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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
1Introduction
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.
2A 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
0 0.51
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.
3Conclusions
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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