Chemical Physics Letters 429 (2006) 310–316
www.elsevier.com/locate/cplett
Second-order integrators for Langevin equations with
holonomic constraints
Eric Vanden-Eijnden
a
a,*
, Giovanni Ciccotti
b,1
Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, United States
b
Dipartimento di Fisica, Università ‘La Sapienza’, Piazzale A. Moro 2, 00185 Roma, Italy
Received 22 May 2006; in final form 24 July 2006
Available online 5 August 2006
Abstract
We propose a numerical scheme for the integration of the Langevin equation which is second-order accurate. More importantly, we
indicate how to generalize this scheme to situations where holonomic constraints are added and show that the resulting scheme remains
second-order accurate.
! 2006 Elsevier B.V. All rights reserved.
1. Introduction
The evolution of a system of interacting particles in the
presence of a thermal bath at temperature T can be
described by the Langevin equation
pffiffiffiffiffiffiffiffiffiffiffiffi
M€xðtÞ ¼ F ðxðtÞÞ $ cM x_ ðtÞ þ 2k B T cM 1=2 gðtÞ
ð1Þ
where xðtÞ 2 R3N denotes the position of the N particles, M
the diagonal mass tensor, F(x) the force, c > 0 the friction
coefficient and gðtÞ ¼ W_ ðtÞ is a white-noise (W(t) being a
Wiener process). As a result, the Langevin Eq. (1) is an useful tool to sample the Boltzmann–Gibbs
distribution
R
q(x,v) = Z$1e$bH(x,v) where Z ¼ R3N &R3N e$bH ðx;vÞ dxdv is
the partition function and H ðx; vÞ ¼ 12 hv; Mvi þ V ðxÞ is the
Hamiltonian (here we assumed that F = $$V).
The question that we investigate here is how to generate
an accurate approximation of a trajectory (x(t), v(t)) via
time-discretization of (2). This question has been addressed
by many authors. Among the most popular integrators are
the one of van Gunsteren and Berendsen (vGB) proposed
in [1], the one of Brooks–Brünger–Karplus (BBK) pro*
Corresponding author. Fax: +1 212 995 4121.
E-mail addresses: eve2@cims.nyu.edu (E. Vanden-Eijnden), giovanni.
ciccotti@roma1.infn.it (G. Ciccotti).
1
Fax: +39 06 4957697.
0009-2614/$ - see front matter ! 2006 Elsevier B.V. All rights reserved.
doi:10.1016/j.cplett.2006.07.086
posed in [2] and the Langevin impulse ðLbIÞ integrator proposed in [3]. These various integrators have been recently
reviewed and compared in [4], and it has been shown that
BBK is first-order accurate, whereas both vGB and LbI
are second-order accurate. More recently, a class of integrators which are up to third-order accurate and also have
some additional nice properties have been proposed in [5].
While all of these integrators have been derived for Langevin equations subject to no constraints, they have been
often used to integrate systems in which holonomic constraints are present: this is usually done by simply applying
an extra step of SHAKE [6] at every time-step. To the best
of our knowledge, however, nobody has ever addressed the
question of whether such a generalization of these integrators to simulate systems with constraints affect their numerical accuracy. This question is pertinent since higher-order
accurate integrators allow for bigger time-steps and
thereby permit to simulate the systems over longer timeintervals. In the deterministic context, because SHAKE
does not affect the general structure of second-order integrators such as Verlet, one can easily show that it must preserve the order of accuracy of the integrator. The situation
is more complex for Langevin integrators, however,
because the structure of these integrators is more complicated and there is a nontrivial interplay between the force
and the random noise at every time-step.
311
E. Vanden-Eijnden, G. Ciccotti / Chemical Physics Letters 429 (2006) 310–316
In this Letter, we investigate the issue of the accuracy of
a Langevin integrator used with SHAKE. Our main results
are: (i) the integrator (21) and its quasi-symplectic version
(23) in Section 2, which give simple and, to the best of
our knowledge, new second-order accurate integrators for
Langevin equations without constraints and (ii) the integrators (34) and (39) in Section 3, which show that SHAKE
can be applied to (21) and (23) to impose holonomic constraints without loosing the second-order accuracy of the
integrator. While we do not do so here, the method used
in Section 3 can in principle be applied to prove (or disprove) that the popular integrators such as vGB, BBK or
LbI keep their order of accuracy when applied in conjunction with SHAKE.
2. Basic algorithms
Written componentwise as a stochastic differential equation, (1) becomes
(
dxi ðtÞ ¼ vi ðtÞdt
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
dvi ðtÞ ¼ ðm$1
2k B T ci m$1
i dW i ðtÞ
i F i ðxðtÞÞ $ ci vi ðtÞÞdt þ
ð2Þ
where (x1, x2, x3) are the x, y and z components of particle 1
(v1, v2, v3) are the components of its velocity, m1 = m2 =
m3 > 0 is its mass and c1 = c2 = c3 > 0 is the friction coefficient it is subject to, and similarly for (x4, x5, x6), etc. To
simplify the formula, it will be convenient to write (2) as
"
dxi ðtÞ ¼ vi ðtÞdt
ð3Þ
dvi ðtÞ ¼ ðfi ðxðtÞÞ $ ci vi ðtÞÞdt þ ri dW i ðtÞ;
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2k B T ci m$1
where we defined fi ðxÞ ¼ m$1
i .
i F i ðxÞ and ri ¼
Going back to vectorial notations, this equation can be
written as the system of integral equations
8
R tþh
>
< xðt þ hÞ ¼ xðtÞ þ t vðsÞds
R tþh
R tþh
ð4Þ
vðt þ hÞ ¼ vðtÞ þ t f ðxðsÞÞds $ c t vðsÞds
>
:
þrðW ðt þ hÞ $ W ðhÞÞ
pffiffi
Using x(s) = x(t) + O(h) and
pffiffi vðsÞ ¼ vðtÞ þ Oð hÞ (recall
that W ðt þ hÞ $ W ðhÞ ¼ Oð hÞ) under the integrals and
neglecting the corrections of order h3/2 and higher, we
arrive at the Euler–Maruyama scheme [7,8]:
" nþ1
x ¼ xn þ hvn
pffiffi
ð5Þ
vnþ1 ¼ vn þ hf ðxn Þ $ hcvn þ hrnn :
n
n
where x is the numerical approximation of x(nh), v that of
v(nh) and nn are independent (for different n) Gaussian variables with mean zero and
covariance Eðnni nnj Þ ¼ dij (here we
d pffiffi n
d
used W ðt þ hÞ $ W ðtÞ ¼ hn , where ¼ denotes equality in
law or distribution). It can be shown [7,8] that the scheme
(5) is first-order accurate, meaning that, given any suitable
observable /,
sup jE/ðxðnhÞ; vðnhÞÞ $ E/ðxn ; vn Þj 6 Ch
06n6T =h
ð6Þ
where (x(nh), v(nh)) is the (exact) solution of (2) evaluated
at t = nh, (xn, vn) is its numerical approximation by (5)
and it is assumed that (x(0), v(0)) = (x0, v0); E denotes
expectation with respect to the noises in (2) and (5), and
the constant C depends on T < 1 and the observable
/(x, v) but not on h. Note that (6) is a finite-time error estimate, hence it says little about the error in computing equilibrium averages, but it indicates that it may be necessary
to use (5) with a very small time-step h when the force field
is stiff. Hence (5) may be insufficient in applications (Fig. 1).
To doR better than (5), go back to (4), and for v(s) in the
tþh
integral t vðsÞds use
Rs
Rs
vðsÞ ¼ vðtÞ þ t f ðxðuÞÞdu $ c t vðuÞdu þ rðW ðsÞ $ W ðtÞÞ
ð7Þ
pffiffi
Using x(u) = x(t) + O(h) and vðuÞ ¼ vðtÞ þ Oð hÞ at this
level we deduce that (recall that s 2 [t, t + h] and hence
s $ t = O(h))
vðsÞ ¼ vðtÞ þ ðs $ tÞðf ðxðtÞÞ $ cvðtÞÞ
þ rðW ðsÞ $ W ðtÞÞ þ Oðh3=2 Þ
ð8Þ
and hence
R tþh
vðsÞds ¼ hvðtÞ þ 12 h2 ðf ðxðtÞÞ $ cvðtÞÞ
t
R tþh
þ r t ðW ðsÞ $ W ðtÞÞds þ Oðh5=2 Þ
ð9Þ
R tþh
Similarly, for f(x(s)) in the integral t f ðxðsÞÞds, use
Rs
f ðxðsÞÞ ¼ f ðxðtÞÞ þ t vðuÞ ' rf ðxðuÞÞdu
ð10Þ
pffiffi
then x(u) = x(t) + O(h) and vðuÞ ¼ vðtÞ þ Oð hÞ to arrive
at
0
10
–1
10
–2
10
–3
10
–4
10
–5
10
2
3
4
5
6
Fig. 1. Numerical errors of the integratorsp(5),
ffiffi (21) and (23) on the
example of x_ ðtÞ ¼ vðtÞ; v_ ðtÞ ¼ $xðtÞ $ vðtÞ þ 2gðtÞ, x(0) = v(0) = 0. The
times steps are h = 2$n for n = 2,3,4,5,6. The quantity monitored for
the error is the estimate of Eðx2 ð1Þ þ v2 ð1ÞÞ ¼ 0:9796111900 . . . computed
by ensemble averaging over 108 independent realizations. The dashed
curves are the graphs of the functions 2$n(=h) and 2$2n(=h2) versus n. The
circles are the errors obtained using the first-order integrator (5) and they
are consistent with the error estimate (6); the diamonds are the errors
obtained using the second-order integrator (21) and they are consistent
with the error estimate (18); the squares are the errors obtained using the
quasi-symplectic second-order integrator (23).
312
E. Vanden-Eijnden, G. Ciccotti / Chemical Physics Letters 429 (2006) 310–316
f ðxðsÞÞ ¼ f ðxðtÞÞ þ ðt $ sÞvðtÞ ' rf ðxðtÞÞ þ Oðh3=2 Þ
ð11Þ
and
R tþh
ð12Þ
t
f ðxðsÞÞds ¼ hf ðxðtÞÞ þ 12 h2 vðtÞ ' rf ðxðtÞÞ þ Oðh5=2 Þ
Recalling that EðWRi ðsÞW j ðs0 ÞÞ ¼ dij minðs; s0 Þ, one sees that
tþh
for fixed t and h, t ðW ðsÞ $ W ðtÞÞds is a Gaussian variable with mean zero and such that
#R
$#R
$
tþh
tþh
E t ðW i ðsÞ $ W i ðtÞÞds
ðW j ðsÞ $ W j ðtÞÞds ¼ 13 h3 dij
t
R tþh
EððW i ðt þ hÞ $ W i ðtÞÞ t ðW j ðsÞ $ W j ðtÞÞdsÞ ¼ 12 h2 dij
ð13Þ
Since (W(t + h) $ W(t)) is also a Gaussian variable with
mean zero and covariance hdij, this means that
d pffiffi
W ðt þ hÞ $ W ðtÞ ¼ hnn
#
$
ð14Þ
R tþh
d
ðW ðsÞ $ W ðtÞÞds ¼ h3=2 12 nn þ 2p1 ffi3ffi gn
t
where (nn, gn) are independent Gaussian variables with
mean zero and covariance
Eðnni nnj Þ ¼ Eðgni gnj Þ ¼ dij ;
Eðnni gnj Þ ¼ 0
ð15Þ
5/2
Inserting (9) and (12) into (4), neglecting terms of order h
and higher and using (14) we arrive at the integrator
(
xnþ1 ¼ xn þ hvn þ An
pffiffi
vnþ1 ¼ vn þ hf ðxn Þ $ hcvn þ hrnn þ 12 h2 vn ' rf ðxn Þ $ cAn
ð16Þ
where
An ¼ 12 h2 ðf ðxn Þ $ cvn Þ þ rh3=2
#
1 n
n
2
þ 2p1 ffi3ffi gn
$
ð17Þ
This integrator is in fact a special case of Milstein’s scheme
[7,8] and it is second-order accurate, i.e. we now have (compare (6))
sup jE/ðxðnhÞ; vðnhÞÞ $ E/ðxn ; vn Þj 6 Ch2
ð18Þ
06n6T =h
where (xn, vn) is the solution of (16). However, (16) is
unpleasant because it involves the term vn Æ $f(xn). To get
rid of the latter, notice that (16) implies that
f ðxnþ1 Þ ¼ f ðxn Þ þ hvn ' rf ðxn Þ þ Oðh3=2 Þ
or, equivalently, that
1 2 n
hv
2
n
' rf ðx Þ ¼
1
hðf ðxnþ1 Þ
2
n
$ f ðx ÞÞ þ Oðh
ð19Þ
5=2
Þ:
ð20Þ
Using (20) in (16) we can rewrite this integrator as
(
xnþ1 ¼ xn þ hvn þ An
pffiffi
vnþ1 ¼ vn þ 12 hðf ðxnþ1 Þ þ f ðxn ÞÞ $ hcvn þ hrnn $ cAn
ð21Þ
To the best of our knowledge, the integrator (21) is new: it
is explicit and, like (16), it is second-order accurate, i.e. (18)
still holds now with (xn, vn) being the solution of (21). It is a
generalization to the Langevin equation of the familiar
velocity-Verlet algorithm, which is obtained from (21) by
setting c = r = 0 (in which case (2) reduces to Hamilton’s
equation of motion)
(
xnþ1 ¼ xn þ hvn þ 12 h2 f ðxn Þ
ð22Þ
vnþ1 ¼ vn þ 12 hðf ðxnþ1 Þ þ f ðxn ÞÞ
Notice that (21) requires no more force field evaluation
than (22), and no more than (5).
There is another way to modify the integrator (21) without losing either accuracy or simplicity (i.e. such that the
computation of $f(xn) is avoided and each time step
requires only one force field evaluation):
pffiffi
8 nþ1=2
v
¼ vn þ 12 hf ðxn Þ $ 12 hcvn þ 12 hrnn
>
>
>
#
$
>
>
1 2
n
n
1 3=2
1 n
>
p1ffiffi gn
>
$
h
cðf
ðx
Þ
$
cv
Þ
$
h
cr
n
þ
>
8
4
2
3
>
<
nþ1=2
3=2
nþ1
n
1
n
x ¼ x þ hv
þ h r 2pffi3ffi g
>
>
pffiffi
>
> vnþ1 ¼ vnþ1=2 þ 1 hf ðxnþ1 Þ $ 1 hcvnþ1=2 þ 1 hrnn
>
>
2
2
2
>
#
$
>
>
:
$ 18 h2 cðf ðxnþ1 Þ $ cvnþ1=2 Þ $ 14 h3=2 cr 12 nn þ p1ffi3ffi gn
n+1
ð23Þ
It can be checked by direct comparison that x
and
vn+1 from (23) agree up to order h2 with xn+1 and vn+1
from either (16) or (21). To the best of our knowledge,
the integrator (23) is also new: it is a second-order generalization of the BBK integrator [2]. Setting c = r = 0 in
(23), this integrator also reduces to velocity-Verlet, now
written as
8 nþ1=2
¼ vn þ 12 hf ðxn Þ
>
<v
ð24Þ
xnþ1 ¼ xn þ hvnþ1=2
>
: nþ1
nþ1=2
1
nþ1
v ¼v
þ 2 hf ðx Þ
The integrator (23) may have some advantage over (21) because, unlike (21), (23) is quasi-symplectic in the sense of
[5]: it reduces to a symplectic integrator when c = r = 0
and the Jacobian J of the transformation from (xn, vn) to
(xn+1, vn+1) is independent of (xn, vn). A direct calculation
&2
Q3N %
gives J ¼ i¼1 1 $ 12 hci þ 18 h2 c2i , to be compared with
P3N
the exact value expð$h i¼1 ci Þ.
3. Generalizations to situations with constraints
Many situations require to generalize the Langevin Eq.
(2) to include hard constraints. Let us assume that K constraints are given in the form of h1(x) = ' ' ' = hK(x) = 0.
Then (2) becomes
8
< x_ ðtÞ ¼ vðtÞ
K
P
ð25Þ
: v_ ðtÞ ¼ f ðxðtÞÞ $ cvðtÞ þ rgðtÞ $
ka ðtÞrha ðxðtÞÞ
a¼1
where k1(t), . . . , kK(t) are Lagrange multiplier added to satisfy the constraints. Since ha(x(t)) = 0 for all a = 1, . . . , K
and all t, it follows that
313
E. Vanden-Eijnden, G. Ciccotti / Chemical Physics Letters 429 (2006) 310–316
0 ¼ h_ a ðxðtÞÞ ¼ vðtÞ ' rha ðxðtÞÞ
ð26Þ
which indicates that at all times v(t) must be orthogonal to
all the $ha(x(t)) (and so the initial velocity v(0) must also
satisfy this condition). Similarly, we must have
0 ¼ €ha ðxðtÞÞ ¼ vðtÞvðtÞ : rrha ðxðtÞÞ þ v_ ðtÞ ' rha ðxðtÞÞ
¼ vðtÞvðtÞ : rrha ðxðtÞÞ þ ðf ðxðtÞÞ $ cvðtÞ þ rgðtÞÞ ' rha ðxðtÞÞ
K
X
$
kb ðtÞrhb ðxðtÞÞ ' rha ðxðtÞÞ;
ð27Þ
b¼1
where, given a; b 2 R3N , ab : rr ¼
is an equation for ka(t). Letting
P3N
ı;j¼1 ai bj o
2
=oxi oxj (27)
K ab ðxÞ ¼ rha ðxÞ ' rhb ðxÞ 2 RK & RK
ð28Þ
and assuming that this matrix is invertible (27) can be
solved in ka(t) as
ka ðtÞ ¼
K
X
b¼1
þ
K $1
ab ðxðtÞÞrhb ðxðtÞÞ
K
X
b¼1
' ðf ðxðtÞÞ $ cvðtÞ þ rgðtÞÞ
K $1
ab ðxðtÞÞrrhb ðxðtÞÞ : vðtÞvðtÞ
ð29Þ
Inserting (29) into (25), one sees that this equation can be
written explicitly as
8
x_ ðtÞ ¼ vðtÞ
>
>
>
< v_ ðtÞ ¼ P ðxðtÞÞðf ðxðtÞÞ $ cvðtÞ þ rgðtÞÞ
K
P
>
>
>
$
rha ðxðtÞÞK $1
:
ab ðxðtÞÞrrhb ðxðtÞÞ : vðtÞvðtÞ
a;b¼1
ð30Þ
where P(x) is the orthogonal projector unto the manifold
h1(x) = ' ' ' = hK(x) = 0: componentwise, it is given by
K
X
oha ðxÞ ohb ðxÞ
K $1
ð31Þ
P ij ðxÞ ¼ dij $
ab ðxÞ
oxi
oxj
a;b¼1
Since the noise g(t) is multiplied by a function of x(t) alone
in (30), there is no problem of interpretation with this equation and the heuristic way we arrived at (30) can be justified
rigorously. Notice that ka(t) contains a martingale part,
K
X
K $1
ð32Þ
dkma ðtÞ ¼ r
ab ðxðtÞÞrhb ðxðtÞÞ ' dW ðtÞ:
b¼1
and a bounded variation part
K
X
dkba ðtÞ ¼
K $1
ab ðxðtÞÞrhb ðxðtÞÞ ' ðf ðxðtÞÞ $ cvðtÞÞdt
b¼1
þ
K
X
b¼1
K $1
ab ðxðtÞÞrrhb ðxðtÞÞ : vðtÞvðtÞdt
ð33Þ
We could write down directly a numerical integrator for
(30) by proceeding as in Section 2. This, however, would
have two disadvantages. First, the resulting integrator
would be rather complicated and, e.g. involve second-order
derivatives of ha(x), which are cumbersome to compute.
Second, and more importantly, even if we write down a second-order accurate integrator, this integrator would only
conserve the constraints h1(x(t)) = . . . = hK(x(t)) = 0 to
O(h2). This is very unsatisfactory since over long times,
the constraints would then drift away.
In the spirit of SHAKE [6], it is much more convenient to
write down an integrator which is accurate to second-order
but conserves the constraints exactly. Below, we show that
the following integrator satisfies these conditions:
8
K
P
>
>
>
xnþ1 ¼ xn þ hvn þ An $
lna rha ðxn Þ
>
>
>
a¼1
<
pffiffi
vnþ1 ¼ vn þ 12 hðf ðxnþ1 Þ þ f ðxn ÞÞ $ hcvn þ hrnn $ cAn
>
>
>
K
>
P
>
>
$h$1 ðl(a rha ðxnþ1 Þ þ lna rha ðxn ÞÞ
:
a¼1
n
Here A is given in (17)
ha ðxnþ1 Þ ¼ 0;
and
ðln1 ; . . . ; lnK Þ
ð34Þ
are chosen so that
a ¼ 1; . . . ; K
ðl(1 ; . . . ; l(K Þ
ð35Þ
are chosen so that
vnþ1 ' rha ðxnþ1 Þ ¼ 0;
a ¼ 1; . . . ; K
ð36Þ
(34) simply means that we can apply RATTLE to (21), remain second-order accurate and satisfy the constraints (26)
exactly. The multipliers lna can be obtained via SHAKE;
the multipliers l(a can be obtained via solution of a linear
system. For future reference, notice that the updating step
for the velocities in (34) can be written as
vnþ1 ¼ P ðxnþ1 Þv(
where
v( ¼ vn þ 12 hðf ðxnþ1 Þ þ f ðxn ÞÞ $ hcvn þ
$ h$1
K
P
a¼1
lna rha ðxn Þ:
ð37Þ
pffiffi n
hrn $ cAn
ð38Þ
In the same way, we can apply RATTLE to (23), remain
second-order accurate and satisfy the constraints (26) exactly. In this case, the integrator reads:
8 nþ1=2
pffiffi
¼ vn þ 12 hf ðxn Þ $ 12 hcvn þ 12 hrnn
v
>
>
>
#
$
>
>
>
1 2
n
n
1 3=2
1 n
p1ffiffi gn
>
$
h
cðf
ðx
Þ
$
cv
Þ
$
h
cr
g
þ
>
8
4
2
3
>
>
>
>
K
>
P n
>
>
$h$1
la rha ðxn Þ
>
>
>
a¼1
>
<
xnþ1 ¼ xn þ hvnþ1=2 þ h3=2 r 2p1 ffi3ffi gn
>
>
pffiffi
>
>
vnþ1 ¼ vnþ1=2 þ 12 hf ðxnþ1 Þ $ 12 hcvnþ1=2 þ 12 hrnn
>
>
>
#
$
>
>
>
1 2
nþ1
nþ1=2
1 3=2
1 n
p1ffiffi gn
>
$
h
cðf
ðx
Þ
$
cv
Þ
$
h
cr
g
þ
>
8
4
2
3
>
>
>
>
K
>
P
>
>
:
l(a rha ðxnþ1 Þ
$h$1
a¼1
ð39Þ
where the Lagrange multipliers
and
ðl(1 ; . . . ; l(K Þ are chosen so that (35) and (37) are satisfied.
It can be checked by direct comparison that xn+1 and
vn+1 from (39) agree up to order h2 with xn+1 and vn+1 from
(34).
ðln1 ; . . . ; lnK Þ
314
E. Vanden-Eijnden, G. Ciccotti / Chemical Physics Letters 429 (2006) 310–316
The remainder of this Letter is devoted to show the consistency of (34) with respect to (30) and establish that it is
second-order accurate; this also establishes the consistency
and second-order accuracy of (39) since this integrator
coincide with (34) to O(h2). Since we were unable to find
a soft argument establishing this claim, we will proceed
the hard way, write down an integrator for (30) which is
second-order accurate and show that, to O(h2) this integrator is equivalent to (34). For simplicity, we will treat the
simpler case where there is a single constraint, K = 1 and
h1(x) = h(x) (the general case can be treated similarly but
the algebra become even more tedious). Under this
assumption, (30) can be written as the following integral
equation (using P(x(t))v(t) = v(t) which follows from (26))
8
R tþh
xðt þ hÞ ¼ xðtÞ þ t vðsÞds
>
>
>
>
R tþh
>
>
>
< vðt þ hÞ ¼ vðtÞ þ t ðP ðxðsÞÞf ðxðsÞÞ $ cvðsÞÞds
R tþh
ð40Þ
þr t P ðxðsÞÞdW ðsÞ
>
>
R
>
tþh
$2
>
>
$ t rhðxðtÞÞjrhðxðsÞÞj rrhðxðsÞÞ
>
>
:
: vðsÞvðsÞds:
Proceeding as in Section 2, we can integrate once and replace
v(s), etc. by their value according to (40), then expand and
keep only the term of order h2 or lower. The relevant expressions are (here for simplicity we denote x(t) ” x and v(t) ” v)
R tþh
t
vðsÞds ¼ hv þ 12 h2 ðP ðxÞf ðxÞ $ cvÞ
R tþh
þ rP ðxÞ t ðW ðsÞ $ W ðtÞÞds
$ 12 h2 jrhðxÞj$2 ðrrhðxÞ : vvÞrhðxÞ
R tþh
t
R tþh
t
þ Oðh5=2 Þ
ð41Þ
P ðxðsÞÞf ðxðsÞÞds ¼ hP ðxÞf ðxÞ
þ 12 h2 v ' rðP ðxÞf ðxÞÞ þ Oðh5=2 Þ
rhðxðsÞÞjrhðxðsÞÞj$2 rrhðxðsÞÞ : vðsÞvðsÞds
ð42Þ
$2
¼ hjrhðxÞj ðrrhðxÞ : vvÞrhðxÞ
þ 12 h2 v ' rðrhðxÞjrhðxÞj$2 rrhðxÞÞ : vv
þ h2 jrhðxÞj$2 ðrrhðxÞ : vðP ðxÞf ðxÞ $ cvÞrhðxÞÞ
R tþh
þ 2rjrhðxÞj$2 ðrrhðxÞ : vP ðxÞ t ðW ðsÞ $ W ðtÞÞdsÞrhðxÞ
$ h2 jrhðxÞj$4 ðrrhðxÞ : vrhðxÞÞðrrhðxÞ : vvÞrhðxÞ
þ 12 h2 r2 jrhðxÞj$2 ðP ðxÞ
5=2
ð43Þ
where we used (this identity holds almost surely under iterations within the scheme)
Rs
Rs
rrhðxðsÞÞ : ðP ðxðsÞÞ t dW ðuÞÞðP ðxðsÞÞ t dW ðuÞÞ
: rrhðxÞÞrhðxÞ þ Oðh
¼ ðs $ tÞP ðxðsÞÞ : rrhðxðsÞÞ
Þ
ð44Þ
and, finally,
R tþh
P ðxðsÞÞdW ðsÞ ¼ P ðxÞðW ðt þ hÞ $ W ðtÞÞ
t
R tþh
þ v ' rP ðxÞ t ðs $ tÞdW ðsÞ þ Oðh5=2 Þ
ð45Þ
R tþh
R tþh
Using t ðs $ tÞdW ðsÞ ¼ hðW ðt þ hÞ $ W ðtÞÞ$ t ðW ðsÞ
$W ðtÞÞds together with (14), inserting these expressions
in (40) and neglecting terms of order h5/2 and higher gives
the following integrator
8 nþ1
$2
x ¼ xn þ hvn þ P n An $ 12 h2 jrhn j ðvn vn : rrhn Þrhn
>
>
>
p
ffi
ffi
>
>
>
vnþ1 ¼ vn þ hP n f n $ hcvn þ hrP n nn
>
>
>
>
>
$hjrhn j$2 ðvn vn : rrhn Þrhn þ 12 h2 vn ' rðP n f n Þ
>
>
>
>
$2
>
>
$cP n An þ 12 h2 cjrhn j ðvn vn : rrhn Þrhn
>
>
<
#
$
þh3=2 rðvn ' rP n Þ 12 nn $ 2p1 ffi3ffi gn
>
>
>
$2
>
>
$ 12 h2 vn ' rðjrhn j ðvn vn : rrhn Þrhn Þ
>
>
>
>
>
>
$2jrhn j$2 ðvn ðP n An Þ : rrhn Þrhn
>
>
>
>
$4
>
þh2 jrhn j ðvn vn : rrhn Þðvn rhn : rrhn Þrhn
>
>
>
:
$ 12 h2 r2 jrhn j$2 ðP n : rrhn Þrhn
ð46Þ
where fn = f(xn), hn = h(xn), Pn = P(xn). Notice that (46) is
second-order accurate, but unlike (34), this integrator does
not preserve the constraints exactly: in fact hn = O(h2) and
Pnvn = vn + O(h2).
Let us now show that (34) is identical to (46) to O(h2).
To this end, notice first that, using (34) and (35) implies
that (using hn = 0 and vn Æ $hn = 0)
0 ¼ hðxnþ1 Þ ¼ hðxn þ hvn þ An $ ln rhn Þ
¼ An ' rhn $ ln jrhn j2
þ 12 ðhvn $ ln rhn Þðhvn $ ln rhn Þ : rrhn þ Oðh5=2 Þ
As a result
$2
$2
ln ¼ jrhn j An ' rhn þ 12 jrhn j ðhvn $ ln rhn Þ
ð47Þ
& ðhvn $ ln rhn Þ : rrhn þ Oðh5=2 Þ
¼ jrhn j An ' rhn þ 12 h2 jrhn j vn vn : rrhn þ Oðh5=2 Þ
$2
$2
ð48Þ
(Notice for later use that even though ln is divided by h in
the equation for vn+1 in (34), we do not need to compute
terms up to O(h3) because vn+1 is obtained via (37) and
Pn+1$hn = O(h).)
(48) implies that the way the positions are updated in
(34) is explicitly
xnþ1 ¼ xn þ hvn þ An $ jrhn j An ' rhn rhn $ 12 h2 jrhn j
& ðvn vn : rrhn Þrhn þ Oðh5=2 Þ
$2
¼ xn þ hvn þ P n An $ 12 h2 jrhn j ðvn vn : rrhn Þrhn þ Oðh5=2 Þ
$2
$2
ð49Þ
which is consistent to second-order with the equation for
xn+1 in (46).
Consider next the way the velocities are updated in (34). In
the present case where K = 1 (38) is explicitly (using (48))
pffiffi
v( ¼ vn þ 12 hðf nþ1 þ f n Þ $ hcvn þ hrnn $ cAn $ h$1 ln rhn
pffiffi
¼ vn þ hf n þ 12 h2 vn ' rf n $ hcvn þ hrnn $ cAn
$ h$1 jrhn j ðrhn ' An Þrhn $ 12 hjrhn j
$2
$2
& ðvn vn : rrhn Þrhn þ Oðh3=2 Þ
3/2
2
ð50Þ
$1 n
where the terms of order h and h come from $ h l $hn
and hence do not contribute to vn+1 since Pn+1$hn = O(h)
315
E. Vanden-Eijnden, G. Ciccotti / Chemical Physics Letters 429 (2006) 310–316
(see the remark after (48)). On the other hand, using (49),
we also have that
P nþ1 v( ¼ P n v( þ hvn ' rP n v( þ P n An ' rP n v(
$ 12 h2 jrhn j ðvn vn : rrhn Þrhn ' rP n v(
$2
þ 12 h2 vn vn : rrP n v( þ Oðh5=2 Þ
ð51Þ
Combining (48), (50) and (51), we deduce that
pffiffi
vnþ1 ¼ vn þ hP n f n þ 12 h2 P n vn ' rf n $ hcvn þ hrP n nn
þh
n n
n
n $2
n
n
n
n
rv ' rP n $ jrh j ðrh ' A Þv ' rP rh
n
$ 12 h2 jrhn j$2 ðvn vn
: rrhn Þvn ' rP n rhn þ P n An ' rP n vn
#
$
$2
þ h2 r2 P n 12 nn þ 2p1 ffi3ffi gn ' rP n nn $ h2 r2 jrhn j
##
$
$ #
$
& 12 nn þ 2p1 ffi3ffi gn ' rhn P n 12 nn þ 2p1 ffi3ffi gn ' rP n rhn
$ 12 h2 jrhn j ðvn vn : rrhn Þrhn ' rP n vn
$2
þ 12 h2 vn vn : rrP n vn þ Oðh5=2 Þ
ð52Þ
To simplify this expression, let us derive first a few useful
identities. Given any two vectors a; b 2 R3N , we have
a ' rP n b ¼ $jrhn j ðab : rrhn Þrhn
$2
$ jrhn j ðb ' rhn Þa ' rrhn
þ 2jrhn j ðarhn : rrhn Þðb ' rhn Þrhn
$4
ð53Þ
when b is perpendicular to $hn (e.g. when b = vn), this
expression simplifies into
ðb ? rhn Þ
ð54Þ
whereas when b = $hn it becomes
a ' rP n rhn ¼ jrhn j$2 ðarhn : rrhn Þrhn $ a ' rrhn
n
ð59Þ
vn þ hP n f n $ hcvn þ
pffiffi n n
hrP n $ hjrhn j$2 ðvn vn : rrhn Þrhn
ð60Þ
The term $cPnAn also appears in both (46) and (52)
(first on the third line in (46) and last at the right
hand-side on the first line in (52)). Next notice that (in
(52) these are the third term at the right hand-side on
the first line, the second and third terms on the second
line, the terms on the third line, and the first term on
the fifth line):
1 2 n n
hP v '
2
3=2
rf n þ h2 vn ' rP n f n $ h2 cvn ' rP n vn
þ h rvn ' rP n nn $ jrhn j ðrhn ' An Þvn ' rP n rhn
þ P n An ' rP n vn
$2
a ' rP n b ¼ $jrhn j$2 ðab : rrhn Þrhn
hvn ' rP n vn ¼ $hjrhn j$2 ðvn vn : rrhn Þrhn ;
and hence all the leading order terms of order O(h) and
lower appear in both (46) and (52): these are
$ cP n An þ hvn ' rP n vn þ h2 vn ' rP n f n $ h2 cvn ' rP n vn
3=2
We now use these relations to expand the terms in the
equation for vn+1 in (46) and show that is identical to
(52) to O(h2). First notice that (this is the first term on
the second line in (52))
ð55Þ
n
we also have (using $h Æ v = 0)
$2
¼ 12 h2 vn ' rðP n f n Þ þ 12 h2 vn ' rP n f n
$ h2 cvn ' rP n vn þ h3=2 rvn ' rP n nn
$ jrhn j ðrhn ' An Þvn ' rP n rhn þ P n An ' rP n vn
$2
¼ 12 h2 vn ' rðP n f n Þ $ 12 h2 cvn ' rP n vn
#
$
þ h3=2 rvn ' rP n 12 nn $ 2p1 ffi3ffi gn þ vn ' rP n An
$ jrhn j ðrhn ' An Þvn ' rP n rhn þ P n An ' rP n vn
$2
¼ 12 h2 vn ' rðP n f n Þ þ 12 h2 cjrhn j ðvn vn : rrhn Þrhn
$2
þ h3=2 rvn ' rP n Þ þ vn ' rP n An
.
vn vn : rrP n vn ¼ $jrhn j$2 ðvn vn vn ..rrrhn Þrhn
$ jrhn j ðrhn ' An Þvn ' rP n rhn þ P n An ' rP n vn
$2
¼ 12 h2 vn ' rðP n f n Þ þ 12 h2 cjrhn j ðvn vn : rrhn Þrhn
#
$
þ h3=2 rvn ' rP n 12 nn $ 2p1 ffi3ffi gn
$2
$ 2jrhn j$2 ðvn vn : rrhn Þvn ' rrhn
þ 4jrhn j$4 ðvn rhn : rrhn Þðvn vn : rrhn Þrhn
ð56Þ
and (these identities holds almost surely under iterations
within the scheme)
#
$
# #
$
$
P n 12 nn þ 2p1 ffi3ffi gn ' rP n nn ¼ E P n 12 nn þ 2p1 ffi3ffi gn ' rP n nn
¼ $ 12 jrhn j ðP n : rrhn Þrhn
$2
ð57Þ
##
$
$ #
$
$ jrhn j$2 12 nn þ 2p1 ffi3ffi gn ' rhn P n 12 nn þ 2p1 ffi3ffi gn ' rP n rhn
#
$
#
$
¼ P n 12 nn þ 2p1 ffi3ffi gn ' rP n ðP n $ 1Þ 12 nn þ 2p1 ffi3ffi gn
# #
$
#
$$
¼ E P n 12 nn þ 2p1 ffi3ffi gn ' rP n ðP n $ 1Þ 12 nn þ 2p1 ffi3ffi gn ¼ 0:
ð58Þ
$ 2jrhn j ðvn ðP n An Þ : rrhn Þrhn
$2
ð61Þ
where in the last equality we used (using (54) and since
both vn and PnAn are perpendicular to $hn):
vn ' rP n An $ jrhn j ðrhn ' An Þvn ' rP n rhn þ P n An ' rP n vn
$2
¼ vn ' rP n An þ vn ' rP n ðP n $ 1ÞAn þ P n An ' rP n vn
¼ vn ' rP n P n An þ P n An ' rP n vn
¼ $2jrhn j$2 ðvn ðP n An Þ : rrhn Þrhn
ð62Þ
The terms in (61) are the second on the second line, the second on the third line, the one on fourth line and the one on
the sixth line in (46).
From (57) and (58) we also have (this is the second term
on the fifth line and the one on the sixth line in (52)):
316
E. Vanden-Eijnden, G. Ciccotti / Chemical Physics Letters 429 (2006) 310–316
#
$
$2
h2 r2 P n 12 nn þ 2p1 ffi3ffi gn ' rP n nn $ h2 r2 jrhn j
##
$
$ #
$
& 12 nn þ 2p1 ffi3ffi gn ' rhn P n 12 nn þ 2p1 ffi3ffi gn ' rP n rhn
4. Concluding remarks
¼ $ 12 h2 r2 jrhn j$2 ðP n : rrhn Þrhn
ð63Þ
which is the term on the eight line in (46).
Finally, the remaining terms unaccounted for in (52) are
(using (54) and (55)):
$2
$ 12 h2 jrhn j ðvn vn : rrhn Þvn ' rP n rhn
¼ $ 12 h2 jrhn j ðvn vn : rrhn Þðvn rhn : rrhn Þrhn
$4
þ 12 h2 jrhn j ðvn vn : rrhn Þvn ' rrhn
$2
$2
$ 12 h2 jrhn j ðvn vn : rrhn Þrhn ' rP n vn
¼ 12 h2 jrhn j$4 ðvn vn : rrhn Þðvn rhn : rrhn Þrhn
and (using (56))
1 2 n n
hvv
2
ð64Þ
ð65Þ
Regarding the numerical integration of Langevin
equations with holonomic constraint, we found the literature quite confusing. We hope that the (courageous)
reader who bore with us until the end will be gratified
to see that, with heavy but straightforward calculations,
a very simple, second-order accurate integrator for these
equations can be derived. At the end, this integrator
requires no more numerical effort than its deterministic
counterpart. We also stress that the calculations above
show that analyzing the accuracy of integrators is more
complicated for Langevin equations than Hamiltonian
systems. As a result, one may easily overestimate the
order of accuracy of a scheme (as was done, e.g. by
one of us in [9]).
: rrP n vn
.
$2
¼ $ 12 h2 jrhn j ðvn vn vn ..rrrhn Þrhn
$ h2 jrhn j$2 ðvn vn : rrhn Þvn ' rrhn
þ 2h2 jrhn j$4 ðvn rhn : rrhn Þðvn vn : rrhn Þrhn
Acknowledgements
ð66Þ
so that
$2
$ 12 h2 jrhn j ðvn vn : rrhn Þvn ' rP n rhn
$ 12 h2 jrhn j$2 ðvn vn : rrhn Þrhn ' rP n vn
References
þ 12 h2 vn vn : rrP n vn
.
¼ $ 12 h2 jrhn j$2 ðvn vn vn ..rrrhn Þrhn
$ 12 h2 jrhn j ðvn vn : rrhn Þvn ' rrhn
$2
n
n
þ 2h2 jrhn j# ðvn rhn : rrhn Þðvn vn : rrh
$ Þrh
n $2 n n
n
n
1 2 n
¼ $ 2 h v ' r jrh j ðv v : rrh Þrh
$4
þ h2 jrhn j ðvn rhn : rrhn Þðvn vn : rrhn Þrhn
$4
We thank Christof Schütte, Tony Lelièvre and Luca
Maragliano for useful discussions. E. V.-E. is partially supported by NSF Grants DMS02-09959 and DMS02-39625,
and by ONR grant N00014-04-1-0565.
ð67Þ
These are the terms on the fifth line and the one on the
seventh line in (46), which were the last ones to account
for in this equation (meaning also that we are finally
done).
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