Applied Mathematics and Mechanics (English …...2. Division of Applied Mathematics, Brown...

Appl. Math. Mech. -Engl. Ed., 39(1), 63–82 (2018)

Applied Mathematics and Mechanics (English Edition)

https://doi.org/10.1007/s10483-018-2257-9

A note on hydrodynamics from dissipative particle dynamics∗

X. BIAN1,†, Z. LI2, N. A. ADAMS1

1. Chair of Aerodynamics and Fluid Mechanics, Department of Mechanical Engineering,

Technical University of Munich, Munchen 85748, Germany;

2. Division of Applied Mathematics, Brown University, Rhode Island 02912, U. S. A.

(Received Jul. 25, 2017 / Revised Nov. 17, 2017)

Abstract We calculate current correlation functions (CCFs) of dissipative particle dy-namics (DPD) and compare them with results of molecular dynamics (MD) and solutionsof linearized hydrodynamic equations. In particular, we consider three versions of DPD,the empirical/classical DPD, coarse-grained (CG) DPD with radial-direction interactionsonly and full (radial, transversal, and rotational) interactions between particles. To fa-cilitate quantitative discussions, we consider specifically a star-polymer melt system at amoderate density. For bonded molecules, it is straightforward to define the CG variablesand to further derive CG force fields for DPD within the framework of the Mori-Zwanzigformalism. For both transversal and longitudinal current correlation functions (TCCFsand LCCFs), we observe that results of MD, DPD, and hydrodynamic solutions agreewith each other at the continuum limit. Below the continuum limit to certain lengthscales, results of MD deviate significantly from hydrodynamic solutions, whereas resultsof both empirical and CG DPD resemble those of MD. This indicates that the DPDmethod with Markovian force laws possibly has a larger applicability than the continuumdescription of a Newtonian fluid. This is worth being explored further to represent gen-eralized hydrodynamics.

Key words dissipative particle dynamics (DPD), fluctuating hydrodynamics, molec-ular dynamics (MD), coarse-graining, Mori-Zwanzig projection

Chinese Library Classification O3522010 Mathematics Subject Classification 82-08, 82C31, 76M28

1 Introduction

Continuum mechanics was formulated more than two centuries ago and since then it has beenapplied successfully to study various materials. One of the main tools of continuum mechanicsis the Navier-Stokes (NS) equations, which are continuum partial differential equations (PDEs)formulating conservation laws for mass and momentum of a Newtonian fluid (e.g., Refs. [1]and [2]). Over a century ago, it was realized that matter consists of discrete molecules. Thecomplex structures and interactions among a large number of molecules manifest continuumbehaviors at the macroscopic level. With advances of computer technology, both NS equations

∗ Citation: Bian, X., Li, Z., and Adams, N. A. A note on hydrodynamics from dissipative par-ticle dynamics. Applied Mathematics and Mechanics (English Edition), 39(1), 63–82 (2018)https://doi.org/10.1007/s10483-018-2257-9

† Corresponding author, E-mail: [email protected]©Shanghai University and Springer-Verlag GmbH Germany, part of Springer Nature 2018

64 X. BIAN, Z. LI, and N. A. ADAMS

and molecular trajectories can be discretized to study hydrodynamic phenomena at distinctscales, from which numerical methods such as computational fluid dynamics (CFD) and molec-ular dynamics (MD) have emerged. Since the former method is fast for simulating macroscopicphenomena and the latter method elaborates on microscopic details, the two methods are notmeant to compete but to complement each other. With the miniaturization of micro-fluidicstechnology and growing interest of fluid phenomena in the biophysics context, the mesoscopicfluid regime has been receiving more and more attention. In this spatial-temporal regime, MDis computationally expensive while a classical CFD is physically inaccurate. As a consequence,several mesoscopic methods have been developed over the last three decades to exploit a propertrade-off between molecular details and computational efficiency. Among the popular meso-scopic methods are coarse-grained molecular dynamics (CG-MD) methods[3], lattice-Boltzmannmethod[4–5], dissipative particle dynamics (DPD)[6–7], and multiple particle colliding dynam-ics/stochastic rotation dynamics[8–9]. CG-MD is a generic name, under which there are manytechniques to generate a CG potential of mean force, such as reverse Monte Carlo[10], iterativeBoltzmann inversion[11], force-matching method[12–13], relative entropy method[14], and Mori-Zwanzig projection[15–16], to name but a few.

In this work, we discuss hydrodynamics of DPD exclusively, but the insights and perspectivemay be relevant to other mesoscopic methods as well. DPD was invented by Hoogerbrugge andKoelman[6] 25 years ago, who postulated pairwise dissipative and random forces between a setof Lagrangian particles to comply with the isotropy and Galilean invariance of a simple fluid.Later, Espanol and Warren[7] added in the pairwise conservative, reformulated the updatingalgorithm as stochastic differential equations (SDEs), and derived the corresponding Fokker-Planck equations (FPEs). Moreover, they obtained the celebrated fluctuation-dissipation the-orem (FDT) for a DPD system so that the Gibbs canonical distribution is the steady statesolution of the FPEs. Thereafter, DPD has found a large variety of applications in simple andcomplex fluids[17–24]. Meanwhile, researchers have been trying to establish rigorous founda-tions for DPD from both bottom-up coarse-graining[25–28] and top-down discretization[29–30].Given an arbitrary input settings of DPD, qualitative results on transport coefficients andequation of state of hydrodynamics can also be derived from both kinetic theory[31] and projec-tion techniques[32]. It is well known from both theories and simulations that DPD reproduceshydrodynamic behavior of NS equations at sufficiently large scales[32–35]. Furthermore, DPDconsiders also thermal fluctuations, which emerge due to finite thermodynamic volume andare hallmarks of mesoscopic phenomena. As the mesoscopic scale is quite vaguely defined andone usually does not explicitly define the applicability range of DPD. If this arbitrariness is ofconcern, one may employ smoothed dissipative particle dynamics (SDPD)[29], which discretizesthe NS equations and considers thermal fluctuations consistently[36]. Correspondingly, SDPDrepresents a discrete version of the Landau-Lifshiftz-Navier-Stokes (LLNS) equations[1], andthe DPD method may be considered as a “poor” version of SDPD method from this top-downperspective. In this work, however, we wish to discuss about the possibility that DPD mayrepresent generalized hydrodynamics[33,37–38] to certain extent, which is not covered by SDPD.

In Section 2, we first introduce the classical DPD method. To have a specific referencesystem for discussion, we employ a toy model system of star polymer described by a Hamilto-nian MD[27–28]. To reproduce static and dynamic properties of MD, we devise three versionsof DPD by two techniques: the first technique is to run DPD simulations for empirical pa-rameters with trials and errors until we have desired properties generated by DPD; the sec-ond technique is to coarse-grain MD trajectories for DPD force fields using the Mori-Zwanzigprojection[15–16,25–28,39]. With the second technique, we consider two levels of degrees of free-dom in the resultant DPD. In Section 3, we introduce the current correlation functions (CCFs),which are significant in time-dependent processes. Meanwhile, we also revisit solutions of CCFsat sufficiently large scales derived from linearized NS equations. In Section 4, we compute CCFsfrom MD and DPD simulations, and compare the results with hydrodynamic solutions. The

A note on hydrodynamics from dissipative particle dynamics 65

aim is to identify to what extent the results of DPD agree with those of MD and continuumhydrodynamics. In the end, we summarize our findings in Section 5.

2 DPD

DPD was originally introduced by Hoogerbrugge and Koelman[6]. Here, we present thereformulated version of Espanol and Warren[7], who introduced the conservative force into theoriginal DPD,

RI = PI/M,

PI = FI =∑

J 6=I

FIJ =∑

J 6=I

(F CIJ + F D

IJ + F RIJ),

(1)

where RI and PI are the position and momentum of particle I, respectively. The mass for eachparticle is taken as an identical constant M . For isotropy and Galilean invariance[6], the forcesare postulated to be pairwise and depend only on relative position RIJ = RI −RJ and relativevelocity VIJ = VI −VJ = (PI −PJ )/M of two particles. To preserve linear momentum, forcesare antisymmetric and satisfy Newton’s third law, that is, FIJ = −FJI . To preserve angularmomentum, forces are always along the radial direction of two particles eIJ = RIJ/RIJ , withRIJ = |RIJ | being their relative distance. Relevant letters are capitalized for a DPD systemto distinguish from its underlying MD system appearing later in Subsection 2.1. Particle indexI ranges from 1 to the total number of particles N .

According to the general coarse-graining theory or the Mori-Zwanzig projectionformalism[15–16,39–41], a generalized Langevin equation (GLE) results from coarse-graining, andthree types of forces emerge at the mesoscopic level of description. Therefore, the total forceon each particle in DPD is decomposed into three parts which are postulated to be[6–7]

F CIJ = αwC(RIJ)eIJ ,

F DIJ = −γwD(RIJ )(eIJ · VIJ )eIJ ,

F RIJ = σwR(RIJ )ξIJeIJ ,

(2)

where α, γ, and σ are strengths of individual forces. Weighting functions wC, wD, and wR

are isotropic and depend only on RIJ . Beyond a cut-off radius Rc, values of wC, wD, and wR

vanish. ξIJ = ξJI is a Gaussian white noise, that is,〈ξIJ (t)〉 = 0,

〈ξIJ (t)ξKL(t′)〉 = (δIKδJL + δILδJK)δ(t− t′),(3)

where δIJ is the Kronecker delta function and δ(t−t′) is the Dirac delta function. If we substitutethe forces of Eq. (2) into Eq. (1), we can write the GLE in a mathematically well-defined formof SDE as

dRI =PI

Mdt,

dPI =( ∑

J 6=I

F CIJ +

∑

J 6=I

F DIJ

)dt+

∑

J 6=I

σwR(RIJ )eIJdWIJ ,(4)

where dWIJ = dWJI are independent increments of the Wiener process, and Ito calculus isassumed. Therefore,

dWIJdWKL = (δIKδJL + δILδJK)dt, (5)


where dWIJ is an infinitesimal of order 1/2 in time[7,18]. Therefore, the total force in Eq. (1)should be rewritten as FI =

∑J 6=I

FIJ =∑J 6=I

(F CIJ + F D

IJ + F RIJ/

√dt).

The FPEs corresponding to the SDEs of DPD can be formulated to acquire a steady statesolution of the canonical form[7]. It turns out that if the following relations

wR(R) = w

1/2D (R),

σ = (2kBTγ)1/2

(6)

are satisfied, a DPD system has an invariant canonical distribution. Equation (6) is the cele-brated FDT of DPD, which was first derived by Espanol and Warren[7].

The central task for a successful DPD simulation is to design a proper setting of Rc, α,γ, wC(R), and wD(R), which may be probed by various techniques. To anchor our discussionon a quantitative analysis, we consider a polymer melt system modeled by MD as reference.This idealized MD system is then taken as the “first principle” in this work. Therefore, anyversion of DPD should refer to the results of MD to evaluate its performance. At first, we shallpresent the details of the MD system. This is followed by elaborations of three versions of DPDresulting from two types of coarse-graining techniques.2.1 MD

We consider an MD system of a polymer melt at temperature kBT = 1.0, with kB beingBoltzmann’s constant. The Hamiltonian defining the phase space trajectories Γ ≡ r1, r2, · · · ,rNt

,p1, p2, · · · ,pNt is the sum of the kinetic energy K and potential energy U of the system

H(Γ) = K(Γp) + U(Γr) =

Nt∑

i=1

p2i

2mi+

1

2

∑

i6=j

V (rij), (7)

where Γp and Γr are momentum and position spaces, respectively. Nt is the total number ofmicroscopic/atomic particles. Each polymer is a molecule with chains of beads connected byshort springs. The polymer is isotropic and has a star shape with a center bead connectingNa arms with Nb monomers per arm. Therefore, each polymer contains Nc = Na × Nb + 1microscopic particles. For simplicity, we only consider Na = 10, Nb = 1, and Nc = 11 in thiswork.

The potential energy is the sum of pairwise potentials, each of which has two components asV (r) = VWCA(r) +VFENE(r). The non-bonded interaction is modeled by a truncated Lennard-Jones (LJ) potential, also named as Weeks-Chandler-Andersen (WCA) potential[42],

VWCA(r) =

4ε

((σr

)12

−(σr

)6

+1

4

), r 6 rc,

0, r > rc,(8)

for which rc = 21/6σ so that only the repulsive part of the LJ potential remains. The bondedinteraction is modeled by a finitely extensible nonlinear elastic (FENE) potential[43],

VFENE(r) =

− 1

2kr20 ln(1 − (r/r0)

2), r 6 r0,

∞, r > r0

(9)

with bond strength k = 30ǫ/σ2 and bond reference length r0 = 1.5σ. We further set the lengthscale σ = 1, the energy scale ǫ = 1, the mass of each bead m = 1, and therefore, the time scaleτ = σ(m/ǫ)1/2 = 1. We take the simulation box to be cubic with length L ≈ 30.184 1, andfill N = 1 000 molecules into the box. Therefore, the total number of microscopic particles isNt = 1 000 ×Nc = 11 000, leading to a number density ρ = Nt/L

3 ≈ 0.4.


The trajectories of microscopic particles in phase-space are determined by the Hamilto-nian/Newtonian mechanics as

dri

dt=∂H∂pi

,dpi

dt= −∂H

∂ri. (10)

In simulations, the trajectories are integrated by the measure-preserving Verlet method[44]

with time step δt = 10−3τ . For a canonical ensemble of MD simulations, we apply the virialequation[37] to calculate the pressure of the system at equilibrium, which measures as P = 0.191.The diffusion coefficient for the center of mass (CoM) for each polymer measures as D = 0.119,by either the time derivative of its mean-squared displacement (MSD) or the integral of itsvelocity autocorrelation function (VACF). The kinematic viscosity of the MD system measuresas ν = 0.965, with either a small-perturbed reversible Poiseuille flow[28,45] or transversal currentcorrelation function (TCCF)[34–35]. These values have also been reported in a previous work[28].Results from different measuring techniques agree with each other.

2.2 Empirical DPD

To achieve certain properties of a target fluid by the classical DPD method, input parametersRc, α, γ (or σ), and weight functions wC(R) and wD(R) (or wR(R)) are calibrated empirically,that is, one performs simulations by trials and error until satisfactory properties are generated.Here, we adopt typical weight functions as suggested[18,20]

wC(R) = 1 −R/Rc,

wD(R) = (1 −R/Rc)s.

(11)

DPD simulations are performed in a cubic box with the same size as that of MD. Then, thenumber density of DPD particles is reduced to be ρ/Nc ≈ 0.036 4. Here, we fix the cut-off radiusas Rc = 3.4 as suggested in Subsection 2.3 before calibrating other input parameters. After aseries of simulations by trials and error[28], we set α = 13.5 to reproduce the pressure of MD.We further determine γ = 32.0 and s = 0.5 to reproduce the diffusion coefficient and viscosityof MD. This set of parameters leads to a DPD system with P = 0.194±0.04, D = 0.119±0.002and ν = 1.012 3± 0.061, closely resembling the properties delivered by the MD simulations. Allquantities in DPD are reported also in LJ units of MD.

Note that the empirical input parameters of DPD may also be probed in a more systematicalapproach with a stochastic optimization process[46–47]. Since radial distribution function (RDF)and VACF for particles in the empirical DPD were not set as the optimized target in thebeginning for deducing the input parameters, it is not surprising that they deviate from thoseof MD[28]. About the effects of arbitrariness from Rc and other input parameters on RDFand VACF of the DPD system, we refer to another thorough study[46]. These later subtleties,however, are not of particular concern of the current study and we shall not discuss themfurther.

2.3 Mori-Zwanzig guided DPD

Using an elegant projection technique named as Mori-Zwanzig formalism in non-equilibriumstatistical mechanics[15–16,39], one may derive closed-form dynamic equations for a set of CGvariables, which are functions of microscopic phase space Γ[25,27]. Here, we give a brief intro-duction following a derivation given in Ref. [25].

1There is a typo in Ref. [28] where it was reported as ν = 0.444 on Table 3.


2.3.1 Mori-Zwanzig formalism

In the case of a polymer melt, the CG variables of interest are quantities associated withthe CoM of each polymer molecule such as its position and translational momentum,

RI =1

MI

Nc∑

i=1

mIirIi, MI =

Nc∑

i=1

mIi, (12)

PI =

Nc∑

i=1

pIi. (13)

We further define the phase-space density for the CG variables as

fs(Γs(t); Γs) = δ(Γs(t) − Γs) =N∏

I

δ(RI(t) − RI)δ(PI(t) − PI), (14)

where the CG phase-space coordinate is defined as Γs = R1,R2, · · · ,RN ,P1,P2, · · · ,PN and

Γs = R1, R2, · · · , RN , P1, P2, · · · , PN are the corresponding field variables. The probabilitydistribution function at equilibrium for the microscopic phase space Γ is taken as the canonicalensemble, that is, φ(Γ) = Z−1 exp(−βH) with β = 1/(kBT ) and Z−1 being the normalizingpartition function. Therefore, the probability distribution function at equilibrium for the CGphase space Γs is

ϕ(Γs) = Z−1

∫dΓδ(Γs − Γs) exp(−βH). (15)

We define configuration and momentum distribution functions as

ω(ΓRs ) =

∫dΓrδ(R − R) exp(−βU)∫

dΓr exp(−βU), (16)

ψ(ΓPs ) =

∫dΓpδ(P − P ) exp(−βK)∫

dΓp exp(−βK), (17)

where ΓRs and ΓP

s are the position and momentum spaces of CG field variables, respectively,

and the integrals are always for Nt dimensions. Then, ϕ(Γs) = ω(ΓRs )ψ(ΓP

s ).

To construct a Hilbert space for dynamical variables, we define the scalar product as thecorrelation function in the canonical ensemble at equilibrium,

(A,B) = 〈A(Γ)B∗(Γ)〉 = 〈A(Γ)B(Γ)〉 =

∫dΓA(Γ)B(Γ)φ(Γ), (18)

where dynamical variables are considered to be real so that B∗(Γ) = B(Γ). If we utilize the

set of phase-space density functions fs(Γs; Γs) as the basis in the Hilbert space, and furtherrequire the completeness condition[25,40]

〈fs(Γs; Γ′s)fs(Γs; Γ

′′s )〉 = δ(Γ′

s − Γ′′s )ϕ(Γ′

s), (19)

then any dynamic variable A(Γ) can be decomposed into two parts by two complementaryprojection operators P + Q = 1 as

A [Γ(t)] = AP [Γ(t)] +AQ [Γ(t)] = PA [Γ(t)] + QA [Γ(t)] . (20)


Since we expect AP(Γ(t)) to be expanded by the basis fs(Γs; Γs) in the Hilbert space, theprojection operator P is naturally defined as

PA [Γ(t)] =

∫dΓ′

s

∫dΓ′′

s fs(Γs; Γ′s)(fs(Γs; Γ

′s), fs(Γs; Γ

′′s ))−1(fs(Γs; Γ

′s), A [Γ(t)]). (21)

Therefore, the definition for Q is Q = 1 − P , or QA [Γ(t)] = A [Γ(t)] − PA [Γ(t)].Applying the definition of projection operators, we may decompose the evolution equation

dfs(Γs(t); Γs)/dt, and thereafter obtain∫

dΓsPIdfs(Γs(t); Γs)/dt. The last term by definitionis the evolution equation for the CG momentum and is expressed as[25]

dPI

dt=

1

β

∂

∂RIlnω(ΓR

s ) − β

N∑

J=1

∫ t

0

ds〈[δFQI (t− s)][δFQ

J (0)]T〉PJ (s)

MJ+ δFQ

I (t) (22)

with the random forces defined as

δFI = FI − 1

β

∂

∂RIlnω(ΓR

s ), (23)

δFI(t) = exp(−iLt)δFI , (24)

δFQI (t) = exp(−QiLt)δFI . (25)

Here, δFI is the fluctuating force, which equals the difference between the instantaneous totalforce FI and mean force on particle I. To verify that the first term on the right-hand side ofEq. (22) is the mean force, we have

1

β

∂

∂RIlnω(ΓR

s ) =1

β

∂ω∂RI

ω(ΓRs )

=

∫dΓR

s δ(R − R)(− ∂U

∂RI

)∫

dΓRs δ(R − R)e−βU

= 〈FI〉Γs, (26)

which is the mean force. We also note the essential difference between δFI(t) and δFQI (t), where

the former evolves with time by the usual microscopic propagator exp(−iLt), whereas the latterevolves with time by the orthogonal propagator exp(−QiLt) and stays always orthogonal to theprojected subspace. By integrating Eq. (22) in time, we have information on PI , and thereforecan determine the position trajectory dRI = PIdt for each CG particle as well.

Although Eq. (22) is insightful and exact, it is still generally difficult to apply for a practi-cal simulation, as it is an integro-differential equation involving many-body effects, with non-Markovian terms and projected dynamics. Firstly, we assume that the mean force is pairwisedecomposable and many-body terms are neglected,

〈FI〉Γs≈

N∑

J 6=I

〈FIJ 〉Γs. (27)

Therefore, the pairwise term can be calculated as a function of distance between CoMs inan ensemble of MD simulations. Secondly, for the convolution (memory) term, we make aMarkovian approximation, i.e.,

〈[δFQI (t− s)][δFQ

J (0)]T〉 = 2ΓIJδ(t− s), (28)

ΓIJ =

∫ ∞

0

dt〈[δFQI (t)][δFQ

J (0)]T〉. (29)

This approximation is justified, as the CoMs are slow variables compared with the MD[28].


The last assumption is to approximate the projected dynamics by the real dynamics, that is,exp (−iQLt) ≈ exp (−iLt). To calculate Γµν , if we simply took the integral for the correlationof real dynamics up-to ∞ as

ΓIJ =

∫ ∞

0

dt〈[δFI(t)][δFJ (0)]T〉, (30)

we would run into the plateau problem[48], i.e., the integral vanishes with ΓIJ ≡ 0[16,49]. How-ever, if we take the integral up-to τ as

ΓIJ =

∫ τ

0

dt〈[δFI(t)][δFJ (0)]T〉, (31)

where an inequality τc ≪ τ ≪ 1/ΓIJ holds. Thus, we do have a plateau region and achievea definite good approximation of ΓIJ

[16,49]. In practice, we apply Eq. (31) with τ dynamicallydetermined so that the integral always reaches the maximum value of all possible τ . In this way,we calculate ΓIJ as a function of distance between CoMs in the ensemble of MD simulations.

If the rotation of the CG particle is of interest, the angular momentum can be defined foreach polymer molecule as

LI =

Nc∑

i=1

(rIi − RI) × (pIi − PI) = IIΩI , (32)

where the averaged moment of inertial is II = 6.55 for the star polymer and ΩI is the angularvelocity. Accordingly, the evolution equation of LI is also available.

2.3.2 Parameters of DPD

Following the Mori-Zwanzig formalism described above, we obtain DPD parameters as α =795.69, γ = γ|| = 146.18, and weight functions as wC(R) = (1 + 4R/Rc)(1 − R/Rc)

4 and

wD(R) = w||D(R) = (1 + 3R/Rc)(1 − R/Rc)

3 with Rc = 3.32. The form and coefficients ofthe polynomials for the weight functions are obtained by fitting the numerical data generatedby MD simulations[28]. For reference, we term this model as Mori-Zwanzig dissipative particledynamics or MZ-DPD for brevity. Given γ and wD(R), we may obtain σ and wR(R) from theFDT of Eq. (6). Finally, we generate Gaussian white noises for the pairwise random force ofMZ-DPD to perform an ensemble of DPD simulations.

It is remarkable that in the MD simulations of polymer melt, the instantaneous forcesbetween different polymer molecules are not along the radial direction eIJ of their CoMs, butin all directions. Since the conservative force depends only on relative positions, its averagevanishes in all directions but the direction along eIJ due to symmetry. However, the dissipativeforce depends not only on relative positions, but also on relative velocities. Therefore, we need tointroduce the transversal component in addition to the radial component between two moleculesto represent the extra degree of freedom in the MD system[50]. Meanwhile, rotational degreesof freedom for each DPD particle also need to be considered so that both linear and angularmomenta of the system are conserved at the CG level. The extra transversal and rotationaldissipative forces have already been hypothesized in a phenomenological way in a previouswork[50]. For the full dynamics of the CG variables, we term the model as Mori-Zwanzig fullDPD, or MZ-FDPD for brevity, and the equation of motion reads[28]

dLI

dt= TI =

∑

J 6=I

RIJ

2× FIJ , (33)


dPI

dt=

∑

J 6=I

FIJ =∑

J 6=I

α · ωC(RIJ )eIJ

−∑

J 6=I

γ‖ · ω‖D(RIJ )(eIJ · VIJ )eIJ +

∑

J 6=I

1√3σ‖ · ω‖

R(RIJ )∆t−1/2 · tr[dWIJ ]eIJ

−∑

J 6=I

γ⊥ · ω⊥D(RIJ)(VIJ − (eIJ · VIJ )eIJ ) −

∑

J 6=I

γ⊥ · ω⊥D(RIJ )

(RIJ

2× (ΩI + ΩJ )

)

+∑

J 6=I

√2σ⊥ · ω⊥

R (RIJ)∆t−1/2 · dW AIJ · eIJ , (34)

where TI is the torque for particle I. Here, γ||, w||D, w

||R, and σ|| are for the dissipative and

random forces, respectively, along the radial direction eIJ , while γ⊥, w⊥D , w⊥

R , and σ⊥ are forthe dissipative and random forces, respectively, on the plane perpendicular to eIJ . dWIJ isa matrix of independent Wiener process with dW A

IJ being its antisymmetric part. Therefore,based on the parameters and weight functions for MZ-DPD, we further have γ⊥ = 110.76 andw⊥

D(R) = (1 + 3.95R/Rc)(1 − R/Rc)3.95 with Rc = 3.32. It is worth noting that MZ-DPD

is just a simplified model by keeping only the forces along the radial direction eIJ betweentwo particles in MZ-FDPD. MZ-DPD does not consider the equation of motion of angularmomentum either.

As the conservative force determines the static properties, both MZ-DPD and MZ-FDPDproduce the same pressure of P = 0.193, close to that of MD. However, due to lack of the extradegree of freedom in the dissipative force, MZ-DPD underestimates the kinematic viscosity asν = 0.851 and overestimates diffusion coefficient as D = 0.138. In contrast, the MZ-FDPD withall necessary degrees of freedom represented on the CG level. It resembles the MD closely withD = 0.12 and ν = 0.954. Before further discussing about the performance of different versionsof DPD, we introduce the concept of correlation functions in the next section, which is crucialin time-dependent processes.

3 CCFs

Temporal correlation functions play a similar role in time-dependent processes as the parti-tion function in equilibrium thermodynamics. For a comprehensive discussion on time-dependentphenomena, we refer to Refs. [37], [38], and [49]. In particular, here, we are concerned with auto-correlation functions, i.e., correlations between the same dynamical variable at different times.More precisely, if A(t) = A[r1(t), r2(t), · · · , rN (t),p1(t),p2(t), · · · ,pN(t)] is a dynamical vari-able of coordinates and momenta of all particles of a system, its autocorrelation function isdefined as

CAA(t′, t′′) = 〈A(t′)A∗(t′′)〉 = 〈A(t)A∗(0)〉 = CAA(t), (35)

where t = t′−t′′ > 0 and A∗ is the complex conjugate of A. Angular brackets denote equilibriumensemble average or time average. We assume that the system is ergodic so that both averagesare equivalent. At equilibrium, the system is stationary so that the correlation functions areinvariant under time translation. Therefore, CAA(t′, t′′) = CAA(t = t′ − t′′). Autocorrelationfunctions are real functions of time[37,49].

For a hydrodynamic variable described by the hydrodynamics equations such as mass den-sity or momentum density, which is a continuous fields variable, we have the correspondingmicroscopic expression of the dynamic variable in an MD or DPD system. The microscopicdefinition is

A(r, t) =N∑

i=1

ai(t)δ(r − ri(t)), (36)


where ai is the mass or momentum of particle i. According to Eq. (36), the microscopic massdensity and momentum density are defined as

ρ(r, t) =

N∑

i=1

mδ(r − ri(t)), (37)

j(r, t) =N∑

i=1

muiδ(r − ri(t)), (38)

where mass for each particle is a constant m and ui is the velocity for particle i. The corre-sponding continuous variables are related to the microscopic variables as

ρ(r, t) =1

v

∫

v

ρ(r′ − r, t)dr′, (39)

j(r, t) =1

v

∫

v

j(r′ − r, t)dr′. (40)

To distinguish, we denote the continuous dynamic variable with an over-bar as A(r, t). Thedefinitions of the continuous variables require a “coarse-graining” explicitly in space over avolume v. The size of v must be macroscopically small to allow for a continuum definition.Meanwhile, it must be microscopically large so that the relative fluctuations of the number ofparticles within v are negligible[37]. When v is sufficiently large, Eqs. (39) and (40) imply acoarse-graining in time as well. However, an exact value of v is problem-dependent and is notdefinitely specified in general.

To compare the autocorrelation functions among the results of MD, DPD of different ver-sions, and hydrodynamic equations, we should avoid the ambiguity of the same dynamic vari-able in physical space, which may be interpreted differently by the three distinct descriptions.Therefore, we consider the Fourier-transformed counterpart of the dynamical variable in k-spacedefined as

Ak(t) =

∫

V

A(t) exp(−ik · r)dr,

Ak(t) =

∫

V

A(t) exp(−ik · r)dr,

(41)

where the integral is taken over the whole volume V of the system and k is the wave vector.Since it is translation invariant at equilibrium, there is no correlation between Ak and Ak′ when

k 6= k′[51]. Accordingly, the corresponding autocorrelation function in k-space with the same k

is defined as

CAA(k, t) = 〈Ak(t)A∗k(0)〉 = 〈Ak(t)A−k(0)〉 =

∫

V

CAA(t) exp(−ik · r)dr. (42)

In particular, when the dynamic variable is Ak(t) = jk(t), the autocorrelation function is a2nd rank tensor and it is named as CCFs. Due to isotropy, it is diagonal and may be furtherdecomposed into transversal and longitudinal components as

Cαβjj (k, t) = kαkβCL(k, t) + (δαβ − kαkβ)CT(k, t), (43)

where α, β = x, y, or z and kα, kβ are Cartesian components of the unit vector k = k/k.Without loss of generality, we take k = (k, 0, 0)T. Therefore, the longitudinal and transversalcurrent correlation functions (LCCFs and TCCFs), are expressed as

CL(k, t) = 〈jx

k (t)jx−k(0)〉,

CT(k, t) = 〈jyk (t)jy

−k(0)〉 = 〈jzk(t)jz

−k(0)〉. (44)


More specifically, if we consider the dynamics at continuum, that is, lim k → 0, we may ig-nore the difference between jk(t) and jk(t) and consider CT(k, t) ≈ CT(k, t) and CL(k, t) ≈CL(k, t)[37–38]. Therefore, we shall neglect the over-bars on the dynamical variables withoutcausing ambiguity at lim k → 0. Moreover, the continuum CCFs can also be expressed by thesolutions of linearized hydrodynamic equations. We shall revisit such solutions as follows.3.1 Hydrodynamic solutions

A dynamic variable is conserved if it satisfies a continuity equation in a differential form as

∂A(r, t)

∂t+ ∇ · jA(r, 0) = 0, (45)

where jA is the current associated with the variable A. Therefore, the conservation laws formass and momentum are expressed as

∂ρ(r, t)

∂t+ ∇ · j(r, 0) = 0, (46)

∂j(r, t)

∂t+ ∇ ·Π(r, 0) = 0, (47)

where j is the mass current or momentum density introduced in Eqs. (38) and (40). Π is themomentum current or stress tensor, and for a Newtonian fluid, it is defined as

Παβ = δαβP (r, t) − η(∂uα(r, t)

∂rβ+∂uβ(r, t)

∂rα

)+ δαβ

(2

3η − ζ

)∇ · u(r, t), (48)

where P (r, t) is the local pressure, and η and ζ are dynamic shear and bulk viscosities,respetively. If we choose a frame of reference where the mean velocity vanishes, that is,u = 〈u(r, t)〉 = 0, Eqs. (46) and (47) may be linearized by considering the variations of thehydrodynamic variables as

ρ(r, t) = ρ+ δρ(r, t), (49)

u(r, t) = u + δu(r, t) = δu(r, t), (50)

j(r, t) = (ρ+ δρ(r, t))(u + δu(r, t)) ≈ ρδu(r, t) = ρu(r, t). (51)

A variation of second order is dropped leading to one “≈” sign in j(r, t). Inserting Eq. (51)into Eqs. (46) and (47), we obtain

∂δρ(r, t)

∂t+ ∇ · j(r, t) = 0, (52)

∂j(r, t)

∂t+ ∇P (r, t) − η

ρ∇2j(r, t) − η/3 + ζ

ρ∇∇ · j(r, t) = 0. (53)

Furthermore, ∇P (r, t) = ∇(P + δP (r, t)) = ∇δP (r, t) = c2T∇δρ(r, t), with cT being theisothermal sound speed. The above equations can be conveniently solved by performing aspace-Fourier transform into k-space introduced in Eq. (41),

∂ρk(t)

∂t+ ik · jk(t) = 0, (54)

∂jk(t)

∂t+ ic2Tρk(t)k + νk2jk(t) + νLkk · jk(t) = 0, (55)


where the shear kinematic and longitudinal kinematic viscosities are introduced as

ν =η

ρ, (56)

νL =1

ρ

(4η

3+ ζ

). (57)

It is apparent from Eq. (54) that the evolution of density ρk(t) is coupled with the longitudinalcomponent of the momentum density jx

k (t) = k · jk(t), but decoupled from the transversalcomponents. Therefore, we do not consider 〈ρk(t)ρ−k(0)〉 separately, as it provides merelya redundant information as CL(k, t). This observation is valid not only at lim k → 0, butat an arbitrary k, as Eq. (54) utilizes the mere fact of mass conservation without any otherassumptions. Therefore, we may rewrite Eqs. (54) and (55) as four equations of scalar variables

∂ρk(t)

∂t+ ikjx

k (t) = 0, (58)

∂jxk (t)

∂t+ ic2Tρk(t)k + νLk

2jxk (t) = 0, (59)

∂jyk(t)

∂t+ νk2jy

k(t) = 0, (60)

∂jzk(t)

∂t+ νk2jz

k(t) = 0. (61)

Equations (60) and (61) are first-oder linear homogeneous ordinary different equations (ODEs),which are readily solved as[52]

jyk (t) = jy

k (0) exp(−νk2t), (62)

jzk(t) = jz

k(0) exp(−νk2t). (63)

Equations (58) and (59) are two coupled ODEs and may be written in a matrix form as

dak(t)

dt= Hak(t), (64)

where a and H are defined as

ak(t) = (ρk(t), jxk (t))T, (65)

H =

[0 −ik

−ic2Tk −νLk2

]. (66)

The general solutions to Eq. (64) are determined by the eigenvalues of H , which can be obtainedby further solving det(H − λI) = 0[52]. It is easy to show that the eigenvalues are

λ1 = −ΓTk2 + isTk, λ2 = −ΓTk

2 − isTk (67)

with

ΓT = νL/2, sT =√

4c2T − ν2Lk

2/2, (68)

where the sound attenuation coefficient is introduced as ΓT. Here, we consider an under-dampedsolution, where sT is real, that is, k < 2cT/νL. In particular, we consider a continuum limitwhere k ≪ 2cT/νL. Therefore sT ≈ cT and we arrive at the solutions as

ρk(t) = exp(−ΓTk2t)(cos(cTkt)ρk(0) − i/cT sin(cTkt)j

xk (0)), (69)

jxk (t) = exp(−ΓTk

2t)(cos(cTkt)jxk (0) − icT sin(cTkt)ρk(0)). (70)


The normalized temporal autocorrelation functions of the fluctuating variables in k-space arereadily expressed as

C′T(k, t) =

〈jyk (t)jy

−k(0)〉〈jy

k (0)jy−k(0)〉 =

〈jzk(t)jz

−k(0)〉〈jz

k(0)jz−k(0)〉 = exp(−νk2t), (71)

C′L(k, t) =

〈jxk (t)jx

k (0)〉〈jx

k (0)jxk (0)〉 = exp(−ΓTk

2t) cos(cTkt), (72)

where⟨jx−k(0)ρk(0)

⟩= 0 is assumed to derive C′

L(k, t). In the subsequent section, we alwayscalculate the normalized quantities C′

T(k, t) and C′L(k, t) and therefore omit the sign “′” without

ambiguity. It is worth noting that the TCCFs and LCCFs are self-similar at continuum scalesand they scale with powers of k. Therefore, self-similar results of MD or DPD at different kwould indicate a continuum behavior, whereas different properties at different k resort to anon-continuum behavior.

4 Results and discussion

Here, we calculate the CCFs for the polymer melt model introduced in Section 2 fromsimulations by MD, empirical DPD, Mori-Zwanzig guided DPD of two versions, and solutionsof the linearized hydrodynamic equations. Thereafter, we are able to compare the results fromDPD simulations of different versions, against that of MD and of hydrodynamic solutions.Since we are focusing on the CCFs in Fourier-transformed k-space, we avoid the difficulty ofone-to-one correspondence between hydrodynamic variables expressed by particles in MD andhydrodynamic variables expressed in the other two coarser descriptions. The input parametersfor MD and DPD simulations are as given in Section 2. The wave vector is taken as kn =(kn, 0, 0) with kn = 2πn/L, where n is a positive integer number and L is the length of a cubicbox. We will consider n = 1, 2, 3, and 4, where for the first two wave numbers, a continuumbehavior is apparent, while for the latter two wave numbers, a particle behavior is prevalent.

4.1 Continuum limit

We compare TCCFs of MD and continuum theory of Eq. (71), where the latter adoptsν = 0.965 as in MD simulations. From Fig. 1(a), it is clear that at length scales L and L/2(wave number k1 and k2 accordingly), MD simulations behave like continuum hydrodynamicsand its TCCFs follow closely exponential decays, where the decay rate is νk2. In continuumhydrodynamics, the determining parameter of TCCFs is the kinematic viscosity ν = 0.965(η = 0.386, ρ = 0.4). Therefore, the classical DPD with empirical input parameters for targetingthe MD viscosity produces almost identical TCCFs as those of MD and continuum theory, asshown in Fig. 1(a). However, with the same number of degrees of freedom as that of the empiricalDPD, MZ-DPD underestimates the decay rate of TCCFs, as shown in Fig. 1(b). This is notsurprising given the fact that MZ-DPD underestimate the viscosity of MD, as reported alreadyin Subsection 2.3. In contrast, with all necessary degrees of freedom on the CG level, MZ-FDPD reproduces well the viscosity of MD, and therefore also the TCCFs of MD, as indicatedin Fig. 1(c).

To recapitulate, TCCFs of DPD resemble those of MD and hydrodynamic solutions atcontinuum scales (k1 and k2) with the same exponentially decaying law characterized by thekinematic viscosity. This is true as long as the version of DPD reproduces the same viscosityof MD or continuum theory, as the viscosity is the only determining factor for TCCFs in thecontinuum scales and TCCFs are self-similar with exponential-decay rates νk2.


Fig. 1 Transversal current correlations of DPD compared with those of MD and continuum theory:continuum limit with wave numbers k1 = 2π/L and k2 = 4π/L

For the LCCFs, the situation is more complicated than that for TCCFs, as the formerpresent themselves with both decaying and oscillating behaviors characterized by ΓT and cT, asshown in Eq. (72). We may fit well the hydrodynamic solution (72) to MD results, as indicatedin Fig. 2(a). After the fitting, parameters ΓT = 1.17 and cT = 1.3 are determined as well. Asfurther by-products, we determine ζ = 0.42, νL = 2.34. Since the empirical DPD is developedto target the pressure and viscosity of the MD system, it is not expected to reproduce ζ andcT of MD well. This deficiency is reflected by the mismatch on LCCFs between DPD and MDin Fig. 2(a). In contrast, the Mori-Zwanzig guided DPD and full DPD reproduce the LCCFs ofMD very well, as indicated in Figs. 2(b) and 2(c). It is also interesting to note that there is noapparent difference between the performance of MZ-DPD and MZ-FDPD on LCCFs, althoughMZ-FDPD has non-radial interactions between particles. This indicates that the longitudinalmodes (ΓT and cT) are purely determined by the radial-direction interactions between particles.

We note that for the approximation made in Eq. (68) for sT ≈ cT, we have assumed thatk1 ≈ 0.208 or k2 ≈ 0.416 ≪ 2cT/νL ≈ 1.11 is valid. From the results shown above, theassumption of “≪” does not lead to an apparent deficiency of the hydrodynamic solutions onLCCFs.4.2 Below continuum limit

If we further consider smaller scales k3 = 6π/L and k4 = 8π/L, we clearly observe thatthe decay of TCCF for MD is no longer strictly exponential, as indicated in Fig. 3. Here, thesolutions of hydrodynamic equations are still exponential with kinematic viscosity taken asν = 0.965. Since we take MD as the first principle, its results are reliable and should be takenas reference. Therefore, the hydrodynamics solutions are incorrect at these small scales. Fromthe development of generalized hydrodynamics[37–38], it is not difficult to realize that we havea crude assumption for our hydrodynamic equations; the fluid is assumed to be Newtonian at


all scales in Eq. (48), that is, Παβ = −η(

∂uα(r,t)∂rβ

+∂uβ(r,t)

∂rα

)for α 6= β. This is a Markovian

relation between the shear stress and instantaneous shear rate. For fluids (not only this starpolymer system, but also simple fluids such as argon) at sufficiently small scales, there is ingeneral a non-Markovian relation, which determines the shear stress from the history of shearrates. This is to say that the fluid is viscoelastic at small scales[37–38,53]. Therefore, if one wantsto compare hydrodynamic solutions with that of MD, one needs to model this non-Markovianrelation phenomenologically (with memory kernels such as exponential, and Gaussian functions)or determine the non-Markovian relation from the projection technique introduced earlier[54–56].Once the memory kernel is available, it is possible to obtain accurate hydrodynamic solutionsgiven that the non-Markovian hydrodynamic equations are still solvable. This is a commonpractice during the development of generalized hydrodynamics and we shall not proceed furtherin this work.

-

-

-

-

-

-

Fig. 2 Longitudinal current correlations of DPD compared with those of MD and continuum theory:continuum limit with wave numbers k1 = 2π/L and k2 = 4π/L

It is remarkable that the TCCFs of the empirical DPD resemble those of MD quite wellat k3 and k4, especially for intermediate and long time (t & 2), as indicated in Fig. 3. This issurprising as the interacting laws between DPD particles are Markovian and the non-exponentialbehavior of TCCFs is non-Markovian in origin. It appears that certain coherent movementsof DPD particles at length scales of 2π/k3 = L/3 and 2π/k4 = L/4 manifest themselves asnon-Markovian effects in the TCCF and show viscoelastic effects. We need to mention that theviscoelastic effects from MD at these small scales are also due to the coherent movements ofMD particles. So far the analogy, if there is any relation at all, between the TCCFs of MD andDPD at k3 and k4 is still not clear.


Fig. 3 Transversal current correlations of MD, continuum theory, and DPD with empirical parame-ters: below continuum limit with wave numbers k3 = 6π/L and k4 = 8π/L

Moreover, the TCCFs of MZ-FDPD resemble those of MD even better, except for t → 0,as indicated in Fig. 4. We note that the interacting laws between particles in MZ-FDPDare also Markovian and their differences from empirical DPD are the actual values of co-efficients and weight functions. These results suggest that even with empirical MarkovianDPD, we do have the possibility to adjust input settings wisely so that we recover non-Markovian/viscoelastic/generalized hydrodynamic behaviors of a reference MD system.

-

-

Fig. 4 Transversal current correlations of fluctuating hydrodynamics theory, MD, and Mori-Zwanzigfull DPD: below continuum limit with wave numbers k3 = 6π/L and k4 = 8π/L

For the LCCF, the results can neither be recovered by the hydrodynamic solutions nor bythe empirical DPD, as indicated in Fig. 5. For the deficiency from the hydrodynamic solutions,the first questionable assumption is the same as in the case of TCCF that a Markovian relationbetween stress and strain rate is invalid, and furthermore η and ζ should be also scale-dependent.Even if the Markovian relation was correct, the assumption about sT ≈ cT is invalid, as it impliesthat k3 ≈ 0.624, k4 ≈ 0.833 ≪ 2cT/νL ≈ 1.11. For the disagreement from the empirical DPDsimulations, it is not surprising, as we never intend to recover the LCCF when deriving theempirical interactions between the particles. The disagreements were already observed for k1

and k2. However, it is remarkable to note that for these small scales at k3 and k4, the LCCFsof MZ-FDPD represent reasonably well of that of MD, as indicated in Fig. 6, except at verysmall times (equivalently higher frequencies). This suggests again that it is possible to adjustinput settings of DPD so that generalized hydrodynamic behavior of a reference MD systemmay be recovered.


Fig. 5 Longitudinal current correlations ofMD, continuum theory, and DPD withempirical parameters: below contin-uum limit with wave numbers k3 =6π/L and k4 = 8π/L

-

-

Fig. 6 Longitudinal current correlations ofMD, continuum theory, and Mori-Zwanzig full DPD: below continuumlimit with wave numbers k3 = 6π/Land k4 = 8π/L

5 Summary

We present three versions of DPD; the first version is empirical so that the pressure, viscosity,and diffusivity of a reference system are simultaneously well reproduced. The force fields of thesecond and third versions are obtained from a systematical coarse-graining procedure within theMori-Zwanzig projection formalism by keeping the properties of the CoM in MD descriptions.The difference between 2nd and 3rd versions are that the former considers only radial-directioninteractions between neighboring particles, just as the empirical/classical DPD formulations,while the latter considers in addition transversal and rotational interactions between particles.They are named MZ-DPD (Mori-Zwanzig DPD) and MZ-FDPD (Mori-Zwanzig full DPD),respectively.

To have some quantitative comparison, we consider a specific system of star-polymer meltat number density ρ = 0.4 modeled by MD as a reference system, which we consider as thefirst principles in this work. Since the coarse-graining level is Nc = 11, that is, eleven atomsare coarse-grained into one DPD particle, time scales between CG variable and MD dynamicsare well separated. Therefore, we may replace the memory kernel with a Dirac delta function,and furthermore the friction forces between DPD particles are Markovian and random forcesare white-colored. We also approximate the random force, which evolves by an orthogonalpropagator, by the usual propagator describing the real dynamics. This is justified for theMarkovian description, as long as we calculate the friction coefficients by taking the maximumvalue from the integral of time correlations of random force. As a result, the typical plateauproblem is avoided.

Previously, we have evaluated the performance of DPD of three versions such as pressure,viscosity, diffusion, and velocity autocorrelation function of the CoM of the polymer, in com-parison with results of MD in physical space[28]. If we want to compare the performances ofDPD further with hydrodynamics solutions, we would have to introduce the definition of hy-drodynamic field variables for MD and DPD. This involves an arbitrary length scale as shownin Eq. (40), which is generally difficult to define. Therefore, in physical space, it is difficult tocompare fairly the dynamic processes at different scales described by MD, DPD, and hydrody-namic equations. However, in the Fourier-transformed k-space, we can focus on a particularwave number or length scale at each time and compare the three descriptions in detail. For suchpurposes, we adopt and revisit the concept of the CCFs. From the calculations of transversal


and longitudinal CCF, or TCCFs and LCCFs for brevity, we observe that at continuum scales,results of DPD of three versions resemble overall well those of MD and hydrodynamic solutionsfor a Newtonian fluid. One notable disagreement is that LCCFs of empirical DPD deviate fromthose of MD and hydrodynamic solution. This is not surprising, as the target for optimizingthe DPD interactions does not include sound speed cT or longitudinal viscosity ΓT. Anothernotable disagreement is that TCCFs of MZ-DPD deviate slightly from those of MD. This is dueto the underestimation of MD viscosity from MZ-DPD, which does not consider transversal orrotational interactions between neighboring particles. The empirical DPD also only considersradial-direction interactions between neighboring particles, but does not have a problem withTCCFs. This is because the friction coefficient γ and weight function wD(R) in empirical DPDare free to tune so that it compensates the missing effects of transversal/rotational interactionson the viscosity, while the friction coefficient and weight function in MZ-DPD are completelydetermined by the projection procedure.

Not only for the polymer fluids of small molecules considered here, but also for simple flu-ids such as argon, the dynamical behavior is viscoelastic/non-Markovian at small scales[38].Therefore, the hydrodynamic equations for a Newtonian fluid do not describe the dynamicsaccurately at small scales. This is reflected by the TCCFs and LCCFs calculated at k3 andk4, where hydrodynamics solutions deviate significantly from those of MD. However, the MZ-DPD and MZ-FDPD are two Markovian approximations of the GLEs. It is remarkable thatthe Markovian interacting laws reproduce quite well the non-Markovian behavior of the TCCFand LCCF from MD. Even more remarkable is that the empirical DPD which has Markovianinteraction laws as well, can also reproduce that of MD at small scales, represented by thecomparison of TCCF between them. These phenomena are no coincidence and we think thatthe coherent movements2 of DPD particles should be explored more so that even an empiricalMarkovian DPD may recover the generalized hydrodynamics to some extent. We report ourfirst observations here and further investigations are on the way.

Acknowledgements Z. LI acknowledges funding support of the U. S. Army Research Laboratorywith Cooperative Agreement No. W911NF-12-2-0023.

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