P ro jectio n o f d i u sio n s o n su b m a n ifo ld s: A p p lica tio n to m...

Projection of di!usions on submanifolds:Application to mean force computation

GIOVANNI CICCOTTIRoma La Sapienza

TONY LELIEVRECERMICS – ENPC

AND

ERIC VANDEN-EIJNDENCourant Institute

Abstract

We consider the problem of sampling a Boltzmann-Gibbs probability distribution whenthis distribution is restricted (in some suitable sense) on a manifold ! of Rn implicitlydefined by N constraints q1(x) = · · · = qN (x) = 0. This problem arises for example insystems subject to hard constraints or in the context of free energy calculations. Weprove that the constrained stochastic di"erential equations (i.e. di"usions) proposedin [W. E and E. Vanden-Eijnden, in: Multiscale Modelling and Simulation, eds. S.Attinger and P. Koumoutsakos (LNCSE 39, Springer, 2004)] and [G. Ciccotti, R.Kapral, and E. Vanden-Eijnden, ChemPhysChem 6, 1809 (2005)] are ergodic withrespect to this restricted distribution. We also construct numerical schemes for theintegration of the constrained di"usions. Finally, we show how these schemes can beused to compute the gradient of the free energy associated with the constraints.

Communications on Pure and Applied Mathematics, Vol. 000, 0001–0045 (2006)c! 2006 John Wiley & Sons, Inc. CCC 0010–3640/98/000001-45

SAMPLING ON SUBMANIFOLDS WITH DIFFUSIONS 2

Contents

1 Introduction 31.1 Summary of the main results 41.2 Comparison with other approaches 51.3 Main assumptions and notations 6

2 A di!usion ergodic with respect to the distribution µ! 72.1 The codimension 1 situation 72.2 The codimension N situation 112.3 Remarks and generalizations 12

3 Numerical schemes 14

4 Free energy calculations 204.1 Definition 204.2 The mean force 214.3 Computational aspects 244.4 A variance reduction method 27

A Some useful identities 30

B The mean curvature vector H 33

C The proof of lim!"0 X!t = Xt 35

D The situation with molecular constraints 39D.1 Definition of the free energy 39D.2 Expression for the mean force 41D.3 The orthogonal case 43


1 Introduction

A standard computational issue in statistical mechanics is the calculationof the expectation

(1.1) I(!) =!

Rn!(x)dµ(x)

of an observable ! : Rn ! R with respect to the Boltzmann-Gibbs distri-bution µ defined as

(1.2) dµ(x) = Z#1 exp(""V (x)) dx.

Here Z ="Rn exp(""V (x)) dx is the partition function, " > 0 is the

inverse temperature, and V : Rn ! R is the potential. When the dimen-sionality of space is high, n # 1, standard numerical techniques basedon discretizing the integral in (1.1) become impractical, and various al-ternative techniques have been developed to evaluate (1.1). By far themost common techniques are the so-called Monte-Carlo sampling tech-niques, which amount to devising a stochastic process ergodic with respectto (1.2), and replace the ensemble average in (1.1) by a time-average overthis process using the ergodic theorem:

(1.3) I(!) = limT"$

IT (!) where IT (!) =1T

! T

0!(X t)dt,

where Xt is a generic sample path of the stochastic process. In the caseat hand, one may for instance consider the di!usion:

(1.4) dXt = "$V (Xt) dt +#

2"#1 dWt,

where Wt is a Brownian motion in Rn. Under mild assumptions on thepotential V (see Section 1.3), the process Xt is ergodic with respect to themeasure µ. Standard numerical schemes exist to compute Xt and thesecan be used to calculate IT (!) which, for T su"ciently large, provides anaccurate estimate of I(!).

Many applications require a more general framework, where the Boltzmann-Gibbs distribution µ is replaced by

(1.5) dµ!(x) = Z#1! e#"V (x)d#!(x)

where

(1.6) Z! =!

!e#"V (x)d#!(x).


Here # is a codimension N submanifold of Rn and the measure #! de-notes the surface element on # (i.e. the Lebesgue measure on # definedfrom the Lebesgue measure in the ambient space Rn % #). In practice,the submanifold # is typically defined as the zero level-set of a smoothfunction q with values in RN .

The distribution µ! may be thought of as the projection (or restric-tion) of µ into #. This distribution arises in applications where the systemis subject to hard constraints, these being either of physical origin or ar-tificially imposed for some numerical purposes, or in the context of freeenergy calculations. In this context, the expectation in (1.1) is replacedby

(1.7) I!(!) =!

!!(x)dµ!(x),

and the questions become how to construct a stochastic process (or, morespecifically, a di!usion) ergodic with respect to µ! to sample this distri-bution and how to design numerical algorithms to compute this di!usionin practice ? In [13], a di!usion ergodic with respect to µ! was proposedand in [6] it was shown how to use it in the context of free energy calcu-lations. In the present paper, we put the results of these two referenceson firm mathematical grounds. We also design some specific algorithmsto compute the di!usion ergodic with respect to µ! and show how to usethese in the context of free energy calculations.

One important feature of these algorithms is that the discretized pro-cess exactly satisfies the constraint to live on the submanifold #. Thisis crucial since we consider ergodic limits, and therefore longtime com-putations. In particular, classical Euler schemes directly applied to thestochastic di!erential equation satisfied by the continuous in time pro-cess which lives on the submanifold are not adequate with respect tothis constraint. Moreover, we propose some algorithms which use onlyfirst derivatives of q, which may be of practical interest when analyticalexpressions for the second derivatives of q are di"cult to compute.

1.1 Summary of the main results

In Section 2, we recall the di!usion constrained on # introduced in [13, 6],and we prove that this constrained di!usion is ergodic with respect to (1.5)and thereby allows to compute expectations like (1.7). These resultsare given in Proposition 2.1 in Section 2.1 (for the simpler case when #has codimension 1) and Proposition 2.2 in Section 2.2 (for the general


case when # has codimension N & 1). In Section 2.3, we also discusssome generalizations when the distribution (1.5) is modified by includingsome extra weight factor (which is relevant in the context of free energycalculations), and we compare the di!usion constrained on # with thatarising when (1.4) is constrained to stay close to # by means of softconstraints whose strength is taken to infinity.

In Section 3, we derive some numerical algorithms to integrate theconstrained di!usion proposed in Section 2: these are given in Proposi-tion 3.1.

In Section 4, we show how to use these results to compute the meanforce, i.e. the gradient of the free energy associated with the reactioncoordinate q, which defines a foliation of Rn by a family of manifolds #.In Section 4.1, we first recall the definition of the free energy, then inSection 4.2, we derive several expressions for its gradient, the mean force.These expressions are given in Lemma 4.1 and Proposition 4.2. Finally,in Section 4.3, we show how to compute the mean force in practice and inSection 4.4 we give a variance reduction technique which enhances the ef-ficiency of these calculations. These results are generalized in Appendix Dfor the situations with molecular constraints.

For the reader convenience, several technical results are deferred toAppendices A, B and C.

1.2 Comparison with other approaches

Traditionally in the molecular dynamics community, the sampling of theBoltzmann-Gibbs distribution (1.2) has been done using Nose-Hoover,hybrid kinetic Monte-Carlo, or (non-zero-mass) Langevin dynamics ratherthan the di!usion (1.4). Similarly, the restricted distribution (1.5) isusually sampled by adding holonomic constraints to the Nose-Hoover,hybrid kinetic Monte-Carlo (HKMC), or Langevin equations of motion(see e.g. [28, 7, 8, 19, 18]). Since these equations of motion involves boththe position x of the system and the momentum p associated with thisposition, adding holonomic constraints not only forces the position of thesystem to remain on the manifold #, but also its momentum to alwaysbe tangent to #. In turns, this means that the restricted distributionsampled by constrained Nose-Hoover, HKMC, or Langevin dynamics isa distribution on a submanifold of Rn ' Rn whose marginal in positionspace is precisely (1.5).

In this paper, we give the counterparts of these results by working inconfiguration space alone and using a constrained version of the di!usion


in (1.4). We also give the counterparts of the various results developedin the context of Nose-Hoover, HKMC, or Langevin equations (see e.g.[5, 29, 11, 10, 9, 19, 18]) to compute the gradient of the free energy (theso-called mean force) as seamlessly as possible.

1.3 Main assumptions and notations

Thorough this paper we assume that the potential V is a C2-functionwhich grows su"ciently fast at infinity so that the di!usion in (1.4) isergodic with respect to the Boltzmann-Gibbs distribution (1.2). Most ofour results generalize straightforwardly to situations where the distribu-tion (1.2) is supported on some connected region $ ( Rn, but we will notconsider these cases for simplicity. We also assume that the manifold #is connected and can be defined as the zero level-set of a smooth vector-valued function q = (q1, . . . , qN ) where 0 < N < n and q# : Rn ! R foreach 1 ) $ ) N and we suppose that

(1.8) rank($q1, ...,$qN ) = N on #,

where rank($q1, ...,$qN ) denotes the rank of the matrix

(1.9)$%q#

%xj

%

1%#%N, 1%j%n

.

We also suppose that

(1.10) * > M & sup1%#%N

&&&N'

$=1

G#1#,$(x)$q$(x)

&&&, +x , #,

where

(1.11) G#,$(x) = $q#(x) ·$q$(x), 1 ) $, & ) N,

and G#1#,$(x) denotes the ($, &)-entry of the inverse matrix (G(x))#1. For a

scalar constraint (N = 1), this amounts to assuming that |$q| is uniformlybounded from below by a positive constant. (1.10) guarantees that theexpectation

!

!

&&&N'

$=1

G#1#,$(x)$q$(x)

&&&2dµ!(x)

is finite, a property that we will use below.


We use Greek indices (varying between 1 and N) to denote the com-ponents of quantities related to constraints. Latin indices vary between 1and n, n being the dimension of the ambient space. For brevity, we usethe summation convention on repeated indices for some long formulae:for Greek indices, the sum is over 1 . . . N and for Latin indices, the sumis over 1 . . . n. We denote by - the tensor product, and by

(1.12) $2u =$

%2u

%xi%xj

%

1%i,j%n

the Hessian matrix of a function u : Rn ! R. We also denote by $iu(resp. $i$ju) its partial derivative %u/%xj (resp. %2u/%xi%xj). Finally,the superscript T denotes the transposition operator.

2 A di!usion ergodic with respect to the distribution µ!

In this section, we give a di!usion that is ergodic with respect to µ! [13, 6].For clarity, we first consider in Section 2.1 the case when # is a manifoldof codimension 1. Then we generalize our result in Section 2.2 to the caseof a manifold of codimension N & 1. Section 2.3 is then devoted to someremarks, especially concerning the case when the constraints are softlyimposed by a constraining potential.

2.1 The codimension 1 situation

Let # := {x : q(x) = 0} be the zero level-set of the scalar valued C2-function q : Rn ! R.

Let us introduce the normal n(x) to # at point x

(2.1) n(x) =$q(x)|$q(x)|

and the orthogonal projector P (x) on the tangent space at point x to #defined by:

(2.2) P (x) = Id " n(x) - n(x).

Notice that P 2 = P and P = P T , since P is an orthogonal projector. Let

(2.3) '(x) = div n(x).

be the mean curvature of # at point x. We have:


Proposition 2.1 The distribution µ! in (1.5) is the unique equilibriumdistribution of the di!usion (written in Ito form):

(2.4) dX t = P (Xt)("$V (Xt) dt+

#2"#1dWt

)""#1'(X t)n(X t) dt.

Proposition 2.1 implies that if ! in Lp(µ!) (with p > 1),

(2.5) I!(!) = limT"$

I!,T (!) where I!,T (!) =1T

! T

0!(X t)dt,

where Xt is a solution of (2.4) and the convergence is a.s. and in Lp.

Proof: First let us note that by assumption on # and V , V viewedas a function from # to Rn is C2 and grows su"ciently fast at infinityso that Z! < *. Therefore µ! is well defined. Moreover, the transitionprobability function is strictly positive so that any invariant measure isequivalent to the Lebesgue measure #!, which implies the uniqueness ofthe invariant measure (see Proposition 6.1.9 p. 188 in [12]).

So it su"ces to prove that µ! is an invariant measure for (2.4). Tothis end let u(t,x) = Ex(f(Xt)), where Xt satisfies (2.4) and Ex denotesthe expectation over this process conditional on X0 = x. Then µ! is aninvariant measure for (2.4) if

(2.6)!

!u(t,x)dµ!(x) =

!

!u(0,x)dµ!(x).

To check (2.6), notice that u(t,x) satisfies the backward Kolmogorovequation

%u

%t= "P (x)$V (x) ·$u + "#1H(x) ·$u + "#1P (x) : $2u

where

P (x) : $2u =n'

i,j=1

Pi,j(x)%2u

%xi%xj


and H = "'n denotes the mean curvature vector. It follows that

d

dt

!

!u(t,x)dµ!(x)

= Z#1!

!

!

("P (x)$V (x) ·$u(t,x) + "#1H(x) ·$u(t,x)

+ "#1P (x) : $2u(t,x))

exp(""V (x)) d#!(x),

= "#1Z#1!

!

!

(div !

*$u(t,x) exp(""V (x))

+

+ H(x) ·$u(t,x) exp(""V (x)))

d#!(x),

= 0.

Here div ! denotes the surface divergence:

(2.7) div !(!) = tr(P$!)

and we used the divergence theorem on manifolds (see [2, 3]):

(2.8) +! , C1c (Rn, Rn) :

!

!div !(!) d#! = "

!

!H · ! d#!.

This shows that (2.6) holds, which concludes the proof.

Componentwise, (2.4) can be written as

d(Xt)i =,

P (X t)("$V (Xt) dt +#

2"#1dWt)-

i

+ "#1n'

j,k=1

Pj,k$jPi,k(X t) dt.(2.9)

where we used the identity.n

j,k=1 Pj,k$jPi,k = "'ni (see (A.1)). From (2.9),we see that (2.4) can also be written in Stratonovich form as

(2.10) dXt = "P (X t)$V (Xt) dt +#

2"#1P (X t) . dWt,

which shows that (2.4) essentially amounts to projecting (1.4) onto #. Inparticular, it implies that

dq(Xt) = $q(Xt) ·,"P (Xt)$V (Xt) dt +

#2"#1P (Xt) . dWt

-= 0,

as necessary since we must have Xt , #.


With a view to the discretization of (2.4) (see Section 3), it is alsoworth mentioning that (2.4) may be obtained by imposing the constraintthat Xt , # using Lagrange multipliers. Indeed, let us modify thestochastic di!erential equation (1.4) in the following way:

(2.11)/

dX t = "$V (X t) dt +0

2"#1dW t + dY t,

with Y t such that P (Xt)dY t = 0 and q(Xt) = 0.

Since we suppose that q(X0) = 0, we set Y 0 = 0. We assume moreoverthat Y t is adapted with respect to the filtration of the Brownian mo-tion W t. Computing dq(X t), and decomposing dY t = dA(t)+S(t)dW t,where A(t) is a process with finite variation, one obtains:

dq(X t) =$q(X t) · ("$V (X t) dt + dA(t))

+ 12$

2q(Xt) :,#

2"#1Id + S(t)- ,#

2"#1Id + S(t)-T

dt

+ $q(Xt) ·,#

2"#1dW t + S(t)dW t

-.

Therefore, in order that d(q(X t)) = 0 we must have (using the fact thatdY t, and hence dA(t) and S(t)dW t, are aligned with the normal direc-tion $q(Xt))

S(t) = "#

2"#1$q -$q

|$q|2(Xt)

and thus

dA(t) =$q ·$V

|$q|2 $q(Xt) dt " "#1 $q

|$q|2$2q : P (Xt) dt,

=$q ·$V

|$q|2 $q(Xt) dt " "#1'n(Xt) dt,

where we used the second equality in (A.1). As a result

(2.12)dY t = (P (X t) " Id)

,"$V (Xt) dt +

#2"#1dW t

-

+ "#1H(X t) dt.

Thus, we recover (2.4).


2.2 The codimension N situation

The result in Section 2.1 can be generalized to the case of a codimension Nmanifold # which is the zero level-set of the vector valued function q =(q1, ..., qN ) with q# : Rn ! R (1 ) $ ) N).

One central object we have considered in Section 2.1 is the orthogonalprojector P onto #. In the case of N constraints, this projector reads(compare (2.2)):

(2.13) P (x) = Id "N'

#,$=1

G#1#,$(x)$q#(x) -$q$(x),

where we recall the definition in (1.11) for the N ' N matrix G:

G#,$(x) = $q#(x) ·$q$(x), 1 ) $, & ) N,

and G#1#,$ denotes the ($, &)-entry of the inverse matrix G#1. To check

that P (x) is the orthogonal projector onto # at point x, notice that forany 1 ) ( ) N ,

P$q% = $q% "N'

#,$=1

G#1#,$$q#$q$ ·$q%

= $q% "N'

#,$=1

G#1#,$G$,%$q# = 0,

while, for any vector u such that +1 ) & ) N , u ·$q$ = 0, we have

Pu = u "N'

#,$=1

G#1#,$$q#$q$ · u = u.

Since P (x) is an orthogonal projector, it is a symmetric matrix. Noticealso that in the special case of orthogonal constraints, namely if $q# ·$q$ = )#,$ |$q#|2, (2.13) simplifies into:

(2.14) P (x) = Id "N'

#=1

n#(x) - n#(x),

where the normal n# is defined by

(2.15) n#(x) =$q#(x)|$q#(x)| , 1 ) $ ) N.


In the case at hand, the equivalent of the curvature ' in (2.3) is

(2.16) '# = |$q#|N'

%=1

G#1%,#

(%q% "$2q% :

* N'

$,&=1

G#1&,$$q& -$q$

+),

and Proposition 2.1 generalizes as

Proposition 2.2 The distribution µ! in (1.5) is the unique equilibriumdistribution of the di!usion:

(2.17)

dXt = P (X t),"$V (Xt) dt +

#2"#1 dWt

-

" "#1N'

#=1

'#(X t)n#(X t) dt.

Equation (2.17) is the generalization of (2.4), and the equivalent of (2.5)holds here as well. The proof of Proposition 2.2 is similar to that of Propo-sition 2.1. It is again based upon the divergence theorem on manifolds,and the fact that the mean curvature vector is

(2.18) H = "N'

#=1

'#n#

(see Appendix B below for further details). We skip it for the sake ofbrevity.

As in the case N = 1, (2.17) can be seen as the projection of (1.4) usingthe projection operator P defined by (2.13), and a Stratonovich integra-tion rule. In other words, (2.9) and (2.10) also hold in the codimensionN situation, since

.nj,k=1 Pj,k$jPi,k = "

.N#=1 '#(n#)i (see (A.3)).

Notice that (2.17) can also be obtained by imposing the constraintthrough Lagrange multipliers, as in the case N = 1. For N > 1, theconstraining force Y t is also defined by (2.11) and similar computationsas in the case N = 1 yield (2.12) (with H defined by (2.18)).

2.3 Remarks and generalizations

Later on in Section 4, we will see that the free energy calculations involvethe following distribution which generalizes (1.5):

(2.19) dµ!,f (x) = Z#1!,f e#"V (x)f(x)d#!(x),


where

(2.20) Z!,f =!

!e#"V (x)f(x)d#!(x),

and f : Rn ! (0,*) is a C2-function to be defined, with a growth condi-tion at infinity consistent with Z!,f < *. Obviously µ! / µ!,1 and

I!,f (!) =!

!!(x)dµ!,f (x) =

I!(f!)I!(f)

.

On the other hand, sampling with respect to (2.19) can also be straight-forwardly performed upon noting that the measure µ!,f associated withthe potential V is simply the measure µ! associated with the potential

(2.21) Vf = V " "#1 ln f.

In other words, a di!usion allowing to sample µ!,f is provided by (2.17)in which Vf is substituted for V , that is

(2.22)

dXt = P (X t)("$(V " "#1 ln f)(Xt) dt +

#2"#1 dWt

)

" "#1N'

#=1

'#(Xt)n#(Xt) dt.

In Section 4, we will see that the measure µ!,|det G|!1/2 (corresponding tothe specific choice f = |detG|#1/2) naturally arises in the context of freeenergy calculations.

Let us describe another instance where the stochastic di!erential equa-tion (2.22) with f = |det G|#1/2 also appears. Consider

(2.23) dX!t = "$V (X!

t ) dt " 12*

N'

#=1

$(q2#)(X!

t ) dt +#

2"#1dW t,

where * > 0 is a parameter. The additional term involving $(q2#) in (2.23)

is a penalty term which constraints Xt in the vicinity of # = {x : q(x) =0}. Letting * ! 0 amounts to imposing the constraint that Xt , # a.s.In fact, we prove in Appendix C (in the case N = 1) that the limit processXt of X!

t when * ! 0 is solution of the stochastic di!erential equation:

dXt = P (Xt)("$

(V + "#1 ln |detG|1/2

)(Xt) dt +

#2"#1 dW t

)

" "#1N'

#=1

'#(X t)n#(Xt) dt.(2.24)


This equation is not (2.17). Rather it is a special case of (2.22) for f =|det G|#1/2, i.e. Xt samples the distribution µ!,|det G|!1/2 .

Notice that the measure µ! depends on q# only through its zero set(which defines #). The values of q# around # are irrelevant. In thissense, µ! is an intrinsic quantity. Accordingly the stochastic di!erentialequation (2.17) (and in particular the mean curvature vector H) canbe defined knowing only #. In contrast, the measure µ!,|det G|!1/2 alsodepends on the values of q# around #. In this sense, it is a non-intrinsicquantity. Because the constraints are softly imposed in (2.24) (and notrigidly as in (2.4)), in the limit as * ! 0 the limiting process Xt still“sees” the variation of q# around #, through the term ln |detG|#1/2 inthe modified potential V| det G|!1/2.

Remark 2.3 (Rigidly and softly constrained dynamics) The fact that thestatistics at equilibrium associated with the rigidly constrained dynam-ics (2.4) and the softly constrained dynamics (2.24) are not identical isrelated to an apparent paradox which has been often discussed in the liter-ature, about the di!erent statistics at equilibrium of rigid and sti! bondsin bead spring models: see [26] p. 228, Section 4.6 in [27], [20], [15], [30],or paragraph 3 in [25]. We here exhibit the di!erence between statisticalproperties of rigidly and softly constrained dynamics in the framework ofover-damped Langevin dynamics, but this question has also been discussedeither in the framework of Hamiltonian systems at equilibrium (in thecanonical ensemble) or in the non-zero-mass Langevin dynamics frame-work (see e.g. [29, 9, 19, 18]).

3 Numerical schemes

In this section, we construct numerical schemes which satisfy exactly theconstraint q(x) = 0 since we have in mind long-time simulations forcomputing I!(!) (defined by (1.7)) by a mean over a sample path. Wesuppose in this section that q(X0) = 0. We have:

Proposition 3.1 The following two schemes are consistent with (2.17):

(3.1)

1222223

222224

Xn+1 = Xn "$V (Xn)%t +#

2"#1%W n

+N'

#=1

+#,n$q#(Xn+1),

where +#,n such that q(Xn+1) = 0,


and

(3.2)

1222223

222224

Xn+1 = Xn "$V (Xn)%t +#

2"#1%W n

+N'

#=1

+#,n$q#(Xn),

where +#,n such that q(Xn+1) = 0,

where %W n = W tn+1 " W tn denotes the Brownian increment.

The proof of the Proposition is given at the end of this section. Thesemi-implicit scheme in (3.1) can in fact be rewritten in a variationalformulation as follows:

(3.3)

13

4X' = Xn "$V (Xn)%t +

02"#1%W n,

Xn+1 = arg minY &Rn

5|X' " Y |2 : q(Y ) = 0

6.

In this case, the +#,n can be interpreted as scalar Lagrange multipliersassociated with the constraint q(Xn+1) = 0. We expect the scheme (3.1)to exhibit better stability properties than the scheme (3.2) since it admitsa variational interpretation.

In practice, in order (3.1), (3.2) and (3.3) to be well-posed, we need %tto be su"ciently small so that X' is not “too far” from the manifold #.To solve Problem (3.1), one can use classical methods for optimizationproblems with constraints. We refer to [16] for a presentation of theclassical Uzawa algorithm, and to [4] for more advanced methods. To solveProblem (3.2), one can use classical methods to solve nonlinear problems(like Newton method, see [4]). We also refer to Chapter 7 of [24] wheresimilar problems are discussed.

In the following, we will admit that the schemes (3.1), (3.2) and (3.3)are well posed and indeed convergent (in the mean square sense for ex-ample), namely that the trajectory (X0, . . . ,XM ) where M = T/%t con-verges when %t ! 0 (for a fixed T ) to (Xs)0%s%T which satisfies (2.17).Since (X t)t'0 is ergodic with respect to µ! (see Proposition 2.2), theschemes (3.1) and (3.2) can be used to sample µ! and to compute quan-tities such as (1.7). In Section 4, we more specifically discuss how theyallow the computation of free energy di!erences and mean forces.

Remark 3.2 (Computation of $V|det G|!1/2) As mentioned earlier, in thecontext of free energy calculations (see Section 4), the distribution µ!,|det G|!1/2

(corresponding to the specific choice f = |det G|#1/2 in (2.19)) arises


naturally and this amount to replacing V by V|det G|!1/2 in (3.1) or (3.2).Since V| det G|!1/2 = V " "#1 ln |det G|#1/2, this requires the computationof the following term:

(3.4)

"#1$ ln(|det G|#1/2

)(Xn)%t

= "12"#1

'

#,$

(G#1

#,$$G#,$

)(Xn)%t,

= ""#1'

#,$

(G#1

#,$$2q#$q$

)(Xn)%t,

where we used Jacobi’s formula: for a given tensor M ,

(3.5) $ ln(det M) ='

#,$

M#1#,$$M$,#.

In order to avoid the computation of $2q#(Xn) which is cumbersome,one can approximate (3.4) by:(3.6)" "#1

'

#,$

(G#1

#,$$2q#$q$

)(Xn)%t

= ""#1N'

#,$=1

G#1#,$(Xn)

*$q# (Xn +%t$q$(Xn)) "$q#(Xn)

++ o(%t).

The proof of Proposition 3.1 relies on the following Lemma which givesexpansions of the +#,n appearing in (3.1) and (3.2).

Lemma 3.3 Let Xn be the solution of (3.1) or (3.2). Then +#,n is suchthat:

(3.7) +#,n = +0#,n

0%t + +1

#,n%t + o(%t),

with

(3.8) +0#,n = "

#2"#1

N'

$=1

G#1#,$$q$(Xn) · wn,

where wn = %W n/0%t are i.i.d. Gaussian variables in Rn with zero


mean and variance Id, and

(3.9)

+1#,n =

N'

$=1

G#1#,$$q$ ·$V (Xn)

+ "#1N'

$,&=1

G#1#,$G

#1%,&$

2q$ : $q% -$q&(Xn)

" "#1N'

$=1

G#1#,$%q$(Xn).

Proof of Lemma 3.3: For the sake of brevity we only present theproof of Lemma 3.3 for the scheme (3.1). The proof for the scheme (3.2)is similar.

The Lagrange multipliers +#,n are obtained by requiring that q(Xn+1) =0 if q(Xn) = 0. Using (3.1) as well as the a priori expansion (3.7) of +#,n,this is equivalent to requiring that: for any 1 ) & ) N ,

(3.10)

0 = q$(Xn+1),

= $q$(Xn) ·("$V (Xn)%t +

#2"#1%W n

+ (+0#,n

0%t + +1

#,n%t)$q#(Xn+1))

+12KT

n$2q$(Xn)Kn + o(%t),

where Kn =0

2"#1%W n + +0#,n

0%t$q#(Xn+1). Since

$q#(Xn+1) = $q#(Xn) + $2q#(Xn)Kn + o(0%t),

where Kn =0

2"#1%W n + +0#,n

0%t$q#(Xn), equating terms of equal

order in %t in (3.10) gives

122223

22224

0 =0

2"#1$q$(Xn) ·%W n + +0#,n

0%t G#,$(Xn),

0 = "$q$(Xn) ·$V (Xn)%t + +1#,nG#,$(Xn)%t

+0%t+0

#,n($q$)T$2q#(Xn)Kn +12K

Tn$2q$(Xn)Kn.

From this, we obtain formula (3.8) for +n0,# and the following expression


for +n1,#:

+1#,n = G#1

#,$$q$ ·$V (Xn)

" G#1#,$+

0%,n+

0&,n$2q% : $q$ -$q&(Xn)

" 12G#1

#,$+0%,n+

0&,n$2q$ : $q% -$q&(Xn)

"#

2"#1 G#1#,$ +

0%,n $2q%(Xn) : $q$ - wn

"#

2"#1 G#1#,$ +

0%,n $2q$(Xn) : $q% - wn

" "#1 G#1#,$$

2q$(Xn) : wn - wn.

We now use (3.8) in this expression together with the fact that in thelimit as %t ! 0, wn - wn = Id since wn is always multiplied by Fn"t

measurable functions. For example, we have in the limit %t ! 0,

+0%,n+

0&,n = 2"#1%t#1G#1

%,%"$q%"(Xn) ·%W nG#1&,&"$q&"(Xn) ·%W n,

= 2"#1 G#1%,%" G#1

&,&" G%",&"(Xn) + o(1) = 2"#1 G#1%,&(Xn) + o(1).

We thus obtain the following expression for +n1,#:

+1#,n = G#1

#,$$q$ ·$V (Xn)

" 2"#1G#1#,$G

#1%,&$

2q% : $q$ -$q&(Xn)

" "#1G#1#,$G

#1%,&$

2q$ : $q% -$q&(Xn)

+ 2"#1 G#1#,$G

#1%,& $

2q% : $q& -$q$(Xn)

+ 2"#1 G#1#,$ G#1

%,& $2q$ : $q& -$q%(Xn)

" "#1 G#1#,$%q$(Xn) + o(1),

from which we deduce (3.9).

We are now in position to prove Proposition 3.1.

Proof of Proposition 3.1: Let us first consider the scheme (3.1)(or, equivalently, (3.3)). To check its consistency with (2.17), we computethe value of the term +#,n$q#(Xn+1)%t using the expressions for +0

#,n and+1

#,n given in Lemma 3.3. and the property that %W n-%W n = Id%t inthe limit as %t ! 0, since %W n is always multiplied by Fn"t measurable


functions. This gives

+#,n$q#(Xn+1)

=(+0

#,n

0%t + +1

#,n%t)$q#(Xn)

+ +0#,n

0%t$2q#(Xn)

,#2"#1%W n + +0

%,n

0%t$q%(Xn)

-+ o(%t),

= "#

2"#1G#1#,$$q#$q$(Xn) ·%W n + G#1

#,$$q#$q$ ·$V (Xn)%t

+ "#1G#1#,$$q#G#1

%,&$2q$ : $q% -$q&(Xn)%t

" "#1 G#1#,$$q#%q$(Xn)%t " 2"#1G#1

#,$$2q#$q$(Xn)%t

+ 2"#1 G#1%,#$2q#$q%(Xn)%t + o(%t),

= (P (Xn) " Id),"$V (Xn)%t +

#2"#1%W n

-

" "#1$q#G#1#,$

("G#1

%,&$2q$ : $q% -$q&(Xn) +%q$(Xn)

)%t + o(%t),

= (P (Xn) " Id),"$V (Xn)%t +

#2"#1%W n

-

" "#1'#n#(Xn)%t + o(%t).

This shows that (3.1) is equivalent to

(3.11)Xn+1 = Xn + P (Xn)

,"$V (Xn)%t +

#2"#1%W n

-

" "#1'#n#(Xn)%t + o(%t),

which is a consistent discretization of (2.17).We now consider the scheme (3.2). In this case, using the expressions

for +0#,n and +1

#,n given in Lemma 3.3, we obtain

+#,n$q#(Xn)

=(+0

#,n

0%t + +1

#,n%t)$q#(Xn) + o(%t),

= "#

2"#1G#1#,$$q#$q$(Xn) ·%W n + G#1

#,$$q#$q$ ·$V (Xn)%t

+ "#1G#1#,$$q#G#1

%,&$2q$ : $q% -$q&(Xn)%t

" "#1 G#1#,$$q#%q$(Xn)%t + o(%t),

= (P (Xn) " Id),"$V (Xn)%t +

#2"#1%W n

-

" "#1'#n#(Xn)%t + o(%t),


which shows that (3.2) is also equivalent to (3.11) and proves that thisscheme is consistent with (2.17).

4 Free energy calculations

In this section, we discuss the computation of free energy di!erences,defined in Section 4.1, and more precisely of the gradient of the free energy,the so-called mean force. In Section 4.2, we give an explicit expressionfor the mean force and exhibit a link between the mean force and theconstraining force Y t defined by (2.11)–(2.12). We then use this link tobuild a numerical scheme to compute the mean force in Section 4.3. Avariance reduction method is proposed in Section 4.4. In this section, weassume that no molecular constraints are present: for completeness, thesituation with molecular constraints is discussed in Appendix D.

In this paper, we describe the computation of free energy di!erencesby imposing the reaction coordinate at a fixed value (this is the so-calledThermodynamic Integration, see [22]). Note that it is also possible tocompute free energy di!erences by prescribing an evolution of the reactioncoordinate, in the spirit of Jarzinski equality (see [21, 23]).

4.1 Definition

Let X , Rn denote the random variable whose distribution is the Boltzmann-Gibbs distribution (1.2). Given q = (q1, . . . , qN ) where q# : Rn ! R foreach 1 ) $ ) N , the quantity

(4.1) Z = q(X)

is a random variable in RN . Let us denote by m(z) the probability densityfunction (with respect to the Lebesgue measure on RN ) of Z. Then, bydefinition, the quantity

(4.2) F (z) = ""#1 ln m(z)

is called the free energy associated with q, which is called the reactioncoordinate. The free energy is directly relevant to compute expectationsof observables depending on x only implicitly via q, since by construction

(4.3)!

Rn!(q(x))dµ(x) =

!

RN!(z)e#"F (z)dz.

Let us now introduce the following generalization of Fubini’s theorem,derived from the co-area formula (see Theorem 2 p. 117 of [14]): if q :


Rn ! RN is Lipschitz and g : Rn ! R is a function in L1(Rn), then

(4.4)!

Rng(x)|det G(x)|1/2 dx =

!

RN

!

!(z)g(x)d#!(z)(x)dz.

Here #(z) = {x : q(x) = z}, #!(z) is the (n " N)-dimensional Hausdor!measure, which reduces in our case to the Lebesgue measure on #(z)since q is regular, and G is the matrix defined in (1.11) which we recallfor convenience

G#,$(x) = $q#(x) ·$q$(x), 1 ) $, & ) N.

Using (4.4) and the definition (1.2) of µ, we have:

(4.5)

!

Rn!(q(x))dµ(x)

=!

RN!(z)

(Z#1

!

!(z)e#"V (x)|detG(x)|#1/2d#!(z)(x)

)dz.

Therefore, by comparing (4.3) and (4.5) we deduce that F (z) is given by

(4.6) F (z) = ""#1 ln(Z#1

!

!(z)e#"V (x)|det G(x)|#1/2d#!(z)(x)

).

Equation (4.6) can also be written as

(4.7) F (z) = ""#1 ln(Z#1Z!(z),|det G|!1/2

)

where Z!(z),|det G|!1/2 is the normalization factor associated with the dis-tribution µ!(z),|det G|!1/2 (see (2.19)). In other words, computing themean force amounts to computing some partition functions.

4.2 The mean force

In practice, a way to compute the free energy F defined by (4.6) (or (4.7))is to compute first its gradient, since the latter can be expressed as anexpectation over the distribution µ!(z),|det G|!1/2 (see Lemma 4.1 below).The gradient of F is usually referred to as the mean force, and it can beexpressed via the following:


Lemma 4.1 The gradient of F (namely the mean force) can be expressedas: for any 1 ) $ ) N ,

$#F (z)

=N'

%=1

!

!(z)

($V · G#1

#,%$q% " "#1$ · (G#1#,%$q%)

)dµ!(z),|det G|!1/2,

(4.8)

=N'

%=1

!

!(z)G#1

#,%$q% ·($V|det G|!1/2 + "#1H

)dµ!(z),| det G|!1/2 .

(4.9)

The proof of this Lemma is given below. From these two expressionsof the mean force, one may use two di!erent methods to compute themean force. The expression (4.8) of $#F is rather complicated since itinvolves the divergence of the inverse of G. However, remarkably enough,we shall show in Section 4.3 that we do not have to compute explicitlythis divergence to evaluate $#F (z). This can be done by suitable nu-merical approximation of (4.8) (together with the approximation (3.6 of$V|det G|!1/2).

On the other hand, the expression (4.9) of $#F can be used to derivean alternative and even simpler procedure. It is based on the followingproposition where $#F (z) is expressed as a mean over the componentalong $q# (taking ($q1, . . . ,$qN ) as a basis of the normal space to #z) ofthe constraining force Y t (defined by (2.12)), using the corrected potentialV|det G|!1/2 (defined by (2.21)) instead of V . This has also been observedin the framework of Hamiltonian dynamics (see [29, 9, 19, 18]).

Proposition 4.2 Consider the processes Xt and Y t defined by (2.11)and (2.12), with V replaced by V| det G|!1/2 . Then for 1 ) $ ) N ,

(4.10) limT"$

1T

! T

0

N'

$=1

G#1#,$$q$(Xt) · dY t = $#F (0),

a.s. and in Lp, p & 1.

Proof of Lemma 4.1: Let & by a C$c (R) function and ! = &(. Let

us consider, for a fixed 1 ) $ ) N ,!

RN!(z#) exp(""F (z)) dz = Z#1

!

Rn!(q#(x))e#"V (x)dx,


where we have used (4.3) and (1.2). The left-hand side can be expressedas follows (using (4.7)):

(4.11)

!

RN!(z#) exp(""F (z)) dz

=!

RN&((z#) exp(""F (z)) dz,

= "!

RN&(z#)$#F (z) exp(""F (z)) dz,

= "Z#1!

RN&(z#)Z!(z),| det G|!1/2$#F (z) dz.

The right-hand side can be expressed as follows:(4.12)

Z#1!

Rn!(q#(x))e#"V (x)dx

= Z#1!

Rn&((q#(x))e#"V (x)dx,

= Z#1!

RnG#1

#,%$q% ·$ (& . q#) (x)e#"V (x)dx,

= "Z#1!

Rn$ ·

(G#1

#,%$q%e#"V

)(x)& . q#(x)dx,

= Z#1!

Rn

("$V · G#1

#,%$q% "$ ·(G#1

#,%$q%

))(x)e#"V (x)&(q#(x))dx,

= Z#1!

RN&(z#)Z!(z),|det G|!1/2

'$!

!(z)

("$V · G#1

#,%$q% "$ ·(G#1

#,%$q%

))dµ!(z),| det G|!1/2

%

dz,

where we have used (4.4) and (2.19) for the last equality. The factthat (4.11) is equal to (4.12) for all functions & completes the proofof (4.8).

To prove (4.9), it remains to show that

$V · G#1#,%$q% " "#1$ · (G#1

#,%$q%)

= $V|det G|!1/2 · G#1#,%$q% + "#1G#1

#,%$q% · H .

Using (2.18) and V|det G|!1/2 = V + "#1 ln(|det G|1/2

), this is equivalent

to show that:

" "#1$ · (G#1#,%$q%)

= "#1G#1#,%$q% ·$ ln |det G|1/2 " "#1|$q#|#1'#,


which is a direct consequence of the expression (A.5) of '# given in Ap-pendix A.

We are now in position to prove Proposition 4.2.

Proof of Proposition 4.2: By replacing V by V| det G|!1/2, the mea-sure sampled by Xt is µ!,|det G|!1/2 . Moreover, the process Y t is thendefined by

dY t = (P (X t) " Id)("$V|det G|!1/2(Xt) dt +

#2"#1dW t

)

+ "#1H(Xt) dt, Y 0 = 0.

Hence, since (P (X) " I) G#1#,$$q$(X) = "G#1

#,$$q$(X), we obtain

G#1#,$$q$(Xt) · dY t =

N'

%=1

G#1#,%$q% ·

($V|det G|!1/2 + "#1H

)(Xt) dt

"#

2"#1N'

$=1

G#1#,$$q$(Xt) · dW t.

In the bounded variation part, we recover the expression (4.9) of the meanforce. Now, (4.10) follows from the ergodicity of Xt (see (2.5)) and thefact that

limT"$

1T

! T

0

N'

$=1

G#1#,$$q$(Xt) · dW t = 0,

by (1.10).

4.3 Computational aspects

We now discuss the computation of the mean force defined by (4.6), usingthe expressions (4.8) or (4.9). In the following, we suppose that z = 0,without loss of generality.

The first method we propose is based on (4.8). The first term at theright hand side of (4.8) can be obtained from

(4.13)

limT"$

lim"t"0

1M

M'

m=1

N'

%=1

($V · G#1

#,%$q%

)(Xm)

=N'

%=1

!

!$V · G#1

#,%$q%dµ!,|det G|!1/2,


where M = T/%t and Xn is the solution to (3.1) or to (3.2), with Vreplaced by V|det G|!1/2 = V + "#1 ln |det G|1/2. As for the second term,we have(4.14)

" "#1 limT"$

lim"t"0

1M%t

M'

m=1

N'

%=1

((G#1

#,%$q%)(Xm +%W m)

" (G#1#,%$q%)(Xm)

)·%W m

= ""#1N'

%=1

!

!$ · (G#1

#,%$q%)dµ!,|det G|!1/2 ,

where we used the fact that %W n-%W n = Id%t in the limit as %t ! 0.Equations (4.13) and (4.14) (together with the approximation (3.6)) allowto estimate $#F (0) without having to compute $2q#.

The second method is based on (4.9), and more precisely, on Propo-sition 4.2. As in the continuous in time case (see (4.10)), the mean force$#F (0) may be computed by averaging the Lagrange multipliers +#,n

entering the algorithms (3.1) or (3.2).

Proposition 4.3 Let Xn be the solution to (3.1) or to (3.2), with Vreplaced by V| det G|!1/2 = V + "#1 ln |det G|1/2. Then,

(4.15) limT"$

lim"t"0

1M%t

M'

m=1

+#,m = $#F (0),

where M = T/%t.

We recall that in practice, in order to compute $V| det G|!1/2 , one can resortto a suitable finite di!erence scheme (see the approximation (3.6)). Thisproposition is a direct consequence of Proposition 4.2 and the followinglemma.

Lemma 4.4 Let Xn be the solution to (3.1) or to (3.2). Assume more-over (1.10). Then, for 1 ) $ ) N , +#,n is such that

(4.16) lim"t"0

1M%t

M'

m=1

+#,m =1T

! T

0

N'

$=1

G#1#,$$q$(X t) · dY t

with Y t defined by (2.12) and where T = M%t is fixed, and thereforeM ! *.


Remark 4.5 (Computation of other energies) All the preceding compu-tations may be generalized to the following energy:

(4.17) Ff (z) = ""#1 ln!

exp(""V (x)) f(x) d#!(z)(x),

where f is a given positive function1 such that Z!(z),f < *. Indeed, inthis case, the expression of the gradient of Ff is given by the followingformula (which is a generalization of (4.9))

$#Ff (z) =!

!(z)

N'

%=1

G#1#,%$q% ·

($Vf + "#1H

)dµ!(z),f ,(4.18)

where the modified potential Vf is defined by (2.21).Suppose now that we use in our numerical schemes (3.1) or (3.2) the

potential Vf instead of V . Then, following the arguments of Section 4.3we obtain that the mean of the Lagrange multipliers converges to the meanforce (written here for z = 0): for 1 ) $ ) N ,

(4.19) limT"$

lim"t"0

1M%t

M'

m=1

+#,m = $#Ff (0),

where M = T/%t.

Proof of Lemma 4.4: Using Lemma 3.3, we have

lim"t"0

1M%t

M'

m=1

+#,m

=1T

! T

0G#1

#,$

($q$ ·$V + "#1G#1

%,&$2q$ : $q% -$q& " "#1%q$

)(Xt) dt

"#

2"#1 1T

! T

0G#1

#,$$q$(X t) · dW t.

Using the fact that (P (X) " I)G#1#,$$q$(X) = "G#1

#,$$q$(X) and

G#1#,%$q% · H = G#1

#,$G#1%,&$

2q$ : $q% -$q& " G#1#,$%q$ = '#

one easily obtains (4.16).

1 With this notation, the free energy defined by (4.6) is F| det G|!1/2 , up to an additiveconstant. Notice that this constant does not intervene in the mean force.


4.4 A variance reduction method

In the numerical scheme we have described to compute the mean force (seeformula (4.15)), there are three sources of errors: the time discretizationerror (%t ! 0), the longtime limit error (T ! *) and the statisticalerror due to the fact that we use a stochastic process. In this section, wefocus on the statistical error. It is linked to the the variance of the result.Let us consider the case N = 1. We have:

1M%t

M'

m=1

+m = "#

2"#1 1T

M'

m=1

|$q|#2$q(Xn) ·%W n

+1T

M'

m=1

+1m%t + o(1).

In the limit %t goes to zero, the first term in the right-hand side convergesto the martingale part

"#

2"#1 1T

! T

0|$q|#2$q(Xs) · dW s

of the constraining force Y t, while the second part converges to thebounded variation part of Y t (see Equation (2.12)). It is the limit, whenT goes to infinity, of the second term which yields the mean force F ((0).The first term goes to 0 in the limit T ! * but this term is responsiblefor a large variance of the result.

Therefore, a natural idea to reduce the variance is to eliminate thefirst term. It is possible to directly compute this term (which is theprojection of the Brownian increment) and to subtract it from the La-grange multiplier. Alternatively, the following scheme, which is very easyto implement, may be used. We consider that t = tn and we denoteby +(%W n) the Lagrange multiplier obtained from (3.1) or (3.2) with aBrownian increment %W n. The next position Xn+1 is defined by (3.1)or (3.2), but the Lagrange multiplier used in formula (4.15) is now definedas:

+n =12

(+(%W n) + +("%W n)) .

One can check that this does not change the value of the bounded varia-tion part +1

n of +n, but “eliminates” the martingale part +0n. This method

reminds us of the antithetic variables variance reduction method classi-cally used in Monte Carlo methods. In practice, this method seems to bevery e"cient (see [23]). This idea can be straightforwardly generalized tothe case N > 1.


Acknowledgments

The authors would like to thank the Centre de Recherches Mathematiques(University of Montreal) and the Institute for Mathematics and its Ap-plications (University of Minnesota) where this work has been initiated.We are grateful to Claude Le Bris for his careful reading of this paperand his suggestions. T. L. would also like to thank M. Rousset andG. Stoltz for stimulating and enlightening discussions. T. L. acknowledgesfinancial support from Action Concertee Incitative Nouvelles Interfacesdes Mathematiques, “Simulation moleculaire”, Ministere de la Recherche,France. E.V.-E. is partially supported by NSF grants DMS02-09959 andDMS02-39625, and by ONR grant N00014-04-1-0565.

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A Some useful identities

Here we give some useful identities that were used in text.

Lemma A.1 (Scalar constraint case, N = 1) We define by (2.2) the or-thogonal projection P (x) on the tangent space of # = {x, q(x) = 0} atpoint x (where q : Rn ! R). The normal n(x) and the curvature '(x)at point x , # are defined by formulas (2.1) and (2.3). The mean cur-vature vector is defined by H = "'n. The following equalities hold: for1 ) i ) n,

(A.1)

n'

j,k=1

Pj,k$jPi,k = |$q|#1n'

j=1

$j(|$q|Pi,j),

= "|$q|#2$iqn'

j,k=1

$j$kqPj,k,

= "'ni = H i.

Moreover, we have: for 1 ) i ) n,

(A.2)n'

j=1

$jPi,j = "n'

j=1

Pi,j$j ln |$q| + H i.

Lemma A.2 (Vectorial constraint case, N & 1) We define the orthogonalprojection P (x) on the tangent space of # = {x, q#(x) = 0, 1 ) $ ) N}(where q# : Rn ! R) at point x by (2.13). The normal n# and the cur-vature '# are defined by (2.15) and (2.16). The mean curvature vector isdefined by H = "

.N#=1 '#n# (see (2.18)). Then, we have: for 1 ) i ) n,

(A.3)

n'

j,k=1

Pj,k$jPi,k = (detG)#1/2n'

j=1

$j((det G)1/2Pi,j),

= "N'

#=1

'#(n#)i = H i.

Moreover, we have: for 1 ) i ) n,

(A.4)n'

j=1

$jPi,j = "n'

j=1

Pi,j$j ln((det G)1/2) + Hi.

For brevity, we only provide the proof of Lemma A.2 below.The following lemma giving another expression for the curvature '#

defined in (2.16) is also useful.


Lemma A.3 Let 1 ) $ ) N . The curvature '# defined by (2.16) can bewritten in the following form:

(A.5) '# = |$q#|(det G)#1/2div

7

8(detG)1/2N'

%=1

G#1#,%$q%

9

: .

Proof of Lemma A.2: Let us start with Pj,k$jPi,k. We have:

(A.6)

Pj,k$jPi,k = $j(Pj,kPi,k) " Pi,k$j(Pj,k),= $j(Pi,j) " Pi,k$j(Pj,k),= ()i,k " Pi,k)$j(Pj,k),

= "G#1#,$$iq#$kq$$j(G#1

%,&$jq%$kq&),

= "G#1#,$$iq#$kq$

($jG

#1%,&$jq%$kq& + G#1

%,&%q%$kq&

+ G#1%,&$jq%$j$kq&

),

= "$iq&$jG#1%,&$jq% "$iq&G

#1%,&%q%

" G#1#,$$iq#$kq$G

#1%,&$jq%$j$kq&,

where the summation convention is from 1 to n for Latin indices and from1 to N for Greek indices. Let us now compute $j(G#1

%,&). We have

0 = $j(G#,$G#1$,&) = $j(G#,$)G#1

$,& + G#,$$j(G#1$,&),

so that

(A.7)$j(G#1

%,&) = "G#1%,#$j(G#,$)G#1

$,& ,

= "G#1%,#G#1

$,& ($j$kq#$kq$ + $j$kq$$kq#)

Therefore, the first term in (A.6) is

"$iq&$jG#1%,&$jq%

= $iq&$jq%G#1%,#G#1

$,& ($j$kq#$kq$ + $j$kq$$kq#) ,

= $iq&$jq%G#1%,#G#1

$,&$j$kq#$kq$ + $iq&$jq%G#1%,#G#1

$,&$j$kq$$kq#

= $iq#$jq%G#1%,&G

#1$,#$j$kq&$kq$ + $iq&$jq$G

#1$,#G#1

%,&$j$kq%$kq#,

where, in the last line, we have swapped $ and ) in the first term and wehave swapped & and ( in the second term. Notice now that the first term


in the last line and the last term in (A.6) cancel, so that:

Pj,k$jPi,k = $iq& G#1%,&

("%q% + G#1

$,#$jq$$j$kq%$kq#

),

= $iq& G#1%,&

("%q% + $2q% : (G#1

#,$$q# -$q$))

,

= "'&$iq&|$q&|#1,

which proves one equality in (A.3).Let us now consider (det G)#1/2$j((det G)1/2Pi,j). Using (3.5), we

have:

(det G)#1/2$j((det G)1/2Pi,j)

= (det G)#1/2$j((det G)1/2)Pi,j + $jPi,j ,

=12$j ln(det G)Pi,j "$j(G#1

%,&$iq%$jq&),

=12G#1

#,$$jG#,$Pi,j "$jG#1%,&$iq%$jq&

" G#1%,&$j$iq%$jq& " G#1

%,&$iq%%q&,

=12G#1

#,$$iG#,$ "12G#1

#,$$jG#,$G#1%,&$iq%$jq&

+ G#1%,#G#1

$,& ($j$kq#$kq$ + $j$kq$$kq#)$iq%$jq&

" G#1%,&$j$iq%$jq& " G#1

%,&$iq%%q&,

= G#1#,$$i$jq#$jq$ " G#1

#,$$j$kq#$kq$G#1%,&$iq%$jq&

+ G#1%,#G#1

$,&$j$kq#$kq$$iq%$jq& + G#1%,#G#1

$,&$j$kq$$kq#$iq%$jq&

" G#1%,&$j$iq%$jq& " G#1

%,&$iq%%q&

= "G#1$,#$j$kq$$kq#G#1

%,&$iq%$jq&

+ G#1%,&G

#1$,#$j$kq&$kq$$iq%$jq# + G#1

%,&G#1$,#$j$kq$$kq&$iq%$jq#

" G#1%,&$iq%%q&,

where, in the last expression we have swapped $ and & in the first termand $ and ) in the third term. Now notice that the third term cancelswith the first term so that we obtain:

(det G)#1/2$j((det G)1/2Pi,j)

= $iq%G#1%,&

(G#1

$,#$j$kq&$kq$$jq# "%q&

),

= "'%$iq% |$q% |#1,


which completes the proof of (A.3).Let us finally consider (A.4). We have:

$jPi,j = (det G)#1/2$j((det G)1/2Pi,j) " (det G)#1/2Pi,j$j((det G)1/2)

= "N'

#=1

'#(n#)i " Pi,j$j(ln(det G)1/2),

which is exactly (A.4).

Proof of Lemma A.3: For a fixed 1 ) $ ) N , we have

(A.8)(det G)#1/2div

((det G)1/2G#1

#,%$q%

)

= G#1#,%$q% ·$ ln

((det G)1/2

)+ $G#1

#,% ·$q% + G#1#,%%q% .

Using (3.5), the first term in (A.8) is:

G#1#,%$q% ·$ ln(det G)1/2 =

12G#1

#,%$q% · G#1$,&$G&,$ ,

=12G#1

#,%$q% · G#1$,&

($2q&$q$ + $2q$$q&

),

=G#1#,%G#1

$,&$2q& : $q$ -$q%.

Using (A.7) the second term in (A.8) is

$G#1#,% ·$q% = "G#1

%,&G#1$,#

($2q&$q$ + $2q$$q&

)·$q% ,

= "G#1%,&G

#1$,#

($2q& : $q$ -$q% + $2q$ : $q& -$q%

).

Therefore, we have:

(det G)#1/2div((det G)1/2G#1

#,%$q%

)

= "G#1$,#$

2q$ :(G#1

%,&$q& -$q%

)+ G#1

#,%%q%,

which yields (A.5), using the definition (2.16) of '#.

B The mean curvature vector H

Here we show that the vector ".N

#=1 '#n# is the so-called mean curva-ture vector H defined as [2]2

(B.1) H = "N'

#=1

div !("#)"#,

2 Depending on the textbooks, the mean curvature vector is defined as ±.N

!=1!!n!,

or as N!1.N

!=1!!n!. The vector H defined by (B.1) is also sometimes called the

additive curvature vector.


where div ! is the tangential divergence and ("1, ...,"N )(x) denotes asmooth orthonormal vector field generating the space normal to # atpoint x. Geometrically, H points in the direction where the area of #decreases most, and intervenes in mean curvature flows or in the diver-gence theorem on manifolds (2.8) (see [2, 3]). This vector only dependson the geometry of the surface # as a submanifold of Rn. In other words,the dynamics (2.17) is intrinsic, like the measure µ! it samples.

To derive the expression H = ".N

#=1 '#n# from (B.1), notice firstthat this definition does not depend on the choice of the vector field("1, ...,"N ). Thus, the mean curvature vector is characterized by the factthat, for any vector " in the normal space to # at point x, H · " ="div !("). Let us then compute: for 1 ) $0 ) N ,

"'#n# · n#0

= " |$q#|G#1%,#

(%q% "$2q% : (G#1

&,$$q& -$q$))

'$q#|$q#|#1 ·$q#0|$q#0 |#1,

= " G#1%,#

(%q% "$2q% : (G#1

&,$$q& -$q$))

G#,#0 |$q#0 |#1,

= "(%q#0 "$2q#0 : (G#1

&,$$q& -$q$))|$q#0 |#1.

But we also have: for 1 ) ) ) N ,

div !(n&) = Id : (P$($q&/|$q&|)) ,

= )i,j()i,k " G#1

#,$$iq#$kq$

)$k

($jq&|$q&|#1)

),

=()i,k " G#1

#,$$iq#$kq$

) ($k$iq&|$q&|#1 "$iq&$kq&|$q&|#2

),

= %q&|$q&|#1 " G#1#,$$iq#$kq$$k$iq&|$q&|#1 " 1

+ G#1#,$$iq#$kq$$iq&$kq&|$q&|#2,

= %q&|$q&|#1 "$2q& : (G#1#,$$q# -$q$)|$q&|#1.

Therefore, for any 1 ) $0 ) N ,

N'

#=1

"'#n# · n#0 = "div !(n#0).

Since (n1, ..., nN )(x) generates the space normal to # at point x, thisproves that H = "

.N#=1 '#n#.


C The proof of lim!"0 X!t = X t

This appendix is devoted to the proof of

lim!"0

X!t = Xt,

where X!t and Xt are such that X!

0 = X0 and respectively satisfy (2.23)and (2.24). For simplicity, we restrict ourselves to the scalar constraintcase, N = 1. Moreover, we suppose in this section that

(C.1) 0 < m ) |$q| ) M and |%q| ) M.

Lemma C.1 Let X!t be the solution of the stochastic di!erential equa-

tion (2.23) with initial condition X!0 = X0. Let us suppose (C.1) and

that:

(C.2) q(X0) = 0,

and

(C.3) |$V | ) M.

Then we have:

(C.4) E

$

supt%T

|q(X!t )|

%

) C2

#2"#1* ln(1 + M2T/*) +

"#1M + M2

m2*,

(C.5) supt'0

E(|q(X!

t )|2)) 2"#1* + 2

$"#1M + M2

m2

%2

*2.

Proof: We have:

q(X!t ) = " 1

*

! t

0|$q|2(X!

s)q(X!s) ds "

! t

0$q(X!

s) ·$V (X!s) ds

+#

2"#1! t

0$q(X!

s) · dW s + "#1! t

0%q(X!

s) ds.

Let us first perform a change of time. The local martingale Mt =" t0 $q(X!

s)·dW s is such that 1M2t =

" t0 |$q|2(X!

s)ds & m2t (by (C.1)). Therefore,1M2$ = * almost surely and thus, by the Dubins-Schwartz Theorem,there exists a Brownian motion B such that Mt = B)M*t . Let us set


,(t) = inf{s, 1M2s > t} and Zt = q(X!((t)). By a change of variable, we

obtain:

Zt = "1*

! t

0Zs ds +

#2"#1Bt +

! t

0

("#1%q "$V ·$q)|$q|2 (X!

((s)) ds.

Therefore, we have:

Zt =#

2"#1

! t

0e#(t#s)/!dBs +

! t

0e#(t#s)/! ("#1%q "$V ·$q)

|$q|2(X!

((s)) ds,

)#

2"#1

! t

0e#(t#s)/!dBs +

"#1M + M2

m2*.

Therefore,

E

$

supt%T

|Zt|%

)#

2"#1E

$

supt%T

&&&&! t

0e#(t#s)/!dBs

&&&&

%

+"#1M + M2

m2*.

The process gt =! t

0e#(t#s)/!dBs is an Ornstein-Uhlenbeck process so

that, by [17], there exist C1, C2 > 0 such that

C10* ln(1 + T/*) ) E sup

t%T|gt| ) C2

0* ln(1 + T/*).

Thus, we have:

E

$

supt%T

|Zt|%

) C2

#2"#1* ln(1 + T/*) +

"#1M + M2

m2*.

This means that

E

$

supt%((T )

|q(X!t )|

%

) C2

#2"#1* ln(1 + T/*) +

"#1M + M2

m2*.

Using the fact that ,(t) & tM2 , we then obtain (C.4).

If we consider another norm, we have:

E|Zt|2 ) 4"#1E&&&&! t

0e#(t#s)/!dBs

&&&&2

+ 2$"#1M + M2

m2

%2

*2.

) 2"#1* + 2$"#1M + M2

m2

%2

*2.

This yields (C.5).


We now introduce the following change of coordinates:

(C.6) & :

123

24

Rn ! Rn

x 3!$

p(x)q(x)

%

where p = (p1, . . . , pn#1) : Rn ! Rn#1 is such that $pi · $q = 0 forall 1 ) i ) n " 1 and for all x , #. In other words, &n = q, while&i = pi, for 1 ) i ) n " 1. We suppose that & is invertible, and thus,Range($p1, ...,$pn#1) = n " 1. It is always possible to build such afunction &, at least locally, by considering a parametrization of #.

For the statement of the next proposition, we suppose that:

(C.7) 0 < m ) |$&| ) M , and $& and %& are Lipschitz functions.

Notice that the assumptions on & are actually some assumptions relatedto the regularity of the surface #, and therefore, some assumptions relatedto the regularity of q. If # is su"ciently smooth, it is possible to define theparametrization & at least in a neighborhood of X0, so that the argumentsgiven below still hold, by a localization procedure on the processes Xt

and X!t .

Proposition C.2 Let X!t be the solution of the stochastic di!erential

equation (2.23) with initial condition X!0 = X0. Let Xt be the solution

of the following stochastic di!erential equation:

(C.8)Xt = X0 "

! t

0P (Xs)$V (Xs) ds +

#2"#1

! t

0P (Xs) dW s

+ "#1! t

0$ · P (Xs) ds.

We suppose that (C.2), (C.1) , (C.3) and (C.7) hold. In addition, wesuppose that:

(C.9) $V is a Lipschitz function.

Then, for small *,

(C.10) supt%T

E |X!t " Xt|2 ) C*,

where C is a constant depending on the data, and on T .


Notice that Xt solution to (C.8) satisfies (2.24) since $·P = "P$ ln |$q|"'n (see (A.2)). All this can be generalized to the case N > 1.

Proof: By rewriting the stochastic di!erential equations in the co-ordinates (p, q), we have, for X!

t ,12223

2224

for all 1 ) i ) n " 1,d&i(X!

t ) = $j&i(X!t )("$jV (X!

t ) dt +0

2"#1dW j(t)) + "#1%&i(X!t ) dt,

dq(X!t ) = " 1

! |$q|2(X!t )q(X

!t ) dt + $q(X!

t ) · ("$V (X!t ) dt +

02"#1dW t)

+ "#1%q(X!t ) dt,

and for Xt,(C.11)123

24

for all 1 ) i ) n " 1,d&i(X t) = $j&i(X t)("$jV (X t) dt +

02"#1dW j(t)) + "#1%&i(X t) dt,

dq(X t) = 0.

Let us prove (C.11). By Ito Formula, we have,

d&i(X t) =$&i · (P$V + "#1$ · P )(Xt) dt

+#

2"#1$&i · (P (X t)dW t) + "#1P : $2&i(Xt) dt

Equation (C.11) can be straightforwardly obtained from this equationusing the fact that: on #,

P$&i =/$&i if 1 ) i ) n " 1,0 if i = n,

and $·(P$&i) =/%&i if 1 ) i ) n " 1,0 if i = n.

Now, by (C.5), we know that, for small *, supt'0 E(|q(X!

t ) " q(X t)|2))

C*. For the other components of &(Xt), we have, by using (C.3), (C.9), (C.7):+0 ) t ) T ,

E

$n#1'

i=1

|&i(X!t ) " &i(Xt)|2

%

) C(T )! t

0E|X!

s " Xs|2 ds.

Therefore, by using (C.7) and (C.5), we obtain: +0 ) t ) T ,

E |&(X!t ) " &(Xt)|2 ) C(T )

! t

0E|X!

s " Xs|2 ds + C*,

and thus

E |X!t " Xt|2 ) C(T )

! t

0E|X!

s " Xs|2 ds + C*,

which yields (C.10).


D The situation with molecular constraints

In many applications, in addition to the constraints associated with thereaction coordinates q(x) whose free energy is of interest, some molec-ular constraints c(x) = (c1(x), . . . , cM (x)) = 0 (M < n) are needed.These correspond to physical constraints on the system, such as fixedbond lengths for example. For completeness, we discuss this case here.We suppose in the following that rank($c1, ...,$cM ) = M and

rank($q1, ...,$qN ,$c1, ...,$cM ) = N + M.

In this section, Latin indices go from 1 to M or from 1 to N + M .When molecular constraints are present, the original Boltzmann-Gibbs

distribution (replacing (1.2)) is

(D.1) dµ#(x) = Z#1# e#"V (x)d##(x)

where ' = {x : c(x) = 0} is the codimension M manifold on which thesystem is constrained due to the presence of the molecular constraints,## is the Lebesgue measure on this manifold, and

(D.2) Z# =!

#e#"V (x)d##(x).

D.1 Definition of the free energy

As in in the case without molecular constraints (see Section 4.1), the freeenergy F# associated with q(x) is such that e#"F!(z) is the probabilitydensity function of the variable Z = q(X) when X is distributed ac-cording to µ#. Thus, the free energy F# is defined by: for all function! : RN ! R,

(D.3)!

#!(q(x))Z#1

# e#"V (x)d##(x) =!

RN!(z)e#"F!(z) dz.

To obtain an explicit expression of F#, we need the following general-ization of the co-area formula (see Theorem 2.93 p. 101 in [1]): for anyfunction g : Rn ! R (possibly defined only on ') and a smooth functionq : Rn ! RN ,

(D.4)!

#g(x)|det G#(x)|1/2d##(x) =

!

RN

!

#+!(z)g(x)d##+!(z)(x)dz,

where the matrix G# is defined by: for 1 ) $, & ) N ,

(D.5) (G#)#,$ = $#q# ·$#q$ .


In (D.5), $# denotes the surface gradient. More explicitly, let us introducethe orthogonal projector Q(x) on the tangent space to ' at point x:

(D.6) Q(x) = Id "M'

i,j=1

K#1i,j (x)$ci(x) -$cj(x),

where, +1 ) i, j ) M ,

(D.7) Ki,j = $ci ·$cj.

Since q# is also defined in the vicinity of ', we can thus express the surfacegradient of q# as:

(D.8) $#q#(x) = Q(x)$q#(x).

We also recall that the surface divergence on ' is the trace of the surfacegradient on ' (see (2.7) with # replaced by ' and P by Q),

Now, using (D.4), by similar computations as those made in Section 4.1to obtain (4.6), we have the following expression for the free energy:(D.9)

F#(z) = ""#1 ln(Z#1

#

!

#+!(z)e#"V (x)|det G#(x)|#1/2d##+!(z)(x)

),

or equivalently

(D.10) F#(z) = ""#1 ln(Z#1

# Z#+!(z),|det G!|!1/2

).

This means that, similarly to the case without constraints, we need to re-place the potential V by V|det G!|!1/2 in order to sample the right measure.Thus, the numerical schemes to consider are (written here for z = 0):

(D.11)

1223

224

Xn+1 = Xn "$(V| det G!|!1/2)(Xn)%t +0

2"#1%W n

+.N

#=1 +#,n$q#(Xn+1) +.M

i=1 µi,n$ci(Xn+1),where +#,n and µi,n such that q(Xn+1) = c(Xn+1) = 0,

and

(D.12)

1223

224

Xn+1 = Xn "$(V| det G!|!1/2)(Xn)%t +0

2"#1%W n

+.N

#=1 +#,n$q#(Xn) +.M

i=1 µi,n$ci(Xn+1),where +#,n and µi,n such that q(Xn+1) = c(Xn+1) = 0.

Notice that $V|det G!|!1/2) involves the Hessian $2#q#, but the computa-

tion of this quantity can be avoided in practice by using an approximationsimilar to (3.6).


D.2 Expression for the mean force

We now turn to the question of the computation of the mean force. Asin the case without molecular constraints, we obtain the following gener-alization of Lemma 4.1:

Lemma D.1 The gradient of F# (namely the mean force) can be ex-pressed as: for any 1 ) $ ) N ,

(D.13)$#F#(z) =

N'

%=1

!

#+!(z)

($#V · (G#1

# )#,%$#q%

" "#1div #((G#1# )#,%$#q%)

)dµ#+!(z),|det G!|!1/2.

As in the case without molecular constraints, this formula can be useddirectly to evaluate $#F#(z). Let Xn be the solution to (D.11) or (D.12).Then, the first term at the right hand side of (D.13) can be obtained from

(D.14)

limT"$

lim"t"0

1M

M'

m=1

N'

%=1

($#V · (G#1

# )#,%$#q%

)(Xn)

=N'

%=1

!

#+!(z)$#V · (G#1

# )#,%$#q% dµ#+!(z),|det G!|!1/2 ,

where M = T/%t. As for the second term, we have(D.15)

" "#1 limT"$

lim"t"0

1M%t

M'

m=1

N'

%=1

(((G#)#1

#,%$#q%)(Xn +%W n)

" ((G#)#1#,%$#q%)(Xn)

)· Q(Xn)%W n

= ""#1N'

%=1

!

#+!(z)div #((G#1

# )#,%$#q%)dµ#+!(z),|det G!|!1/2 .

Proof of Lemma D.1: Let & by a C$c (R) function and ! = &(.

Then for a fixed 1 ) $ ) N ,

(D.16)

!

RN!(z#) exp(""F#(z)) dz

=!

RN&((z#) exp(""F#(z)) dz,

= "!

RN&(z#)$#F#(z) exp(""F#(z)) dz

= "Z#1#

!

RN&(z#)Z#+!(z),|det G!|!1/2$#F#(z) dz.


On the other hand, using successively (D.4), (2.8), and the fact that themean curvature vector of ' is orthogonal to $#q% , for any 1 ) ( ) N wehave(D.17)!

RN!(z#) exp(""F#(z)) dz

= Z#1!

RN

!

#+!(z)!(q#(x))e#"V (x)|det G#(x)|#1/2d##+!(z)(x)dz

= Z#1!

#&((q#(x))e#"V (x)d##(x),

= Z#1!

#(G#1

# )#,%$#q% ·$# (& . q#) (x)e#"V (x)d##(x),

= "Z#1!

#div #

((G#1

# )#,%$#q%e#"V)

(x)& . q#(x)d##(x),

= Z#1!

#

("$#V · (G#1

# )#,%$#q% " div #

((G#1

# )#,%$#q%

))(x)

' e#"V (x)&(q#(x))d##(x),

= Z#1!

RN&(z#)Z#+!(z),| det G!|!1/2

'!

#+!(z)

("$#V · (G#1

# )#,%$#q% " div #

((G#1

# )#,%$#q%

))

' dµ#+!(z),|det G!|!1/2 dz.

The fact that (D.16) is equal to (D.17) for all functions & completes theproof of (D.13).

Notice that, as a generalization of (4.9), $#F#(z) can also be expressedas

(D.18)

$#F#(z)

=!

#+!(z)

N'

%=1

(G#1# )#,%$#q% ·

($#V|det G!|!1/2 + "#1H#

)

' dµ#+!(z),|det G!|!1/2,

where H# is defined by:

(D.19) H# = "N'

#=1

('#)#(n#)#,


with(D.20)

('#)# = |$#q#||det G#|#1/2div #

7

8|detG#|1/2N'

%=1

(G#1# )#,%$#q%

9

: ,

and

(D.21) (n#)#(x) =$#q#(x)|$#q#(x)| .

D.3 The orthogonal case

By Proposition 4.3, we know that for 1 ) $ ) N ,(D.22)

limT"$

lim"t"0

1T

T/"t'

m=1

+#,m

=!

#+!

N+M'

i=1

L#1#,i$ri ·

($V|det G!|!1/2 + "#1H

)dµ#+!,|det G!|!1/2,

where we have used the (N + M) dimensional constraints vector

r = (q1, . . . , qN , c1, . . . , cM )

and the (N + M) ' (N + M) matrix L:

(D.23) Li,j(x) = $ri(x) ·$rj(x).

Notice that in (D.22), H is the mean curvature vector of the surface '4#:

H = "N+M'

i=1

'ini

with ni = ,ri|,ri| and (see (A.5))

'i = |$ri||det L|#1/2div(|det L|1/2

N+M'

j=1

L#1i,j $rj

).

In the case the molecular constraints and the constraints related to thereaction coordinates are orthogonal in the sense that: +1 ) i ) M and+1 ) $ ) N , $ci(x) ·$q#(x) = 0 for x , ' 4 #, (D.22) indeed gives the


Giovanni CiccottiDepartment of PhysicsUniversity of Roma “La Sapienza”P.le A. Moro 200185 Roma (Italy) Department of Mathematics and PACME-mail: [email protected]

Tony LelievreEcole Nationale des Ponts et Chaussees (ParisTech)6 et 8 avenue Blaise PascalCite Descartes - Champs sur Marne77455 Marne la Vallee Cedex 2 (FRANCE)E-mail: [email protected]

Eric Vanden-EijndenCourant Institute251 Mercer StreetNew York, NY 10012E-mail: [email protected]

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