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arXiv:cond-mat/0602523v1 [cond-mat.soft] 22 Feb 2006

A stochastic Trotter integration scheme for

dissipative particle dynamics

M. Serranoa,∗G. De FabritiisbP. Espa˜ nolaP.V. Coveneyb

aDepartamento de F´ ısica Fundamental, UNED

Senda del Rey 9, 28040 Madrid, Spain

bCentre for Computational Science, Department of Chemistry,

University College London, 20 Gordon street, WC1H 0AJ, London, UK

Abstract

In this article we show in detail the derivation of an integration scheme for the dis-

sipative particle dynamic model (DPD) using the stochastic Trotter formula [1]. We

explain some subtleties due to the stochastic character of the equations and exploit

analyticity in some interesting parts of the dynamics. The DPD-Trotter integrator

demonstrates the inexistence of spurious spatial correlations in the radial distribu-

tion function for an ideal gas equation of state. We also compare our numerical

integrator to other available DPD integration schemes.

Key words: Trotter stochastic formula, dissipative particle dynamics

PACS: 05.40.-a, 05.10.-a, 02.50.-r

1 Introduction

Mesoscopic models require the use of stochastic differential equations (SDEs)

to include the effects of thermal fluctuations so the selection of an efficient

stochastic integration scheme is crucial to simulate correctly these systems. In

this article we focus on dissipative particle dynamics (DPD) [2,3] as one of the

most simple and widely used model. Although nowadays the conventional DPD

model has a good theoretical basis, in the past few years there has been quite

a controversy reported in literature about practical aspects of the simulations:

the appearance of spurious effects related to time discretization, in concrete,

the unphysical systematic drift of the temperature from the value predicted

by the fluctuation-dissipation theorem and uncontrolled spatial correlations

∗Corresponding author. E-mail address: mserrano@fisfun.uned.es

Preprint submitted to Elsevier Science6 February 2008

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among particles. This is the reason of the increasing interest in developing good

integrator methods for the DPD model. Several authors [4,5,6] have considered

improvements to the basic stochastic Euler scheme through the use of solvers

that have been successfully employed for deterministic dynamical systems in

molecular dynamics (MD) simulations [7] such as the velocity Verlet algorithm

(DPD-VV [4]).

Recently in [1] we investigated the applicability of the Trotter formula (widely

used in molecular simulations) to a general SDEs and discuss the optimum

way to split the dissipative-stochastic generators. It resulted that the Trotter

formula cannot be applied without considering the special stochastic character

of the equations. In general, different variables depending on the same noise

should not be split. In the DPD equations this does not happen which allowed

us to write a integration scheme. In [1] we concluded that, considering the

accuracy of the equilibrium temperature and the computational cost, DPD-

Trotter is among the best integrators for the DPD equations.

In this article we explain in detail how to apply the stochastic Trotter formula

to the particular case of DPD. The aim is to furnish a non-trivial example

to be used as a reference when one wishes to derive new integration schemes

based on the stochastic Trotter formula for a general set of SDEs. We also

test the behavior of the radial distribution function in the DPD model. For

an ideal gas equation of state we find the DPD-Trotter scheme presents no

spatial correlations at any scale.

2 A Trotter integration scheme for dissipative particle dynamics

The DPD model consists of a set of N particles moving in continuous space.

Each particle k is defined by its position rkand its momentum pkand mass

m. The dynamics is specified by a set of Langevin equations very similar to

the molecular dynamics equations, but where in addition to the conservative

forces there are dissipative and fluctuating forces as well

drk=pk

mdt,

dpk=?N

where Fc(r) is the conservative pair interaction force weighted by positive and

symmetric parameter akl, rkl= rk− rlis the distance between the particle k

and particle l, rklits length and ekl= rkl/rkl. The weight function ω usually

has a finite range rc. A typical selection is ω(r) = 1 − r/rcfor r < rcand

ω(r) = 0 for r ≥ rc. The conservative force is usually chosen to be of the

form Fc(rkl) = w(rkl). This system has a well defined Gibbsian equilibrium

l?=kekl

??

aklFc(rkl) −

γ

mω2(rkl)(ekl· pkl)

?

dt +√2γkBT0ω(rkl)dWt

kl

?

(1)

,

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state at a temperature T0. Because the stochastic term in conventional DPD

does not depend on the momenta, note the Itˆ o or Stratonovich interpretation

are exactly equivalent (additive noise) and we can apply the standard rules of

ordinary calculus formally treating dW as′′dt′′.

The global state is x = (r1,...,rN,p1,...,pN) and the SDEs (1) can be ex-

pressed as dx = L[x]dt with formal solution x(t) = T eLt[x](x0) where T is

the time-ordered operator (see [8]). The global time generator is divided in two

operators generating “orthogonal dynamics” L = Lr+ Lp, where Lr=?

and Lp=?

particles k and l satisfy action-reaction (Newton’s third law), the momentum

is locally (and totally) conserved. For each partial dynamics, the generator

can be subdivided in components Lk

kLk

r

k,l>kLkl

p. Because in the DPD model the forces between interacting

r=?

µLrµ

kand Lkl

p=?

µLpµ

klwith

Lrµ

k=pµ

k

m∂rµ

?

k,

Lpµ

kl= Dpµ

kl+ Spµ

kl(t),

Dpµ

kl=aklFc(rkl) −γ

?

mωD(rkl)(ekl· pkl)

2γkBT0ω(rkl)eµ

?

eµ

kl(∂pµ

k− ∂pµ

l),

Spµ

kl(t)=f(t)

kl(∂pµ

k− ∂pµ

l).(2)

Note that the momentum operator has two contributions, the deterministic

and the stochastic which is an explicit function of time with f(t) = dWt

kl/dt.

The formal solution of our system x(t) = T eLt[x](x0) corresponds to a con-

tinuous time evolution. In order to devise any integrator scheme we must

discretize the continuous time in finite steps. The continuum time propaga-

tor can be approximated by discrete time steps of size ∆t = t/P, recursively

applying P times the exponential operator eLt≡

Note that when the generator depends explicitly on time L(t) ≡ Lt, the time-

ordered exponential is relevant and the recursively nested exponentials become

T eLtt≈ eLt+P∆t∆t···eLt+2∆t∆teLt+∆t∆t([8]). At this point we must provide

some approximation of the discrete time propagator of a “generic” global dy-

namics eL∆t. As we have mentioned in the previous paragraph, in general the

generator L is formed by many generators

to a particular dynamics i. The generalized Trotter formula (Strang [9]) gen-

erates a straightforward approximation to the time propagator exact up to

second order in time

?

eLt/P?P

≈ eL∆t···eL∆t.

?Lieach of them corresponding

e

?M

i=1Ait=

1

?

i=M

eAi∆t

2

M

?

j=1

eAj∆t

2

P

+ O(∆t3).(3)

Because it can be performed in many possible ways, the most important prac-

tical issue to apply formula (3) is the selection of a particular splitting of the

global dynamics. One reasonable criteria is to keep the minimum number of

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generators and exploit analyticity for each of them whenever possible. For the

stochastic equations of DPD, the splitting we propose consist in 1 +N(N−1)

operators: the global Lrand a Lkl

2

pfor each pair.

The Baker-Campbell-Haussdorff (BCH) formula reads

eAeB= eA+B+1

2[A,B]+1

12[A,[A,B]]+1

12[B,[B,A]]+···

(4)

so for [A,B] = 0 we have the exact formula eA+B= eAeB= eBeA. The DPD

position generator Lr =

generators per particle and components that satisfy [Lrµ

particles k,l (k ?= l) and components µ,ν except µ ?= ν. Therefore we can use

the exact formula

?

k,µLrµ

kis composed by many simple individual

k,Lrν

l] = 0 for all

eLr∆t= e

?

kLk

r∆t≡

N

?

k=1

d

?

µ=1

e

Lrµ

k∆t

(5)

with d the dimensionality. In MD (DPD without dissipative and random

forces) the momentum generator Lp=?

for all pairs of particles kp,lq, with k ?= l, p ?= q, p ?= q and components

µ,ν,µ ?= ν, such that the ordering of the individual-component momentum

generators is absolutely irrelevant. On the contrary in DPD, the forces de-

pend on the other components of the velocity of the particle and also on other

particles velocities and the operator eLp∆t= e

integrated and has to be approximated in some way. This is the reason for the

splitting it in

2

momentum operators. Due to this splitting and formula

(3), the DPD scheme is finally given by the following Trotter integrator

k,µLpµ

klalso satisfies [Lpµ

kp,Lpν

lq] = 0

?

k,l>kLkl

p∆tcannot be globally

N(N−1)

x(t + ∆t) =

N

?

q=1,r>1

eLqr

p

∆t

2

?N

i=1

?

eLi

r∆t

?

1?

k=N,l<N

eLkl

p

∆t

2

x(t). (6)

The propagator that corresponds to the generator Lµ

update which is analytically given by

rkproduces the position

e

Lk

rµ

k

∆t[x] :rµ

k(t + ∆t) = rµ

k(t) +pµ

k(t)

m

∆t(7)

because the momentum is a constant in this step of the scheme. The next

step is to solve the propagator of the momenta of the interaction pair k, l

(corresponding to the generator Lkl

mentioned before that DPD forces satisfy action-reaction, so for a particular

interacting pair k,l we propose to make a change of variables from pk,pl

to pk+ pl,pkl = pk− pl. The new system to solve is d(pk+ pl) = 0 and

dpkl= 2dpk. Because the positions of the particles are “frozen” at this step

of the Trotter scheme, the equation for dpklcan be solved more easily for the

p) independently of the positions. We have

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projection on the radial direction pe

kl= pkl· ekl

kldt + CdWt

dpe

kl= Adt − Bpe

kl, (8)

where A = 2aklFc(rkl), B = 2γ/mω2and C = 2√2γkBT0ω. This equation is

an Ornstein-Uhlenbeck process with analytical solution [10]

pe

kl(t) = e−B∆tpe

kl(t0) + A

?t

t0eB(s−t)ds + C

?t

t0eB(s−t)dWs, (9)

where ∆t = t − t0, t0being the initial time. The solution of (9) requires the

generation of colored noise based on a numerical scheme itself. A version of

the method to generate coloured noise [1,11] adapted to Eq.(9) results

∆pe

kl=

?

pkl· ekl−aklFc

τ

??

e−2∆t

τ − 1

?

+

?

2kBT0m

?

1 − e−4∆t

τ

?

ξkl,(10)

where τ = γ/mω2, ξkl= ξlkare normal distributed with zero mean and

variance one (N(0,1)) and ∆pe

pkand plgives

kl= pe

kl(t) − pe

kl(t0). The propagator eLkl

p∆tfor

eLkl

p∆t[x] : (pt+∆t

k

,pt+∆t

l

) =

?

pk(t) +∆pe

kl

2

ekl(t),

pl(t) −∆pe

kl

2

ekl(t)

?

. (11)

So in DPD we can solve the dynamics corresponding to the generator Lkl

(globally for all components at the same time) without the need to go to the

scalar operator Lpµ

integration algorithm (6) consists of the following steps: for the interaction

pairs k,l update the momentum half timestep according to the propagator

(11) with a noise ξkl; iterate over particles k updating the position according

to (7); finally, update pairs k,l in reverse order again using the propagator

(11) but with new noises ξ′

pair-list only once per iteration and has the same complexity as a simple DPD

velocity-Verlet scheme (DPD-VV [4]).

p

klcorresponding to the coordinate µ. In practice the Trotter

kl. This algorithm requires the calculation of the

We tested in [1] this integration scheme using the open-source code mydpd

[12] for the equilibrium temperature with N = 4000 particles, γ = 4.5,kBT0=

1,m = 1,rc= 1 in a three dimensional periodic box (L,L,L) with L = 10

with periodic boundary conditions. These settings give a particle density ρ =

4. Here, we show in left Fig.1 the radial distribution function for akl = 0

(corresponding to an ideal gas equation of state) and a time step ∆t = 0.05.

We compare the results for three methods: the velocity Verlet (DPD-VV) [4],

the Shardlow scheme [13] and DPD-Trotter [1]. We find good agreement with

the theoretical value 1 for Shardlow and DPD-Trotter integrators but DPD-

VV is notably wrong displaying spurious spatial correlations at distances less

than the finite range rc. In right Fig.1 we show the radial distribution function

for a simulation with akl= 25 and a time step ∆t = 0.01. As we see the three

methods perform very similarly.

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0.98

1

1.02

1.04

1.06

1.08

1.1

1.12

00.51

R

1.52

g(R)

0

0.2

0.4

0.6

0.8

1

1.2

00.51

R

1.52

g(R)

Fig. 1. Radial distribution function for three integrator methods. Velocity Verlet

with (△) symbols, Shadlow scheme with (2) symbols and Trotter DPD with (•)

symbols. Left figure corresponds to an ideal gas simulation akl= 0. Right figure

corresponds to simulations including conservative forces with akl= 25.

3 Conclusions

The stochastic Trotter formula can be successfully applied to the DPD model

and the procedure to tailor the integrator scheme has been explained in detail.

In the scheme we have also exploited the exact integration of important parts

of the dynamics like the conservation of total momentum of an interacting

pair of particles. The DPD-Trotter integrator displays correctly the radial

distribution functions for an ideal gas (no conservative forces among particles)

and also for a non ideal gas. Following this important example and [1] it should

be strathforward to apply the stochastic Trotter formula to new mesoscopic

models and more general SDEs.

Acknowledgements M.S. and P.E. are supported by the Spanish Ministerio

de Educaci´ on y Ciencia project FIS2004-01934 and GDF by the EPSRC Inte-

grative Biology project GR/S72023. PVC & MS thank the EPSRC (UK) for

funding RealityGrid under grant number GR/R67699; this project supported

MS’s 6 month visit to the CCS at UCL during 2005.

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