arXiv:cond-mat/0602523v1 [cond-mat.soft] 22 Feb 2006
A stochastic Trotter integration schem e for
dissipative particle dynam ics
M .Serrano a; G .D e Fabritiisb P.Espa~
nola P.V .C oveney b
aD epartam ento
de F sica Fundam ental,U N ED
Senda delRey 9,28040 M adrid,Spain
b C entre
for C om putationalScience,D epartm ent ofC hem istry,
U niversity C ollege London,20 G ordon street,W C 1H 0A J,London,U K
A bstract
In this article we show in detailthe derivation ofan integration schem e for the dissipative particle dynam ic m odel(D PD )using the stochastic Trotterform ula [1].W e
explain som e subtleties due to the stochastic character ofthe equations and exploit
analyticity in som e interesting parts ofthe dynam ics.T he D PD -Trotter integrator
dem onstrates the inexistence ofspurious spatialcorrelations in the radialdistribution function for an ideal gas equation of state. W e also com pare our num erical
integrator to other available D PD integration schem es.
K ey words: Trotter stochastic form ula,dissipative particle dynam ics
PA C S: 05.40.-a,05.10.-a,02.50.-r
1
Introduction
M esoscopic m odels require the use ofstochastic di erentialequations (SD Es)
to include the e ects of therm al uctuations so the selection of an e cient
stochastic integration schem e iscrucialto sim ulate correctly these system s.In
thisarticle we focuson dissipative particle dynam ics(D PD )[2,3]asone ofthe
m ostsim pleand w idely used m odel.A lthough nowadaystheconventionalD PD
m odelhasa good theoreticalbasis,in the pastfew years there hasbeen quite
a controversy reported in literature aboutpracticalaspectsofthe sim ulations:
the appearance ofspurious e ects related to tim e discretization,in concrete,
the unphysicalsystem atic drift ofthe tem perature from the value predicted
by the uctuation-dissipation theorem and uncontrolled spatialcorrelations
C orresponding author.E-m ailaddress:m serrano@ sfun.uned.es
Preprint subm itted to Elsevier Science
25 A ugust 2016
am ong particles.T hisisthereason oftheincreasing interestin developing good
integratorm ethodsfortheD PD m odel.Severalauthors[4,5,6]haveconsidered
im provem ents to the basic stochastic Euler schem e through the use ofsolvers
that have been successfully em ployed for determ inistic dynam icalsystem s in
m oleculardynam ics(M D )sim ulations[7]such asthevelocity Verletalgorithm
(D PD -V V [4]).
R ecently in [1]we investigated the applicability ofthe Trotterform ula (w idely
used in m olecular sim ulations) to a generalSD Es and discuss the optim um
way to split the dissipative-stochastic generators.It resulted that the Trotter
form ula cannotbeapplied w ithoutconsidering thespecialstochastic character
ofthe equations.In general,di erent variables depending on the sam e noise
should notbe split.In the D PD equationsthisdoesnothappen w hich allowed
us to w rite a integration schem e. In [1]we concluded that,considering the
accuracy ofthe equilibrium tem perature and the com putationalcost,D PD Trotter is am ong the best integrators for the D PD equations.
In thisarticle we explain in detailhow to apply the stochastic Trotterform ula
to the particular case of D PD .T he aim is to furnish a non-trivialexam ple
to be used as a reference w hen one w ishes to derive new integration schem es
based on the stochastic Trotter form ula for a generalset of SD Es.W e also
test the behavior ofthe radialdistribution function in the D PD m odel.For
an idealgas equation of state we nd the D PD -Trotter schem e presents no
spatialcorrelations at any scale.
2
A Trotter integration schem e for dissipative particle dynam ics
T he D PD m odelconsists ofa set ofN particles m oving in continuous space.
Each particle k is de ned by its position rk and its m om entum pk and m ass
m .T he dynam ics is speci ed by a set ofLangevin equations very sim ilar to
the m olecular dynam ics equations,but w here in addition to the conservative
forces there are dissipative and uctuating forces as well
drk =
dpk =
pk
dt;
m
P
N
l6
= k ekl
h
aklF c(rkl)
! 2(rkl)(ekl
pkl) dt+
p
i
2 kB T0!(rkl)dW
t
kl
;
(1)
w here F c(r)isthe conservative pairinteraction force weighted by positive and
sym m etric param eter akl,rkl = rk rl is the distance between the particle k
and particle l,rkl its length and ekl = rkl=rkl.T he weight function ! usually
has a nite range rc.A typical selection is !(r) = 1 r=rc for r < rc and
!(r) = 0 for r
rc.T he conservative force is usually chosen to be of the
form F c(rkl) = w (rkl).T his system has a wellde ned G ibbsian equilibrium
m
2
state at a tem perature T0.Because the stochastic term in conventionalD PD
doesnotdepend on the m om enta,note the It^
o orStratonovich interpretation
are exactly equivalent(additive noise)and we can apply the standard rulesof
ordinary calculus form ally treating dW as 00dt00.
T he globalstate is x = (r1;:::;rN ;p1;:::;pN ) and the SD Es (1) can be expressed as dx = L[x]dt w ith form alsolution x(t) = T eL t[x](x0) w here T is
thetim e-ordered operator(see[8]).T heglobaltim egeneratorisdivided in two
P
operatorsgenerating \orthogonaldynam ics" L = L r + L p,w here L r = k L kr
P
and L p = k;l> k L kl
p .Becausein theD PD m odeltheforcesbetween interacting
particles k and lsatisfy action-reaction (N ew ton’s third law ),the m om entum
is locally (and totally) conserved. For each partialdynam ics, the generator
P
P
L p kl w ith
L rk and L kl
can be subdivided in com ponents L kr =
p =
L rk =
pk
@ ;
m rk
L p kl = D p kl + Sp kl(t);
D p kl = aklF c(rkl)
q
m
!D (rkl)(ekl
Sp kl(t)= f(t) 2 kB T0!(rkl)ekl(@p k
pkl) ekl(@p k
@p l );
@p l ):
(2)
N ote that the m om entum operator has two contributions,the determ inistic
t
and the stochastic w hich is an explicit function oftim e w ith f(t)= dW kl
=dt.
T he form alsolution ofour system x(t) = T eL t[x](x0) corresponds to a continuous tim e evolution. In order to devise any integrator schem e we m ust
discretize the continuous tim e in nite steps.T he continuum tim e propagator can be approxim ated by discrete tim e steps ofsize t= t=P ,recursively
P
L t
applying P tim es the exponential operator eL t
eL t=P
eL t
e.
t
N ote thatw hen the generatordependsexplicitly on tim e L(t) L ,the tim eordered exponentialisrelevantand therecursively nested exponentialsbecom e
t
t+ P t t
L t+ 2 t t L t+ t t
T eL t
eL
e
e
([8]). A t this point we m ust provide
som e approxim ation ofthe discrete tim e propagatorofa \generic" globaldynam ics eL t.A s we have m entioned in the previous paragraph,in generalthe
P
generator L is form ed by m any generators L i each ofthem corresponding
to a particular dynam ics i.T he generalized Trotter form ula (Strang [9]) generates a straightforward approxim ation to the tim e propagator exact up to
second order in tim e
0
P
e
M
i= 1
A it
= @
Y1
i= M
t
eA i 2
YM
1P
eA j
t
2
A
+ O ( t 3):
(3)
j= 1
Because itcan be perform ed in m any possible ways,the m ostim portantpracticalissue to apply form ula (3) is the selection ofa particular splitting ofthe
globaldynam ics.O ne reasonable criteria is to keep the m inim um num ber of
3
generatorsand exploitanalyticity foreach ofthem w heneverpossible.Forthe
stochastic equations ofD PD ,the splitting we propose consist in 1 + N (N2 1)
operators:the globalL r and a L kl
p for each pair.
T he Baker-C am pbell-H aussdor (BC H ) form ula reads
1
1
1
eA eB = eA + B + 2 [A ;B ]+ 12 [A ;[A ;B ]]+ 12 [B ;[B ;A ]]+
(4)
so for [A ;B]= 0 we have the exact form ula eA + B = eA eB = eB eA .T he D PD
P
position generator L r =
k; L rk is com posed by m any sim ple individual
generators per particle and com ponents that satisfy [L rk ;L rl ] = 0 for all
particles k;l(k 6
= l)and com ponents ; except 6
= .T herefore we can use
the exact form ula
1
0
P
t
eL r
= e
Lk
k r
YN
t
@
k= 1
Yd
t
Lr
e
k
A
(5)
=1
w ith d the dim ensionality. In M D (D PD w ithout dissipative and random
P
forces) the m om entum generator L p = k; L pkl also satis es [L p kp ;L p lq ]= 0
for all pairs of particles kp;lq, w ith k 6
= l, p 6
= q, p 6
= q and com ponents
;; 6
= ,such that the ordering of the individual-com ponent m om entum
generators is absolutely irrelevant. O n the contrary in D PD ,the forces depend on the othercom ponentsofthe velocity oft
ticle and also on other
P he par
L kl
t
Lp t
p
k;
l>
k
particles velocities and the operator e
= e
cannot be globally
integrated and hasto be approxim ated in som e way.T hisisthe reason forthe
splitting itin N (N2 1) m om entum operators.D ue to this splitting and form ula
(3),the D PD schem e is nally given by the follow ing Trotter integrator
1
0
x(t+ t)= @
YN
t
L qr
p 2
e
q= 1;r> 1
A
YN
L ir
e
i= 1
!0
t @
1
Y1
t
L kl
p 2
e
A
x(t):
(6)
k= N ;l< N
T he propagator that corresponds to the generator L rk produces the position
update w hich is analytically given by
Lk
e
r
k
t
[x]: rk (t+ t)= r k (t)+
pk (t)
t
m
(7)
because the m om entum is a constant in this step of the schem e. T he next
step is to solve the propagator of the m om enta of the interaction pair k,l
(corresponding to the generatorL kl
p )independently ofthe positions.W e have
m entioned before that D PD forces satisfy action-reaction,so for a particular
interacting pair k;l we propose to m ake a change of variables from pk ;pl
to pk + pl;pkl = pk pl.T he new system to solve is d(pk + pl) = 0 and
dpkl = 2dpk.Because the positions ofthe particles are \frozen" at this step
ofthe Trotterschem e,the equation fordpkl can be solved m ore easily forthe
4
projection on the radialdirection pekl = pkl
e
kl
t
dpekl = A dt B pekldt+ C dW kl
;
(8)
p
w here A = 2aklF c(rkl),B = 2 =m ! 2 and C = 2 2 kB T0!.T his equation is
an O rnstein-U hlenbeck process w ith analyticalsolution [10]
Z
pekl(t)=
e
B
t
pekl(t0)+
Z
t
A
B (s t)
e
t
ds+ C
t0
eB (s t) dW s;
(9)
t0
w here t= t t 0,t0 being the initialtim e.T he solution of(9) requires the
generation ofcolored noise based on a num ericalschem e itself.A version of
the m ethod to generate coloured noise [1,11]adapted to Eq.(9) results
p
e
kl =
pkl
e
kl
aklF c
r
2
e
t
1 +
2kB T0m 1
4
e
t
kl
;
(10)
w here = =m ! 2, kl = lk are norm al distributed w ith zero m ean and
kl
variance one (N (0;1)) and p ekl = pekl(t) pekl(t0).T he propagator eL p t for
pk and pl gives
kl
eL p
t
[x]:(ptk+
t
;ptl+
t
)=
p ekl
ekl(t); pl(t)
2
pk (t)+
p ekl
ekl(t) : (11)
2
So in D PD we can solve the dynam ics corresponding to the generator L kl
p
(globally for allcom ponents at the sam e tim e) w ithout the need to go to the
scalaroperatorL p kl corresponding to the coordinate .In practice the Trotter
integration algorithm (6) consists of the follow ing steps: for the interaction
pairs k;l update the m om entum half tim estep according to the propagator
(11) w ith a noise kl;iterate over particles k updating the position according
to (7); nally,update pairs k;l in reverse order again using the propagator
0
(11) but w ith new noises kl
.T his algorithm requires the calculation ofthe
pair-listonly once periteration and hasthe sam e com plexity asa sim ple D PD
velocity-Verlet schem e (D PD -V V [4]).
W e tested in [1] this integration schem e using the open-source code m ydpd
[12]forthe equilibrium tem perature w ith N = 4000 particles, = 4:5;kB T0 =
1;m = 1;rc = 1 in a three dim ensionalperiodic box (L;L;L) w ith L = 10
w ith periodic boundary conditions.T hese settings give a particle density =
4. H ere, we show in left Fig.1 the radial distribution function for akl = 0
(corresponding to an idealgas equation ofstate) and a tim e step t= 0:05.
W e com pare the results forthree m ethods:the velocity Verlet (D PD -V V )[4],
the Shardlow schem e [13]and D PD -Trotter[1].W e nd good agreem ent w ith
the theoreticalvalue 1 for Shardlow and D PD -Trotter integrators but D PD V V is notably w rong displaying spurious spatialcorrelations atdistances less
than the nite range rc.In rightFig.1 we show the radialdistribution function
fora sim ulation w ith akl = 25 and a tim e step t= 0:01.A swe see the three
m ethods perform very sim ilarly.
5
1.12
1.2
1.1
1
0.8
1.06
g(R)
g(R)
1.08
1.04
0.6
0.4
1.02
0.2
1
0.98
0
0
0.5
1
R
1.5
2
0
0.5
1
R
1.5
2
Fig. 1. R adial distribution function for three integrator m ethods. Velocity Verlet
w ith (4 ) sym bols,Shadlow schem e w ith (2 ) sym bols and Trotter D PD w ith ( )
sym bols.Left gure corresponds to an ideal gas sim ulation akl = 0.R ight gure
corresponds to sim ulations including conservative forces w ith akl = 25.
3
C onclusions
T he stochastic Trotterform ula can be successfully applied to the D PD m odel
and the procedure to tailorthe integratorschem e hasbeen explained in detail.
In the schem e we have also exploited the exactintegration ofim portantparts
of the dynam ics like the conservation of totalm om entum of an interacting
pair of particles. T he D PD -Trotter integrator displays correctly the radial
distribution functionsforan idealgas(no conservative forcesam ong particles)
and also fora non idealgas.Follow ing thisim portantexam pleand [1]itshould
be strathforward to apply the stochastic Trotter form ula to new m esoscopic
m odels and m ore generalSD Es.
A cknow ledgem ents M .S.and P.E.are supported by the Spanish M inisterio
de Educacion y C iencia projectFIS2004-01934 and G D F by the EPSRC Integrative Biology project G R /S72023.PV C & M S thank the EPSRC (U K ) for
funding R ealityG rid under grantnum ber G R /R 67699;this project supported
M S’s 6 m onth visit to the C C S at U C L during 2005.
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