a function argu-ment in order to minimize the number of indices. In fact, wecould assume k to take on continuous values in accordancewith the continuous FE basis functions underlying CGMD,but this would just obscure the act of coarse-graining as a re-duction of degrees of freedom of a discrete system. The in-dices μ, ν denote in this notation the primitive Bravais latticecell and, for later use, we introduce indices b, b′ which takethe values 1 or 2 denoting the first or the second node in theunit cell (cf. Fig. 1). In the notation, we follow the rule that,if indices b, b′ are explicitly given, the Greek indices μ, ν re-fer to the lattice cell and not to the node in the cell. In Eqs.(55) and (56), summation is performed only over μ, ν andTb
μ(k) ≡ T (Rbμ, k)δαβ with T (Rb
μ, k) as defined in (47) andTb
μ(−k) is its conjugate transpose (and inverse at the sametime32).
Since, on the CG-level, we are dealing with a triangularlattice of CGMD-nodes (cf. Sec. III A), the smallest unit-cellin a rectangular periodic domain with size Lx × Ly is againrectangular and contains two nodes (cf. Fig. 1). If there areN
cgx unit-cells in x-direction and N
cgy unit-cells in y-direction,
then we may choose the admissible k-vectors as
k =∑
α
2πk
cgα
Lα
kα, (57)
where kα is a unit-k-vector in α-direction, α = x, y,and k
cgα = −(N cg
α − 1)/2..(N cgα − 1)/2 for odd N
cgα and k
cgα
= −Ncgα /2 + 1..N
cgα /2 if N
cgα is even. Note that, due to the
coarse-graining, the largest k-vectors from the atomistic sys-tem are now excluded, since N
cgα < Nα (cf. Sec. IV B).
With Eq. (54) we have split the system of Eq. (44)into NCG
k independent small 4 × 4 systems (4 = 2 nodesper CG unit cell times 2 Cartesian DOFs), where NCG
k= Nn
x + Nny = Nn/2 is the number of admissible wave vec-
tors and Nn is the number of nodes in the CG-system. The fourequations of the 4 × 4 systems are still coupled. To uncouplecompletely, we now project onto the four normalized eigen-vectors Vbα(k) (in our problem, b = 1, 2 and α = x, y) of thematrix K′
bb′ (k). This will only be useful if we make the criti-cal modeling assumption that these eigenvectors are also theeigenvectors of the (to be found) friction matrix �′
bb′ (k). Wejustify this assumption by the fact that the dissipative effectsthat we are looking for and which are contained in �′
bb′ (k)are very small compared to the elastic effects contained inK′
bb′ (k), i.e., the system is highly underdamped, as is to beexpected for a lattice. Hence the mode shapes will be alteredvery weakly and we may approximate them by those of theharmonic system. A consequence of this assumption is that
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234103-10 Kauzlaric et al. J. Chem. Phys. 137, 234103 (2012)
we restrict �′bb′ (k) to a specific class of matrices with similar
nodal connectivity as the stiffness matrix. It can be shown thatthere exist matrices �′
bb′ (k) which, if introduced into a purelyconservative system, keep the original eigenvectors invariantbut alter the eigenvalues. Let Kbα(k) be the scalar eigenvaluesof K′
with the vector X b(k) containing the four scalars Xbα(k), thematrix Qbb′ (k) containing the four normalized eigenvectorsVbα(k) as columns, the diagonal matrix Kbb′ (k) containingall the eigenvalues Kbα(k), and the diagonal matrix �bb′ (k)containing all the eigenvalues �bα(k). Inserting a solution ofthe form
Xbα(k, t) = Xbα(k, 0) exp(−zbα(k)t) (60)
into Eq. (58) we get
(zbα(k))2 = −Kbα(k) + zbα(k)�bα(k) (61)
or, with zbα(k) ≡ γbα(k) + iωbα(k),
γ 2bα(k) − ω2
bα(k) + 2iωbα(k)γbα(k)
= −Kbα(k) + (γbα(k) + iωbα(k))�bα(k). (62)
Equating real and imaginary parts gives
�bα(k) = 2γbα(k), Kbα(k) = ω2bα(k) + γ 2
bα(k), (63)
which yields the desired stiffness and friction matrices if in-serted into Eq. (59). The second of Eq. (63) implies that thedifference between the stiffness matrices Kbα(k) of a purelyharmonic system (as already computed in Sec. V A) and ofa harmonic system with dissipation is of the order of γ 2
bα(k),hence very small, since we expect γbα(k) � ωbα(k) for allk, b, α. Therefore, it is safe to use the already computedCGMD-stiffness matrix from Sec. V A.
We compute the required quantities ωbα(k) and γbα(k)from MD-simulations of shear and pressure wave decay for afew k vectors and fit an analytical function to the results al-lowing to approximate the whole spectrum. As a final step thematrices �′bb′
k are then assembled into one big block diago-nal matrix and backtransformed by inverting Eq. (56) (with �
instead of K).More specifically, after an equilibration period we instan-
taneously add on the thermal velocities of the atoms a cosinedisturbance and assume it to be approximately equivalent tothe nodal wave, i.e.,
Xdμ(k, α, b) ≡ Vd
μ(k, α, b) = Re[V d
0 Vbα(k) exp(ik · Rμ)]
(64)
for one specific choice of k, α, b where V d0 is a suitable veloc-
ity amplitude and the eigenvectors Vbα(k) have been definedimplicitly in Eq. (59) as the columns of Qbb′
(k). The approxi-mation of atomistic by nodal eigenmodes can be shown to be
valid for large wavelengths λ = 2π/|k| compared to the nodaldistance. These are the most relevant modes. For larger k theapproximation is not totally correct, but still the best we cando being dependent on MD-results, since all functions deviat-ing from the microscopic eigenmodes will not produce uniquefrequencies ωbα(k) and γbα(k) in the MD-simulations but amixture which is not useful. The choice between acoustic(b ≡ 1) and optic (b ≡ 2) modes is determined by the index band the choice between transversal and longitudinal modes isdetermined by the index α. Note that for arbitrary k-vectors,the association of the terms “transversal” and “longitudinal”to respective α-indices is not rigorously possible.32 Therefore,we only apply waves with k-vectors along reciprocal latticevectors bα and along the diagonal bx + by which are direc-tions of high symmetry. In these cases, the association is pos-sible since Vbα is either exactly parallel (“longitudinal”) orperpendicular (“transversal”) to k. After applying the wavewe track in the MD-run the time evolution of the quantity
Vbα(k, t) ≡ Xbα(k, t) ≡ d
dtXbα (k, t)
= d
dt
(Vbα(k) · Tμ(k)Xμ(t)
)= Vbα(k) · Tμ(k)Vμ(t) (65)
for the same k, α, b that was used in Eq. (64). Usually, weperform 10 to 100 MD-runs depending on the computationalcost to compute this quantity as an average over initial ther-mal velocities. The computational cost per k, α, b depends onthe number of time steps required to estimate the time evolu-tion Eq. (65) and the latter varies for reasons explained in thefollowing.
First of all a dependence of the decay rate γbα(k) on thespecific mode, i.e., on the specific k, α, b is to be expected.But this fact does not greatly influence the number of requiredtime steps because, anyway, we are not capable of computingthe whole mode-decay with a MD-simulation but have to esti-mate it from the initial time evolution. Much more importantis the fact that Eq. (44) and the Ansatz (60) represent onlyan approximation and can only capture the exponential decaybut no additional details of the time evolution which stronglyinfluence how many time steps are needed to extract an expo-nential approximation.
Figure 7 gives a good representation of the different kindsof time evolution of the modes Vbα(k, t) from Eq. (65) that canbe observed in the simulations. All plots consist of many sinu-soids as shown in the inset in Fig. 7(b). The plotted time scalesqueezes the sinusoids so much that they cannot be identifiedindividually. The inset as well as the numbers on the verticalaxes in general show that the mode relaxation is usually verysmall over the time periods observable by MD-simulation.The blue (dark) semi-transparent plot in Fig. 7(a) for the low-est transversal-acoustic mode for k‖ky is typical for a plotthat was averaged over many (here 48) initial thermal veloc-ities. Due to the effort put into the averaging, the number oftime steps is more limited (here 6 × 106) than without av-eraging. The curve shows a decaying behavior, but at a firstglance not of an exponential as required by the Ansatz (60).On the other hand, the red (non-transparent) plot in Fig. 7(a)
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234103-11 Kauzlaric et al. J. Chem. Phys. 137, 234103 (2012)
-2
1
time [t∗]
amplitu
de
[10−
3l∗
/t∗ ]
2.69
2.7
2.71
2.72
2.73
0
0
0
2
−1
250 500
10000 20000
(b)
(a)
time [t∗]
amplitu
de
[10−
3l∗
/t∗ ]
1.98
1.97
1.96
0 4000020000
(c)
FIG. 7. The function Vbα(k, t) from Eq. (65) for exemplary acoustic modes(b = 1) from top to bottom for: (a) kx = 0, ky = 1, α = x, (b) kx = 0,ky = 4, α = x, (c) kx = 0, ky = 2, α = y. Hence, (a) represents one ofthe two lowest transversal-acoustic modes, while (c) represents the secondlongitudinal-acoustic mode in y-direction. Note the different time scales inthe plots. Averaging has been performed over 48 (opaque), 99, 28 initial con-ditions, respectively, and in (a) the longer red (non-opaque) curve representsthe result of a single run, i.e., without any averaging. The almost straight lineis its exponential fit.
shows the behavior of the same mode, but instead of averag-ing, one simulation was run for a much longer time. Sincethis is a simulation starting from an initial state which is aperturbation from equilibrium, time-averaging is excluded aswell. That this mode, besides the thermal noise and suppos-
edly additional dynamic effects, shows a behavior quite closeto exponential decay, and that, additionally, the fitted expo-nential (solid line) is also a quite good approximation of anexponential fit of the short averaged version, makes us con-fident that also the shorter averaged curves may be fitted toan exponential. There is certainly dynamics that we neglectwithin this approximation but that should not be essential forthe dissipative behavior of the resonator. Another exampleconfirming this view is shown in Fig. 7(b). This is the 4thtransversal mode for k‖ky . Since it both oscillates and decaysfaster, more of its relevant dynamics can be observed with thesame effort. We see that, even though the decay is not mono-tonic, it goes on, also for longer times. Hence, we deducefrom the faster, and hence more easily observable dynamicsof the higher order modes, that also the more relevant lowestorder modes show a decaying behavior that we are allowed toapproximate by an exponential. Note that in any case, since,within reasonable MD-simulation times, we have only ac-cess to the initial decay, we basically see the linear shorttime behavior of the exponential. A few longitudinal modesshow additional pronounced low-frequency components suchas the plotted second longitudinal acoustic mode for k‖ky inFig. 7(c).
Our aim is to develop a scheme that allows to repro-duce the dissipation of technologically relevant modes withina coarse-grained description. Such a mode is, e.g., one of thelowest transversal modes as in case (a) in Fig. 7. Using theMarkovian model Eq. (44) implies that all modes decay ex-ponentially. For example, in Fig. 7(c), we neglect that the en-velope function contains at least one additional trigonometricfunction in addition to an exponential. But note that the decayrate in Fig. 7(c) is still correct in our model, as far as our mea-surement of the initial behavior permits us to judge. Note thatan error in this mode will not affect the aforementioned modeof interest, because the matrix �bb′
k in Eq. (59) is diagonal andtherefore the resolved modes are uncoupled in the model byconstruction. The only coupling is to the bath of modes thathave been neglected as irrelevant variables in the CG-system.
To obtain γbα(k) for the specific mode characterized bythe triple (k, b, α), we perform a direct least-squares fittingto an exponential for types (a) and (b), whereas we have toextract the maxima of the envelope for behavior of type (c)before fitting. Figure 8 shows the values for acoustic modesas obtained from MD and fitting curves which we use to ob-tain those γbα(k) that have not directly been measured. Alsohere we observe a non-trivial behavior. In addition to a cleartendency of γbα(k) to increase with |k|, different kinds of reg-ular and irregular deviations can be observed that are too largeto be of statistical nature. We make the decision to fit only theincrease with |k| without modeling the detailed behavior. Dur-ing this fitting we also exclude in both longitudinal curves inFig. 8(a) the most strongly deviating peak in each case. Weare not able to explain the source of these peaks. None of thedifferent kinds of deviations from the monotonic increase canbe simply associated with exclusively one of the main decaytypes of Fig. 7 or to multiples of certain wave numbers.
It is encouraging to observe that, applying the mentionedsimplifications, the analytic functions giving a satisfactory fitare simple and physical. In both cases, it is sufficient to
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234103-12 Kauzlaric et al. J. Chem. Phys. 137, 234103 (2012)
γbα
(k)
[10−
6/t
∗ ]
kx,y [2π/Lx,y]
00
2
4
6
66 1212 18
(a)
simulation: kx-directionsimulation: ky-direction
fit: kx-directionfit: ky-direction
γbα
(k)
[10−
6/t
∗ ]
kx,y [2π/Lx,y]
00
1
2
3
4
5
66 1212 18
(b)
simulation: kx-directionsimulation: ky-direction
fit: kx-directionfit: ky-direction
10
10
γbα
(k)
[10−
6/t
∗ ]
|k| [2π√
7/216 l∗]
00
5
5
(c)
simulation: longitudinal
simulation: transversalfit: longitudinal
fit: transversal
FIG. 8. The symbols represent constants γbα(k) for acoustic modes obtained from fitting MD-results for wave decay to an exponential according toEq. (60). The straight lines represent parabolic fits to the MD-results. Note that in the first two plots, the left (red/gray) part represents the ky -direction,while the right (black) part represents the kx -direction, and that different (inverse) length scales are used left and right in terms of multiples of the smallestwave vector in the respective direction. (a) Longitudinal modes and quadratic fit along kx . (b) As (a) but transversal modes. (c) Longitudinal and transversalmodes along kx + ky . For details see the text. The error bars only characterize the fitting procedure and do not take systematic errors due to deviations fromexponential decay into account.
consider a function of the form fl(kx, ky) = a|kx |2+ b|ky |2 + c|kx‖ky | and we obtain in the longitudinalcase (a) al = 5.03282 × 10−6, bl = 3.03896 × 10−6, cl
= −1.41231 × 10−7 in the units l*2/t* (cf. Sec. II). In thetransversal case (b), we obtain at = 4.0182 × 10−6, bt =1.09241 × 10−5, ct = 1.68101 × 10−5. Therefore, the maincontribution to dissipation seems to stem from terms pro-portional to |k|2, very well known from hydrodynamics.34 Inaddition it is interesting to observe that, while in the longitudi-nal case, the mixed term |kx‖ky | seems to be negligible to firstorder (cl is less than 5% of bl and less than 3% of al) it is im-portant for the transversal modes which is an intuitive result.
The reason for ignoring the optic modes in MD is thatthey are not represented in the CGMD-lattice. A scheme todo so was proposed in Ref. 15. It is not used here since wewish to keep the slow dynamics and to average over the highoptic frequencies.
Note that the observed relaxation behavior of the modesis specific to the used microscopic potential for graphene and
will most likely change by using another potential, such asthe more realistic Brenner-potential.21 On the other hand, theCG-scheme described here, can be used for any potential. Toinvestigate more deeply the specific relaxation behavior ob-served here is therefore neither our focus nor of big relevance.
C. Random forces
In order to allow for thermal fluctuations at a well-defined temperature, we now consider the term F′b
μ (t)= (Mbb′
μν )−1Fbμ(t) from Eq. (44). In the following, the b-
indices are unnecessary and will be omitted. The noise shouldbe defined in terms of a stochastic differential equation andmodeled as a linear combination of Wiener processes, e.g.,dWμ(t) obeying the Ito rule
dWμ(t)dWν(t ′) = dWμα(t)dWνβ(t ′) = δμνδαβδ(t − t ′)dt,
(66)
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234103-13 Kauzlaric et al. J. Chem. Phys. 137, 234103 (2012)
that is,
Fbμ(t)dt ≡ Fμ(t)dt ≡ dPμ(t) =
∑ν
BμνdWν(t). (67)
From the fluctuation-dissipation theorem (43) or
dPμdPν = 2kBT �μν (68)
and the Ito rule it follows then that∑σ
Bμσ Bσν = 2kBT �μν (69)
or
Bμν =√
2kBT√
�μν, (70)
where the second square root is a matrix square root, definedby giving that matrix Bμν that satisfies Eq. (69). Since weknow from the way we constructed the matrix �μν that it isdiagonalizable and semi-positive definite, real square roots ofthe matrix can be easily computed from square roots of theeigenvalues as √
�μν = Vμν�1/2μν V−1
μν (71)
and �1/2μν ≡ +
√�
μbα
νb′βδμνδbb′δαβ . The “+”-sign indicates thatwe compute the so-called principal square root for everyeigenvalue, but this is only required for making an unambigu-ous choice and any other choice works as well since it is thesquare of Bμν that affects the equation of motion. In addition,the matrix square root preserves the symmetry of �μν and theproperty ∑
ν
Bμν = 0μ (72)
for every μ. For the equations of motion (44) we found it morepractical to compute
F′bμ (t)dt ≡ F′
μ(t)dt ≡ dP′μ(t) =
∑ν
B′μνdWν(t) (73)
with
B′μν = M−1
μσ
√2kBT
√�σν =
√2kBT
√M−1
μσ�′σν. (74)
VI. COMPARISON OF MD AND (D-)CGMD RESULTS
Having computed the CGMD mass, stiffness, friction,and noise matrices, we are now able to compare predictionsproduced by CGMD and MD for the nodal values. First, wecompare momentum-correlation functions, and then the de-cay of shear and pressure waves
A. Correlation functions
Figure 9 shows the nodal velocity autocorrelation func-tion (VACF) 〈Vμ(t) · Vμ(0)〉 for Ncg = 24 as computed fromfull MD and from the CGMD model, and Fig. 10 correspond-ingly for Ncg = 96. The curves match very well in the initialdecaying part but show significant deviations for later times.These deviations could be expected for conservative CGMDbut, in addition, its CFs are indistinguishable from those of
time [t∗]
AC
F[k
BT
tr(M
−1 μμ)]
−0.5
0.5
1
0
0 50 100
nodal MD-VACF for Ncg = 24nodal CGMD-VACF for Ncg = 24
FIG. 9. Nodal velocity autocorrelation function (VACF) for Ncg = 24 ascomputed from full MD and from CGMD. The ACFs are given in units ofkBT tr(M−1
μμ).
our D-CGMD-model. This means, on the level of the CFs thedissipation and thermal noise included in D-CGMD have noeffect. For the initial decay it is remarkable that the agreementof CGMD is better than for the comparison of COM-ACFsfrom MD with COM-ACFs from a harmonic “N-DPD” modelwith a CG potential and no dissipation as shown in Figure 11.But note that a CG DPD-model for COMs with dissipationstill gives the best overall reproduction of ACFs.13
In order to quantify the deviation of CGMD and N-DPDfrom MD for the ACFs in Figs. 10 and 11, we apply the L2-norm to the differences of the values at the first 400 availablediscrete time steps until t = 100 and normalize to the initialvalue of the MD-ACF. We get
∣∣∣∣�CPP24
∣∣∣∣CPP
24 (0)≈ 1.93,
∣∣∣∣�CPP96
∣∣∣∣CPP
96 (0)≈ 1.75
∣∣∣∣�C ′PP96
∣∣∣∣CPP
96 (0)≈ 2.28, (75)
time [t∗]
AC
F[k
BT
tr(M
−1 μμ)]
−0.5
0.5
1
0
0 50 100
nodal MD-VACF for Ncg = 96nodal CGMD-VACF for Ncg = 96
FIG. 10. Nodal velocity autocorrelation function (VACF) for Ncg = 96 ascomputed from full MD and from CGMD. The ACFs are given in units ofkBT tr(M−1
μμ).
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234103-14 Kauzlaric et al. J. Chem. Phys. 137, 234103 (2012)
time [t∗]
AC
F[2
kBT
/M]
−0.5
0.5
1
0
0 50 100
blob MD-VACF for Ncg = 96blob N-DPD-VACF for Ncg = 96
FIG. 11. Velocity autocorrelation function (VACF) of COM-blobs for Ncg= 96 as computed from full MD and from harmonic N-DPD. The ACFs aregiven in units of 2kBT/M, where M is the blob mass.
where �CPPNcg
is the vector collecting all the differences be-
tween the MD and CGMD VACFs for Ncg, and �C ′PPNcg
correspondingly collects all the differences between the MDand N-DPD blob-ACFs. Higher coarse-graining numbers aswell as CGMD compared to N-DPD lead to slightly decreas-ing deviations from MD. The former is consistent with theexpectation that dissipation and noise become less importantfor larger Ncg, but the decay is admittedly very slow. Similarobservations hold for the cross correlation functions (CCFs),e.g., Fig. 12 shows the nodal CCFs along the connecting linefor the nearest neighbors.
The fact that deviations to MD are smaller for theCGMD-variables than for the COM-variables shows that dis-sipation in the CGMD-model is weaker.
B. Shear and pressure wave decay
While the dissipative force seems to have a negligible ef-fect on the correlation functions, it is crucial for wave decay
time [t∗]
CC
F[1
0−6
MD
-unit
s]
0
0−3
−2
3
2
1
−1
50 100
nodal MD-VCCF for Ncg = 96
nodal CGMD-VCCF for Ncg = 96
FIG. 12. Nodal velocity cross correlation function (VCCF) of nearest neigh-bors for the velocity component parallel to their connecting line for Ncg = 96as computed from full MD and from CGMD.
FIG. 13. Comparison of MD and D-CGMD simulations with Ncg = 96 forthe decay of the excited lowest order transversal mode with wave vectork‖ky . The red (light) MD-result was made semi-transparent to make theunderlying D-CGMD-plot visible as well. Again, the individual oscillationperiods are unresolvable as in Fig. 7.
and makes the difference between a quasi-infinite and a finiteand technologically relevant Q-factor of the resonator.
The D-CGMD-model developed in Sec. V including thedissipation matrix �μν and the noise term F′
μ in Eq. (44) isnow applied to the same simulation setup which was usedto construct the fluctuating and dissipative parts, i.e., afterthermal equilibration at the same temperature T = 298 K asin the MD-simulation, a sinusoidal wave as in Eq. (64) isadded as an instantaneous disturbance. The results presentedhere are for a D-CGMD-model with coarse-graining numberNcg = 96. Figure 13 shows for the lowest order transversalmode with wave vector k‖ky , that the D-CGMD model re-produces exactly, as expected, the decay rate that was ob-served in the MD-simulation. Since no averaging over initialthermal velocities was performed neither in the MD nor inthe D-CGMD simulation, the fluctuations can be observed aswell and match very well in a statistical sense. While the MD-simulation is limited in the reachable simulation time and thisspecial simulation took many days, the D-CGMD-simulationcan easily run for much more time steps and needed only afew hours for the shown time span. The speedup factor isroughly 100 in the computational cost per time step �t, i.e.,on the order of Ncg and could be pushed by at least an otherfactor of 10 by which we may increase the MD-simulationtime step �t in the CG-simulation. The non-locality of theCGMD-matrices as opposed to the local MD-potential doesnot seem to have a significant impact at the used relativelysmall system sizes.
In addition to the slightly better reproduction of at leastthe initial behavior of the velocity correlation functions, theD-CGMD-model presented here, by construction, approxi-mates the MD-behavior of wave decay much better than theCOM-based DPD-model presented in Ref. 13 which was or-ders of magnitude too dissipative for long-wavelength shearand pressure waves. In addition, CGMD is better in reproduc-ing the frequency of the modes. Its error is only ≈5% com-pared to MD, whereas the COM-variables gave an error of≈30%.
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234103-15 Kauzlaric et al. J. Chem. Phys. 137, 234103 (2012)
VII. SUMMARY AND DISCUSSION
In the present work, we apply the CGMD procedure toa simple atomistic 2D model of graphene. The focus is onthe dynamics of the relevant nodal variables in terms of theircorrelation functions and the decay of long wavelength shearand pressure waves.
The chosen finite element mesh which is required forCGMD is regular and triangular, such that the nodes formthe COMs of previously investigated hexagonal CG COMvariables.13 This allows a comparison with this model.
We have shown, that choosing CGMD nodal vari-ables as the relevant variables, the Mori projection opera-tor formalism16 leads on one hand to the finite tempera-ture reversible part of the dynamics as known from standardCGMD15 and, additionally, the formalism delivers the dissi-pative part as well in terms of a memory kernel and the corre-sponding thermal noise.
The inverse of the singular displacement covariance ma-trix occurring in a system with periodic boundary conditionswas shown to be computable as a pseudoinverse. This gives awell-defined CGMD-stiffness matrix as the minimal solutionin least-squares sense.
CGMD nodal momenta in the graphene model turn outto obey clearly non-Markovian equations of motion. This wasshown by searching for a time scale separation between themomentum-CFs and the corresponding force-CFs. This timescale separation is non-existent in contrast to findings forCOM-variables for the same underlying graphene model.13
It was shown that this difference between CGMD-nodes andCOMs can be explained by the fact that the CGMD-weightingfunction suppresses more strongly high-frequency modes.This was demonstrated by computing the Fourier transformof the weighting function matrix. This property leads toan assimilation of the decay times of momentum-CFs andforce-CFs.
Omitting dissipative and stochastic forces in CGMD, wewere able to show that CGMD reproduces the short-time be-havior of the momentum-CF better than a corresponding har-monic non-dissipative (N-DPD) model for COM-variables.But the overall reproduction of the CFs is expectedly bad dueto the missing dissipation and noise. On the other hand, thisshows that CGMD is less dissipative than N-DPD, in the sensethat, without dissipative terms, CGMD reproduces the truedynamics already better than N-DPD. In the harmonic frame-work, i.e., limiting ourselves to harmonic CG-potentials (as isdone in CGMD) the missing part of the true dynamics is whatthe linear Mori-theory16 adds to the harmonic CG-potential,which is the dissipation (and noise). This missing dissipationis smaller for CGMD than for N-DPD.
From a macroscopic point of view, the decay of long-wavelength modes is of interest. A strong argument for choos-ing CGMD is that it reproduces the corresponding frequenciesbetter than N-DPD or DPD. As the Markovian assumption isnot valid for the CGMD variables in a strict sense, and sincewe wanted to avoid here the development of a sophisticatednon-Markovian model, we focused on the basic behavior ofdecaying shear and pressure waves as observed from MD-simulations. The measured decay times were directly plugged
into the eigenmode representation of an assumed MarkovianCGMD model so that the CG model reproduces the decaytimes by construction, ignoring further details in the wave de-cay that can clearly be observed. This enables us to simulategraphene based resonators with the correct average Q-factoron much larger length and time scales than accessible withMD. Note that this approach can be applied equally well toother CG variables such as the COMs. The computational ef-ficiency that we gain for a coarse-graining number of 96 isabout a factor of 1000 in total.
In summary, we have monitored two different types ofmotion of the microscopic graphene lattice which live on dif-ferent time scales: (i) the local thermal motion of the CGMD-nodes observable by correlation functions and involving alleigenmodes of the system, and (ii) the macroscopic globalmotion mainly represented by the lowest eigenmodes. Whilemotion (i) turns out to be complex in the sense that it is, e.g.,non-Markovian and hence requires a sophisticated CG modelfor the motion of the CGMD-nodes, motion (ii), to a first ap-proximation, shows very much the behavior of hydrodynamicmodes with a k2 dependence of their damping. As a conse-quence, we were able to construct a D-CGMD model con-taining these modes and their correct long time behavior. Wehave clearly seen that the dynamics of the eigenmodes con-tains more details that most likely reflect their coupling tomotion (i).
The presented CG scheme is general and remains valid ifwe introduce out-of-plane motion of the Graphene sheet, i.e.,the full 3D problem. On the other hand, the specific choice ofrelevant variables may change. We expect out-of-plane modesto be considerably slower than in-plane modes for a givenwave vector k. If the aim is to minimize non-Markovian ef-fects, we should choose the slowest modes as relevant degreesof freedom. This requires to pick many out-of-plane and just afew in-plane modes. In terms of CGMD or COM nodes, thiswould require a subgrid with high resolution and a subgridwith low resolution for the many out-of-plane and the fewin-plane modes, respectively. Work along these interesting di-rections is currently in progress.
It is obvious that the developed D-CGMD model is nota universal model but was constructed for the specific needof simulating more efficiently resonators with the correct Q-factor. The dissipation required for this purpose keeps the CFsof the CGMD-variables basically unaltered and does not re-duce the error as compared to the MD-CFs. This indicates thatfor a better reproduction of the CFs, it is more sophisticatedreversible dynamics that is missing. This can be incorporatedby a memory function within the Mori-formalism used hereand/or by nonlinear CG potentials. The latter was performedin a Markovian framework in Ref. 13 leading to a good re-production of CFs but wrong Q-factors. Combining these twoapproaches would probably lead to even better CG models.
ACKNOWLEDGMENTS
The authors acknowledge funding by the Universityof Freiburg through the German excellence initiative. D.K.thanks the Universidad Nacional de Educación a Distancia(UNED) in Madrid for its kind hospitality and acknowledges
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234103-16 Kauzlaric et al. J. Chem. Phys. 137, 234103 (2012)
funding by the German Research Foundation (DFG) via theproject KA 3482/2 Simulation graphenbasierter Nanores-onatoren: Systematische Reduktion von Freiheitsgraden.P.E. thanks the support of Institute for Biocomputationand Physics of Complex Systems (BIFI) and the Min-istry of Science and Innovation through Project No.FIS2010-22047-C05-03. Additionally, we gratefully thankthe bwGRiD project (Ref. 35) for the computationalresources.
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