Coarse-graining stiff bonds

Eur. Phys. J. Special Topics 200, 107–129 (2011)c© EDP Sciences, Springer-Verlag 2011DOI: 10.1140/epjst/e2011-01521-1

THE EUROPEANPHYSICAL JOURNALSPECIAL TOPICS

Regular Article

Coarse-graining stiff bonds

P. Espanol1,a, J.A. de la Torre1, M. Ferrario2, and G. Ciccotti3,4

1 Departamento de Fısica Fundamental, UNED, 28040 Madrid, Spain2 Dipartimento di Fisica, Universita di Modena e Reggio Emilia, via Campi 213/A,41100 Modena, Italy

3 Dipartimento di Fisica and CNISM, Universita La Sapienza, P.le A. Moro 2, 00185 Rome,Italy

4 School of Physics, Room 302B EMSC-UCD, University College Dublin, Belfield,Dublin 4, Ireland

Received 09 September 2011 / Received in final form 10 October 2011

Published online 13 December 2011

Abstract. The method of constraints in molecular dynamics is usefulbecause it avoids the resolution of high frequency motions with verysmall time steps. However, the price to pay is that both the dynamicsand the statistics of a constrained system differ from those of the un-constrained one. Instead of using constraints, we propose to disposeof high frequency motions by a coarse-graining procedure in whichfast variables are eliminated. These fast variables are thus modeled asfriction and thermal fluctuations. We illustrate the methodology witha simple model case, a diatomic molecule in a monoatomic solvent,in which the bond between the atoms of a diatomic molecule is stiff.Although the example is very simple and does not display the inter-esting effects of “wrong” statistics of the constrained system (i.e. thewell-known issue connected to the Fixman potential), it is well suitedto give the proof of concept of the whole procedure.

1 Introduction

When a system has bond potentials that are stiff, the motion has high frequencycomponents and a numerical solution of the system is hampered by the small timestep required to resolve this high frequency motion. Multiple time step integrationmethods [1] alleviate the problem, but, more usually, in order to treat these bondswithout the risk of introducing instabilities and spurious resonances [2], one resorts tothe introduction of holonomic constraints [3], as the freezing of the bond length allowssafely the use of a substantially larger time step for the numerical integration of theequations of motion. Constrained dynamics, however, has an impact on the probabilitydensity function of the system [4]. This may result in a measurable modification bothof some equilibrium properties, such as average bending and torsion angles, and ofsome dynamical properties.

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108 The European Physical Journal Special Topics

While a good solution to recover static equilibrium properties is provided by theintroduction of the ‘Fixman potential’ [5–8], which permits to sample the configura-tion space in accord with the classical statistical mechanics of the spring model, it isnot at all predictable to what extent and in which directions a dynamics driven byconstraints plus these extra ‘Fixman potential’ terms alters the timescale of fluctua-tions in the system and its dynamical properties (i.e., time correlation functions).In the present paper, we present an alternative approach to deal with the fast

degrees of freedom induced by stiff bonds. The idea is to average out these fast de-grees of freedom by following a standard and dynamically respectful coarse-graining(CG) procedure based on the Mori-Zwanzig formalism [9,10]. The effect of the elim-inated fast motion appears in the remaining comparatively slow degrees of freedomin terms of dissipative and stochastic forces. The method provides an explicit expres-sion for the friction matrix while the stochastic forces have amplitudes dictated bythe Fluctuation-Dissipation theorem.At variance with constrained dynamics, the statistical properties (both static and

dynamics) of the soft degrees of freedom will be, by construction, the properties ofthe original system with springs, while the problem of the need of a very short timestep to resolve the stiff, high frequency, motion is avoided. The price to pay is that, inprinciple, one has now to solve numerically a stochastic rather than a deterministicset of equations of motion. Our aim in this note is to provide the proof of conceptof this CG approach by considering perhaps the simplest example of a single stiffbond, that of a diatomic molecules immersed in a solvent constituted by monoatomicparticles (e.g. a nitrogen molecule in liquid argon). While this system, when treatedwith constraints, it is too simple to require the introduction of the Fixman potential,we show how one can treat flexibility effects in an explicit manner.We should mention that the idea of dealing with constraints through elimination

of fast variables was already advanced by Helfand in 1979 [6]. In this original paper,a discussion of the Mori-Zwanzig projection method is made on a simple generic twodimensional problem. In this way, dissipation and noise emerge on the dynamics ofthe retained slow degree of freedom. However, no explicit calculation of the frictioncoefficient is attempted in Ref. [6] and the discussion is limited to dimensional argu-ments.The remainder of this paper is organized as follows. In Sec. 2 we review the

Mori-Zwanzig based CG procedure in the Markovian approximation which leads toa set of stochastic equations of motion, i.e. the coarse-grained dynamics. In Sec. 3we present the description of the microscopic model, a solvated dimer, introducingthe generalized coordinate description. In Sec. 4 the CG dynamics for the model isderived in general terms, with the projection procedure made explicit using numericalquadrature. There we also discuss the difficulties which arise in the implementation ofthe random forces in the most general case. This has lead us to detail, in Sec. 5, how,by taking advantage of the stiffness of the bond, it is possible to further simplify thefriction matrix in such a way that the random force terms satisfying the fluctuation-dissipation relations are straightforward to obtain. In Sec. 6 we just summarize theoverall result while, finally, in Sec. 7 we give a few concluding remarks together withsome outlook considerations.

2 The coarse-grained dynamics

In this section we review Mori-Zwanzig route [10] which provides a very elegant andgeneral formulation of the theory of coarse-graining [11].Classically, at a microscopic level, a molecular system is described by the coor-

dinates and momenta of all the atoms in the system. We denote with z this full set

Constraints: From Physical Principles to Molecular Simulations and Beyond 109

of particle coordinates and momenta which define the microstate of the system. Theevolution of the microstate is governed by Hamilton’s equations of motion

dz

dt= J∂H

∂z(z) (1)

where J is the symplectic matrix and H(z) the Hamiltonian function.The first step in the CG procedure is to select the set of CG variables which will

be generally defined by a set of functions X(z) in phase space. In the present work,these CG variables will include all microscopic variables except those correspondingto the stiff bonds. Zwanzig projection operator technique leads to a Fokker-Planckequation (FPE) for the time evolution of the probability density P (x, t) that thephase functions X(z) take numerical values x [10,11]. This FPE has the form

∂

∂tP (x, t) = − ∂

∂xμ

[Vμ(x) +Mμν(x)

∂F

∂xν

]P (x, t)+kBT

∂

∂xμMμν(x)

∂

∂xνP (x, t). (2)

Hereafter, Einstein’s repeated index summation convention is assumed. The reversibledrift Vμ(x) is defined as the projected time derivatives of the CG variables (L is theLiouville operator)

Vμ(x) = 〈LXμ〉x (3)

where the conditional average 〈· · · 〉x, which acts as the Mori-Zwanzig projection op-erator P by taking the conditional equilibrium expectation of an arbitrary phasefunction on the manifold X(z) = x, is defined in terms of the equilibrium probabilitydensity ρeq(z) as

〈· · · 〉x = 1

Ω(x)

∫dzρeq(z)δ(X(z)− x) · · · (4)

The normalization factor Ω(x) appearing here is the equilibrium probability distrib-ution of the CG variables

Ω(x) =

∫dzρeq(z)δ(X(z)− x) (5)

which in turn defines the free energy function F (x) appearing in Eq. (2) via the usualstatistical mechanics relation

F (x) = −kBT lnΩ(x). (6)

The divergence of the reversible drift vanishes,∂Vμ∂xμ(x) = 0, and, therefore, it can be

shown that indeed the CG equilibrium distribution Ω(x) is the stationary solution ofthe Fokker-Planck equation (2).Finally it remains to give the explicit expression of the friction matrix Mμν(x),

which is defined in the Markovian approximation as

Mμν(x) =1

kBT

∫ ∞

0

〈QLXν exp{QLQt}QLXμ〉xdt. (7)

The projector operator Q = 1 − P in Eq. (7) is defined through its effect on anarbitrary phase function G(z),

QG(z) = G(z)− 〈G〉x=X(z) (8)

that is, its action is to subtract from the function its conditional average evaluatedon the manifold x = X(z).


One can write down a mathematically equivalent formulation of Eq. (2) in termsof a set of stochastic differential equations (SDE). In this case, the general (Ito) SDEgoverning the dynamics of the set of CG variables x is

dxμ = 〈LXμ〉xdt−Mμν(x) ∂F∂xν(x)dt+ kBT

∂Mμν

∂xν(x)dt+ dxμ (9)

where the random terms dx are linear combinations of Wiener processes of the form

dxμ = (2kBT )1/2

∑m

Bμm(x)dWm. (10)

The amplitudes of the noise terms, Bμm are related to the friction matrix through

∑m

Bμm(x)Bνm(x) =Mμν(x) (11)

which is just the fluctuation-dissipation theorem which sometimes is written in theheuristic form

dxdx

2kBTdt=M(x) (12)

where the mnemotechnical Ito rule dWmdWn = δmndt is used.

2.1 Relevant variables are microscopic degrees of freedom

When applied to the problem of stiff bond potentials, the selected CG variables arejust a subset of the microscopic degrees of freedom and, therefore, it is convenientto write z = (x, y) where x are the slow atomic variables and y are the fast atomicvariables. We will assume that both sets, x and y, contain the coordinates and theirrespective momenta. As opposed to virtually all applications of coarse-graining whereone keeps a small number of relevant variables, here the number of slow variables isalmost identical to the number of degrees of freedom. Only a few microscopic degreesof freedom y are supposed to be fast and are, then, eliminated.When the relevant variables are microscopic degrees of freedom themselves, the

above SDE takes a particularly simple form. By assuming that the equilibrium en-semble is given by the canonical ensemble, we have that the equilibrium distributionin Eq. (5) for the relevant variables is given by

Ω(x) =

∫dy1

Zexp {−βH(x, y)} (13)

because in this case it is trivial to perform the integrals over the Dirac delta functions.Here, Z is the classical canonical partition function

Z =

∫dx dy exp {−βH(x, y)} . (14)

When the relevant variables are microscopic degrees of freedom, the free energy F (x)is usually referred to as the coarse-grained (CG) Hamiltonian HCG(x), because it hasthe form

F (x) = −kBT ln∫dy exp {−βH(x, y)}+ kBT lnZ

≡ HCG(x). (15)


In this way we can write the equilibrium distribution of the relevant variables in theform

Ω(x) =exp{−βHCG(x)}∫dx exp{−βHCG(x)} . (16)

The reversible part of the dynamics 〈LX〉x also simplifies

〈LX〉x = 1

Ω(x)

∫dy1

Zexp {−βH(x, y)} J ∂H

∂x(x, y) = J

∂HCG

∂x(x) (17)

with the CG Hamiltonian generating the reversible part of the CG dynamics. TheHamiltonian structure is respected as far as the reversible part of the CG dynamicsis concerned.

2.2 Exponential approximation of the memory kernel

The evolution operator exp{QLQt} in the definition of the friction matrix M(x) inEq. (7) is not directly computable in a molecular dynamics simulation due to thepresence of the projector Q. For this reason it is usually assumed that this evolu-tion operator may be approximated, during the decay time of the correlation by theordinary evolution exp{Lt}. In the present work we take a different route and ap-proximate the friction matrix by assuming that the correlation function inside thetime integral decays exponentially, this is

〈QLX exp{QLQt}QLX〉x ≈ 〈(QLX)(QLX)〉x exp{−t/τ} (18)

where τ is a correlation time. The approximation is, of course, exact at t = 0. Inprinciple, we could have a different correlation time for every matrix element, butwe will simplify matters and assume that all the correlations decay with the samecorrelation time. By performing the time integral in Eq. (7) the friction matrix, inthe exponential approximation, becomes simply

M(x) ≈ τ

kBT〈(QLX)(QLX)〉x. (19)

The symmetric and positive definite character of the friction matrix is fully respectedin this exponential approximation. We expect that this approximation captures allthe state dependence of the friction matrix, while still being sufficiently simple toadmit treatment. Indeed, the main reason for writing the exponential approximationis that we need to include the state dependence of the friction matrix. In the usualtreatment of the CG theory, one selects a few collective variables X(z) and eliminatesmassively all the others degrees of freedom. The treatment of stiff bonds throughcoarse-graining requires just the opposite, to keep most of the degrees of freedom andeliminate just a few of them, those corresponding to stiff bonds. It is expected that theeffect of the dependence on the configuration x of the friction matrix should play animportant role in this case. The exponential approximation captures the configurationdependence and yet allows for an explicit calculation of the friction matrix.As we will see, the main difficulty in setting up the numerical solution of the

derived stochastic dynamics in Eq. (9) is in the implicit definition through thefluctuation-dissipation relations Eq. (11) of the amplitudes of the thermal noise Bμmin terms of the friction matrix Mμν . The high dimensionality of the space of the slowCG variables makes this a formidable problem, unless some simplifying assumption istaken. In the next two sections we will introduce the solvated dimer model and derivein a general way the associated coarse-grained stochastic dynamics. Subsequently, we


will show that the friction matrix simplifies enormously in the so-called stiff limit(when the time scale separation of the stiff bond vibration dynamics with respect tothe slow CG variables is large). This makes trivial the calculation of the stochasticterms satisfying the fluctuation-dissipation.

3 Diatomic molecule at the microscopic level

In this section we introduce the particular system we model in this work which consistsof a diatomic molecule with a stiff bond interacting with a monoatomic (Lennard-Jones type) solvent. The microstate z of the system is given by the positions andmomenta of each atom of the molecule {R1,P1,R2,P2} plus the positions and mo-menta of each atom of the solvent r, p = {ri,pi} with i = 1, · · · , N . The Lagrangefunction L = T − V of the system in these coordinates is

L =1

2m1R

21 +1

2m2R

22 +

∑i

mi

2r2i − V (R1,R2, r) (20)

where the potential energy is given by

V =κ

2(ρ− ρ0)2 + φ(R1,R2, r) + φs(r). (21)

Here, ρ0 is the (in vacuo) equilibrium value of the length ρ = |R1−R2| of the springjoining the two atoms of the dimer, φ is the interaction potential between the solventand the atoms of the dimer, and φs(r) is the potential energy of the solvent.If κ→∞, the distance ρ will be a fast variable. We will assume that this variable

and its associated momentum are the only fast variables in the system and will becalled the stiff coordinates. In that case, we select as CG variables all the atomicvariables (solvent and molecular) except the stiff coordinates. We will then write themicroscopic state as z = (x, y) where, moving to generalised coordinates as shownbelow, x = R,P, θ, πθ, ϕ, πϕ, r, p are the slow variables and y = ρ, πρ are the fastvariables.In fact, in order to integrate out the stiff degree of freedom, we need to transform

coordinates to a most suitable set and it seems natural to use as coordinates of themolecule the center of mass R, and the relative vector ρ of the molecule. Thesecoordinates are given by

R =m1R1 +m2R2

M(22)

ρ = R2 −R1.In order to write the Lagrangian in the new coordinates we need to express R1,R2in terms of R,ρ which are

R1 = R− m2Mρ

(23)

R2 = R+m1

Mρ.

The kinetic energy in the new variables is

T =∑i

mi

2r2i +

M

2R2 +

μ

2ρ2 (24)


where M = m1 + m2 and μ = m1m2/M is the reduced mass. We can now makethe final change of variables from ρ to the usual spherical coordinates ρ, θ, ϕ. Ifρ = (ρ1, ρ2, ρ3), then

ρ1 = ρ sin θ cosϕ

ρ2 = ρ sin θ sinϕ (25)

ρ3 = ρ cos θ

and the kinetic energy becomes

T =∑i

mi

2r2i +

M

2R2 +

μ

2

(ρ2 + ρ2θ2 + ρ2ϕ2 sin2 θ

)(26)

while the Lagrangian is L = T −V where in the potential term (R1,R2) are replacedby their expressions, Eqs. (24), in terms of (R, ρ, θ, ϕ). The conjugate momenta are

P ≡ ∂L∂R=MR

πρ ≡ ∂L∂ρ= μρ

πθ ≡ ∂L∂θ= μρ2θ

(27)

πϕ ≡ ∂L∂ϕ= μρ2 sin2 θϕ

pi ≡ ∂L∂ri= miri.

The Hamiltonian H = T + V can be explicitly written as

H(x, y) = Hsol(r, p)+P2

2M+π2ρ

2μ+κ

2(ρ− ρ0)2+K(θ, πθ, πϕ, ρ)+φ(R,ρ, r) (28)

where we have introduced the shorthand expressions for the solvent Hamiltonian

Hsol(r, p) =∑i

p2i2mi

+ φs(r), (29)

the rotational kinetic energy

K(θ, πθ, πϕ, ρ) =π2θ2μρ2

+π2ϕ

2μρ2 sin2 θ(30)

and the interaction potential between the solvent particles and the diatomic molecule

φ(R,ρ, r) ≡∑i

φLJ(ri −R+ m2

Mρ)+

∑i

φLJ(ri −R− m1

Mρ)

(31)


Hamilton’s equations of motion can now be derived from the Hamiltonian in Eq. (28)

R =P

MP = − ∂φ

∂R

θ =πθ

μρ2πθ = −∂φ

∂θ− π2ϕ cos θ

μρ2 sin3 θ

ϕ =πϕ

μρ2 sin2 θπϕ = −∂φ

∂ϕ

ρ =πρ

μπρ = −κ(ρ− ρ0)− ∂φ

∂ρ

ri =pimi

pi = Fsoli −

∂φ

∂ri.

(32)

As we have considered a point transformation (i.e. the simple change from the carte-sian coordinates R1,R2 to the generalised coordinates R, ρ, θ, ϕ), the transformationis canonical and the Jacobian of the transformation is unity.

3.1 Constrained dynamics

If we were to solve the dynamics with the method of constraints, we would proceedby fixing in the Lagrangian L expressed in generalised coordinates, the values of thecoordinate ρ = ρ0 and the velocity to ρ = 0. The constrained Lagrangian reads

Lconst =∑i

mi

2r2i +

M

2R2 +

μ

2

(ρ20θ

2 + ρ20ϕ sin2 θ

)− V (33)

and the associated Hamiltonian generating the constrained dynamics becomes (in thissimple case πρ is also zero),

Hconst(x; ρ0) = Hsol(r, p) +

P2

2M+π2θ2μρ20

+π2ϕ

2μρ20 sin2 θ+ φ(R, ρ0, θ, ϕ, r) (34)

from which the resulting constrained dynamic equations can be derived

R =P

MP = −∂φ0

∂R

θ =πθ

μρ20πθ = −∂φ0

∂θ− π2ϕ cos θ

μρ20 sin3 θ

ϕ =πϕ

μρ20 sin2 θ

πϕ = −∂φ0∂ϕ

ri =pimi

pi = Fsoli −

∂φ0

∂ri.

(35)

Compared with Eqs. (32), please note that we have now introduced the shorthandnotation for the potential terms evaluated at ρ0

φ0 = φ0(R, θ, ϕ, r) = φ(R, ρ0, θ, ϕ, r). (36)

We have so far set up the classical mechanics description of our model and of theconstrained model. In the next section we will make use of all the notations hereintroduced when deriving the description of the dynamics at the coarse-grained level.


4 Diatomic molecule at the CG level

At the CG level, we take as relevant variables the slow variables in the system whichare all the degrees of freedom except ρ, πρ, i.e. those corresponding to the fast bondmotion. The whole problem is now to write down explicitly the stochastic dynamicsequations, from the (SDE) Eq. (9), for the selected relevant variables. To achieve this,in the next subsections we will compute explicitly the equilibrium distribution Ω(x),and all terms which appears in the SDE, i.e. the associated free energy function F (x),the reversible drift of 〈LX〉x and the friction matrix M(x), discussing the thermalnoise terms, whose definition will be left implicit until Section 5.

4.1 The equilibrium distribution

We first compute the equilibrium distribution Ω(x) of the relevant variables defined inEq. (13), where Z in Eq. (14) is the partition function that normalizes the canonicaldistribution function. This amounts to explicitly integrate over the conjugate variablesρ, πρ. The integral over πρ can be explicitly performed because the Hamiltonian isseparable and it involves a Gaussian integral. Therefore, we can write the equilibriumdistribution function and the related CG Hamiltonian, i.e. the free energy Eq. (15)in the form

Ω(x) =

√2πμkBT

Zexp

{−βHsol(r, p)− β P

2

2M

}

×∫ ∞

0

dρ exp{−β κ2(ρ− ρ0)2

}exp {−β [K(θ, πθ, πϕ, ρ) + φ(R, ρ,n, r)]} (37)

HCG(x) = Hsol(r, p) +P2

2M+ kBT ln

Z√2πμkBT

− kBT ln∫ ∞

0

dρ exp{−β κ2(ρ− ρ0)2

}exp {−β(K(x, ρ) + φ(x, ρ))} . (38)

The integral in the CG Hamiltonian can be computed numerically with a Gauss-Hermite quadrature formula. Gauss-Hermite quadrature approximates the followingintegral ∫ +∞

−∞e−s

2

f(s)ds ≈n∑i=1

wif(si) (39)

where n is the number of sample points to use for the approximation. The si are theroots of the Hermite polynomial Hn(s), (i = 1, 2, ..., n) and the associated weights wiare given by [12]

wi =2n−1n!

√π

n2[Hn−1(si)]2. (40)

In order to apply this quadrature, we make the change of variable ρ → s in theintegral

ρ = ρ0 + ερ0s(41)

s =1

ε

(ρ

ρ0− 1

)


where the dimensionless parameter ε is defined as

ε ≡√2kBT

κρ20. (42)

This parameter measures the ratio between the amplitude of the fluctuations of thespring length over the equilibrium spring length. It also governs the separation of timescales between the fast degree of freedom ρ and all the other microscopic degrees offreedom, because the larger the spring constant, the faster the motion for the relativedistance of the dimer. With the change of variables in Eq. (42), Eq. (38) becomes

HCG(x) =P2

2M− kBT ln

∫ ∞

−ε−1ds

(e−s

2

exp {−β(K(x, ρ0 + ερ0s) + φ(x, ρ0 + ερ0s)})

+Hsol(r, p) + kBT lnZ√

2πμkBT. (43)

As we will be typically dealing with values of the ε parameter which are small (stiffsprings), we can safely assume that the error committed by extending the lowerlimit of integration to −∞ is exponentially minor. Therefore, the CG Hamiltonianexpressed in terms of the Gauss-Hermite quadrature becomes

Heff(x) =P2

2M− kBT ln

n∑i=1

wi exp {−β(K(x, ρi) + φ(x, ρi)}

+Hsol(r, p) + kBT lnZ√

2πμkBT(44)

where the discretization points of the integral in the ρ variable are

ρi ≡ ρ0 + ερ0si (45)

Eq. (44) is an explicit analytical form for the CG Hamiltonian which, of course, de-pends on the level of approximation dictated by the number of quadrature points n.The equilibrium distribution with the same notation for the Gauss-Hermite quadra-ture is given by

Ω(x) =

√2πμkBT

Z

n∑i=1

wi exp

{−β

(Hsol(r, p) +

P2

2M+K(x, ρi) + φ(x, ρi)

)}. (46)

4.2 Conditional averages

Conditional averages can be computed in a way similar to the case of the CG Hamil-tonian. The conditional average Eq. (4) of an arbitrary function A(x, ρ) that dependson the slow variables and the relative distance (but do not depend on the conjugatemomentum πρ) takes the form

〈A〉x =∫∞0dρA(x, ρ) exp

{−β κ2 (ρ− ρ0)2} exp {−β(K(x, ρ) + φ(x, ρ))}∫∞0dρ exp

{−β κ2 (ρ− ρ0)2} exp {−β(K(x, ρ) + φ(x, ρ))} . (47)

Therefore, the conditional expectations are simply averages over the relative distancevariable. Again, by applying a Gauss-Hermite quadrature we obtain the followingexpression for conditional averages

〈A〉x =n∑i=1

Wi(x)A(x, ρi) (48)


where we have introduced for convenience the following shorthand notation for theconfiguration dependent weights

Wi(x) =wie

g(x,ρi)∑nj wje

g(x,ρj), (49)

where g(x, ρi) = −β(K(x, ρi) + φ(x, ρi)). Note that the conditional averages dependon the parameter ε through the quadrature points ρi (see Eq. (45)).

4.3 Reversible part of the dynamics

Next we need to compute the reversible part of the dynamics, which is determinedby the reversible drift 〈LX〉x. As discussed in Sec. 2.1, the CG Hamiltonian is thegenerator of the reversible part of the CG dynamics when the relevant variables aremicroscopic degrees of freedom themselves. Therefore, by using either the form of theCG Hamiltonian or by using the time derivatives Eqs. (32) together with the form ofthe conditional averages Eq. (48), we obtain for the reversible part of the dynamics

〈LR〉x = ∂HCG

∂P(x) =

P

M

〈LP〉x = −∂HCG

∂R(x) = −

n∑i=1

Wi(x)∂φ

∂R(x, ρi)

〈Lθ〉x = ∂HCG

∂πθ(x) =

πθ

μρ20

n∑i=1

Wi(x)

(1 + εsi)2

〈Lπθ〉x = −∂HCG

∂θ(x) = −

n∑i=1

Wi(x)∂φ

∂θ(x, ρi)−

π2ϕ cos θ

μρ20 sin3 θ

n∑i=1

Wi(x)

(1 + εsi)2

(50)

〈Lϕ〉x = ∂HCG

∂πϕ(x) =

πϕ

μρ20 sin2 θ

n∑i=1

Wi(x)

(1 + εsi)2

〈Lπϕ〉x = −∂HCG

∂ϕ(x) = −

n∑i=1

Wi(x)∂φ

∂ϕ(x, ρi)

〈Lri〉x = ∂HCG

∂pi(x) =

pimi

〈Lpi〉x = −∂HCG

∂ri(x) = Fsoli −

n∑i=1

Wi(x)∂φ

∂ri(x, ρi)

where Fsoli is the force on solvent atom i due to the rest of solvent atoms.

4.4 Irreversible part of the dynamics

The calculation of the friction matrix Eq. (19) requires to evaluate first the projectedcurrentsQLX defined in Eq. (7). Starting from the definition of the projector operator


Q in Eq. (8) in terms of conditional averages and by observing that the kinetic energyterm in Eq. (30) does not depend on the coordinate ϕ while the dependence on thecoordinate ρ goes simply as 1

ρ2, we obtain in the present case

QLR = 0

QLP = − ∂∂Rφ(x, ρ) +

⟨∂

∂Rφ(x, ρ)

⟩x(51)

QLθ = ∂K0∂πθ

[ρ20ρ2−

⟨ρ20ρ2

⟩x]

QLϕ = ∂K0∂πϕ

[ρ20ρ2−

⟨ρ20ρ2

⟩x]

QLπθ = − ∂∂θφ(x, ρ) +

⟨∂

∂θφ(x, ρ)

⟩x− ∂K0∂θ

[ρ20ρ2−

⟨ρ20ρ2

⟩x]

QLπϕ = − ∂∂ϕφ(x, ρ) +

⟨∂

∂ϕφ(x, ρ)

⟩x(52)

QLri = 0

QLpi = − ∂∂riφ(x, ρ) +

⟨∂

∂riφ(x, ρ)

⟩x.

Here K0 is the kinetic energy (30) evaluated at ρ = ρ0. The non-zero terms for theprojected currents in Eqs. (53) can be summarized as

QLα = ∂K0∂παδρ

QLpx = δFqx (53)

QLπα = δFα − ∂K0∂αδρ

where the index α is either θ or ϕ and we have introduced yet another shorthandnotation

δρ ≡[ρ20ρ2−

⟨ρ20ρ2

⟩x](54)

δFqx ≡ −∂

∂qxφ(x, ρ) +

⟨∂

∂qxφ(x, ρ)

⟩x

where qx is any variable in the set {θ, ϕ,R, ri} and px is the associated conjugatemomentum. The non-zero correlation terms appearing in the friction matrix can now


be expressed in a closed form

〈QLαQLβ〉x = ∂K0∂πα

∂K0

∂πβ〈δρδρ〉x

〈QpxQpx′〉x =⟨δFqxδFqx′

⟩x〈QLαQLpx〉x = ∂K0

∂πα〈δρδFqx〉x

〈QLπαQLβ〉x = ∂K0∂πβ

〈δFαδρ〉x − ∂K0∂πθ

∂K0

∂α〈δρδρ〉x (55)

〈QLπαQLpx〉x = 〈δFαδFqx〉x −∂K0

∂α〈δρδFqx〉x

〈QLπαQLπβ〉x = 〈δFαδFβ〉x − ∂K0∂β〈δFαδρ〉x

−∂K0∂α〈δρδFβ〉x + ∂K0

∂α

∂K0

∂β〈δρδρ〉x .

We point out that these correlations involve just the following conditional averages

〈δρδρ〉x =⟨ρ40ρ4

⟩x−

⟨ρ20ρ2

⟩x⟨ρ20ρ2

⟩x

〈δρδFqx〉x = −⟨ρ20ρ2∂φ

∂qx

⟩x+

⟨ρ20ρ2

⟩x⟨∂φ

∂qx

⟩x(56)

〈δFqxδFqx′ 〉x =⟨∂φ

∂qx

∂φ

∂qx′

⟩x−

⟨∂φ

∂qx

⟩x⟨∂φ

∂qx′

⟩x

for which the Gauss-Hermite numerical quadrature results in the following explicitexpressions

〈δρδρ〉x =n∑i=1

Wi(x)

(1 + εsi)4−

[n∑i=1

Wi(x)

(1 + εsi)2

]2

〈δρδFqx〉x = −n∑i=1

Wi(x)

(1 + εsi)2∂φ

∂qx(x, ρi)

+

[n∑i=1

Wi(x)

(1 + εsi)2

][n∑i=1

Wi(x)∂φ

∂qx(x, ρi)

]

〈δFqxδFqx′ 〉x =n∑i=1

Wi(x)∂φ

∂qx(x, ρi)

∂φ

∂qx′(x′, ρi)

−[n∑i=1

Wi(x)∂φ

∂qx(x, ρi)

][n∑i=1

Wi(x)∂φ

∂qx′(x, ρi)

]. (57)

By collecting all results (54)–(57), we may write down the friction matrix M(x) inEq. (19) explicitly. By construction the friction matrix is symmetric and positivedefinite. The structure of this matrix is that of a first upper 10× 10 block involvingthe dimer variables R,P, θ, πθ, φ, πφ and a lower 6N ×6N block involving the solventmolecules. As such it is a very large matrix. Note, however that this matrix will bevery sparse because only the solvent molecules which are close enough to the dimer dointeract with a corresponding non-zero element in the friction matrix. This intuitive


idea will be substantiated further in Sec. 5. Overall, the effective size of the matrixwill be of the order of (10 + 6m) × (10 + 6m) where m is the average number ofinteracting solvent neighbors to the dimer, its value will depend on the nature of thepotential but in general it will be substantially less than the total number of solventmolecules.In order to complete the construction of the SDE for the slow variables, we need

to compute the divergence terms kBT∂Mμν∂xν. Because we have an explicit expression

for the friction matrix, these are straightforward to compute, although laborious. Wewill do this calculation after discussing the stiff limit in Sec. 5.The final requested item is the formulation of the random noise terms dxμ. A brute

force approach to obtain the random terms from the friction matrix would require tocompute the eigenvalues λi and the normalized eigenvectors v

i of the M matrix insuch a way that the spectral decomposition of the matrix would lead to the particularform

Mμν =∑i

λiviμviν . (58)

In this way, a simple implementation of the random terms is

dxμ =∑i

(2kBTλi)1/2viμdW

it (59)

where dW it , i = 1, · · · , are independent increments of the Wiener processes withdW it dW

jt = δijdt. Indeed, we have that the dyadic product of the vector in Eq. (59)

is given by the matrix in Eq. (58) and, therefore, the fluctuation dissipation theorem,Eq. (12), is satisfied.A full numerical implementation of the solution of the SDE would require to diag-

onalize (at every time step!) the large matrixM(x). This is pointless but, fortunately,it is not necessary. As we will discuss in the next section in the limit of large stiffness,(i.e. ε small), all other eigenvalues are much smaller than the largest one and onlyone term in the sum in Eq. (58) survives.

5 The stiff limit

The conditional averages defining the terms of the SDE depend on the dimensionlessinverse spring constant parameter ε through the integration points ρi = ρ0 + ερ0si.When the results of Eq. (57) are expanded to second order in the small parameter εthe covariances

〈δρδρ〉x = ε2

24

⟨δFqxδFqx′

⟩x=ε2

2

(−ρ0 ∂

2φ

∂ρ∂qx

)(−ρ0 ∂

2φ

∂ρ∂qx′

)(60)

〈δρδFqx〉x =ε2

22

(−ρ0 ∂

2φ

∂ρ∂qx

)

where the derivatives of the potential in the left hand side are evaluated at ρ = ρ0. Asketch of how Eqs. (61) emerge is illustrated in the Appendix. Two important featuresemerge from this calculation. The first one is that the friction matrix scales as ε2 and,therefore, for the dimer problem dissipative effects will be necessarily small. Thesecond point worth noting is that the structure of the covariance matrix in Eq. (61)is just that of a dyadic product. In the language of Eq. (58), only one eigenvalue is


different from zero in the stiff limit. As a consequence, in that limit it is possible towrite the friction matrix in the form

Mμν(x) =ε2τ

2kBTDμ(x)Dν(x) (61)

where the vector Dμ is explicitly

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

DR

DP

Dθ

Dπθ

Dϕ

Dπϕ

Dri

Dpi

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

=

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0

−ρ0 ∂2φ

∂ρ∂R

2∂K0∂πθ

−ρ0 ∂2φ∂ρ∂θ

− 2∂K0∂θ

2∂K0∂πϕ

−ρ0 ∂2φ

∂ρ∂ϕ

0

−ρ0 ∂2φ

∂ρ∂ri

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(62)

with all the derivatives of the potential functions evaluated at the value ρ = ρ0. Thederivatives of the kinetic energy can be made explicit and are given by

∂K0

∂θ= − π

2ϕ

μρ20

cos θ

sin3 θ

∂K0

∂πθ=πθ

μρ20(63)

∂K0

∂πϕ=πϕ

μρ20

1

sin2 θ.

5.1 The random noise

When the friction matrix has a dyadic structure it becomes very easy to identify thestructure of the random terms. These thermal noise terms satisfy the Ito rule, Eq.(12), and, therefore, we can introduce the following stochastic processes

dxμ = (kBTε2τ)1/2DμdWt (64)

where dWt is a single Wiener process, the same for all values of μ. In the stiff limitonly one term in the sum in Eq. (59) survives, that of the largest eigenvalue.In view of the enormous simplification in the form of the friction matrix and in

the calculation of the random terms, reminding the fact that the dissipation matrixis of order ε2 and, therefore, small, it seems reasonable to adopt the stiff limit of thefriction matrix instead of its explicit Gauss-Hermite quadrature. We should remind aswell that the friction matrix has already been the subject of several approximations,most prominently of the exponential approximation, for which the parameter τ stillremains undetermined. This parameter needs to be conveniently fitted.Of course, we should also take the stiff limit for the reversible part of the dynamics

which is given by the stiff limit of the CG Hamiltonian. A similar expansion in the


parameter ε of the CG Hamiltonian, Eq. (44), shows that the leading term, which isof order ε0, is given by the constrained Hamiltonian in Eq. (34). It seems, therefore,that for this simple model in order to obtain anything different from the constraineddynamics it is necessary to use the full CG Hamiltonian for the reversible part togetherwith the contribution coming from the irreversible part of the SDE in order to ensurethat the equilibrium distribution sampled is given correctly by Eq. (46).

5.2 The term kBT∂M∂xin the stiff limit

We still need to compute the term kBT∂M∂xin order to be able to write down the

explicit form of the SDE, Eq. (9). By taking the derivative of Eq. (61), we have

kBT∂

∂xνMμν(x) =

ε2τ

2

∂

∂xνDμ(x)Dν(x)

=ε2

2τ

[Dμ(x)

∂

∂xνDν(x) +Dν(x)

∂

∂xνDμ(x)

]. (65)

The first divergence term vanishes because none of the elements in Eq. (62) dependon the corresponding variable, that is

∂

∂xνDν(x) =

∂

∂RDR +

∂

∂PDP +

∂

∂θDθ +

∂

∂πθDπθ +

∂

∂ϕDϕ +

∂

∂πϕDπϕ

+∂

∂riDri +

∂

∂piDpi = 0. (66)

Let us compute now the second term. We find

Dν(x)∂

∂xνDμ =

[DR

∂

∂R+DP

∂

∂P+Dθ

∂

∂θ+Dπθ

∂

∂πθ+Dϕ

∂

∂ϕ

+Dπϕ∂

∂πϕ+Dri

∂

∂ri+Dpi

∂

∂pi

]Dμ

=

[Dθ∂

∂θ+Dϕ

∂

∂ϕ+Dπθ

∂

∂πθ+Dπϕ

∂

∂πϕ

]Dμ (67)

where we have made use of the fact that Dμ(x) does not depend on the translationalmomenta P,pi. Therefore, we arrive finally at the result

Dν(x)∂

∂xνDμ(x)→

[Dθ∂

∂θ+Dϕ

∂

∂ϕ+Dπθ

∂

∂πθ+Dπϕ

∂

∂πϕ

]

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

DR

DP

Dθ

Dπθ

Dϕ

Dπϕ

Dri

Dpi

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠


= 2

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 + 0

[πθμρ20

∂∂θ+

πϕμρ20 sin

2 θ∂∂ϕ

] (−ρ0 ∂

2φ∂ρ∂R

)+ 0

0 + Dπθ1μρ20[

πθμρ20

∂∂θ+

πϕμρ20 sin

2 θ∂∂ϕ

] (−ρ0 ∂

2φ∂ρ∂θ

)+ Dπϕ

2πϕμρ20

cos θsin3 θ

−Dθ π2ϕ

μρ20

2+cos 2θsin4 θ

0 + Dπϕπϕ

μρ20 sin2 θ− 2Dθ π

2ϕ cos θ

μρ20 sin3 θ

[πθμρ20

∂∂θ+

πϕμρ20 sin

2 θ∂∂ϕ

] (−ρ0 ∂

2φ∂ρ∂ϕ

)+ 0

0 + 0

[πθμρ20

∂∂θ+

πϕμρ20 sin

2 θ∂∂ϕ

] (−ρ0 ∂

2φ∂ρ∂ri

)+ 0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠(68)

6 The CG dynamic equations

We can now fully write down the equations of motion for the CG dynamics givenby Eq. (9) for the solvated dimer model in the stiff limit case. We can distinguishfour qualitatively different contributions. We call Rμ(x) the reversible term, Iμ(x)the irreversible part, Cμ(x) the divergence of the friction matrix, and finally dxμ isthe random term. With all the assumptions made so far, these contributions can bequalitatively summarized in terms of the Dμ vector introduced in Eq. (62)

Rμ(x) = 〈LXμ〉x

Iμ(x) = −ε2τ

2A(x)Dμ(x) with A(x) =

1

kBTDν(x)

∂HCG

∂xν

Cμ(x) =ε2τ

2Dν(x)

∂Dμ(x)

∂xν

dxμ = (kBTε2τ)1/2DμdWt (69)

where we have explicitly used the fact that the friction matrix is the dyadic productin Eq. (61) when writing the irreversible term Iμ(x).

The stochastic equation of motion for the CG dynamics can be expressed in thisnotation in a compact form as

dxμ = Rμ(x)dt+ Iμ(x)dt+ Cμ(x)dt+ dxμ (70)

and, now, inserting in it the reversible vector in Eq. (51), the Dμ terms as given inEq. (62) and the derivatives of the CG Hamiltonian, the full equations of motion can


be finally displayed

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

dR

dP

dθ

dπθ

dϕ

dπϕ

dri

dpi

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

=

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

PM

−∑ni=1Wi(x)

∂φ∂R (x, ρi)

πθμρ20

∑ni=1

Wi(x)(1+εsi)2

−∑ni=1Wi(x)

∂φ∂θ(x, ρi)− π2ϕ cos θ

μρ20 sin3 θ

∑ni=1

Wi(x)(1+εsi)2

πϕμρ20 sin

2 θ

∑ni=1

Wi(x)(1+εsi)2

−∑ni=1Wi(x)

∂φ∂ϕ(x, ρi)

pimi

= Fsoli −∑ni=1Wi(x)

∂φ∂ri(x, ρi)

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

dt

+2ε2τ

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0[πθμρ20

∂∂θ+

πϕμρ20 sin

2 θ∂∂ϕ

] (−ρ0 ∂

2φ∂ρ∂R

)

Dπθ1μρ20[

πθμρ20

∂∂θ+

πϕμρ20 sin

2 θ∂∂ϕ

] (−ρ0 ∂

2φ∂ρ∂θ

)−Dθ π

2ϕ

μρ20

2+cos 2θsin4 θ

+Dπϕ2πϕμρ20

cos θsin3 θ

−2Dθ π2ϕ cos θ

μρ20 sin3 θ+Dπϕ

πϕμρ20 sin

2 θ[πθμρ20

∂∂θ+

πϕμρ20 sin

2 θ∂∂ϕ

] (−ρ0 ∂

2φ∂ρ∂ϕ

)

0

[πθμρ20

∂∂θ+

πϕμρ20 sin

2 θ∂∂ϕ

] (−ρ0 ∂

2φ∂ρ∂ri

)

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

dt

− ε2τ2 A(x)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

DR

DP

Dθ

Dπθ

Dϕ

Dπϕ

Dri

Dpi

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

dt+ (kBTε2τ)1/2

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

DR

DP

Dθ

Dπθ

Dϕ

Dπϕ

Dri

Dpi

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

dWt. (71)


The set of dynamical equations of motion, Eq. (71) given above may appear formi-dable, but apart from their algebraic complexity all terms are explicitly computable.They obviously simplify for ε→ 0 when, thanks to the fact that the Fixman’s poten-tial in this particular case of a solvated dimer is a constant, these equations coincidewith the constrained dynamics in Eq. (35). Algebraically the main complexity comein the terms associated to dynamics of the θ and ϕ degrees of freedom, which alsorequire some numerical care to get around the mapping singular points for θ at thepoles. Also, note that the dynamics of the solvent molecules ri and of the center ofmass R is affected by the dynamics of θ, ϕ only through terms that are governedby the potential φ0. This means that for solvent molecules beyond the range of theinteraction in φ0 the dynamics is unaffected by the elimination of the stiff spring. Inthat sense, the whole complexity is limited to only those degrees of freedom that aredirectly interacting with the dimer.

7 Conclusions

In this paper we have delineated, on a simple model of a dimer molecule, a procedureby which stiff degrees of freedom, characterized by a fast oscillatory dynamics, can beeliminated by using the coarse-graining procedure based on the Mori-Zwanzig projec-tion theory. This allows one to treat stiff bonds in a way that has the same advantagesof the constraints method while conserving statistical and dynamical properties of theflexible system. In particular, with a transformation to generalized coordinates, wehave shown how to make explicit the action of the projector operators, which con-sist in taking conditional averages on the manifold in which the slow CG variableshave a fixed value. The effects of the eliminated degrees of freedom show up in theslow variables in the form of an CG Hamiltonian, which takes care of the properstatistics of the retained degrees of freedom, and in the introduction of dissipativeand stochastic forces, which account for the correct dynamics of the slow degreesof freedom. Note that the validity of the Markovian assumption through which theMori-Zwanzig method leads to a useful CG dynamics [13] is ensured only in the limitof clear separation of time scales between the fast and slow degrees of freedom. Thisseparation of time scales is governed by the parameter ε2 = 2kBT

κρ20(i.e. the inverse

of the frequency of the stiff spring which models the bond in the diatomic molecule,in the dimensionless terms introduced in Sec. 4). For a nitrogen molecule solvated inliquid argon ε2 is of order of 10−4 [14].In working out the model two main difficulties have been singled out and sur-

mounted. The first problem comes from the projected time evolution in the frictionmatrix, which we propose to bypass by the exponential assumption in Eq. (19), withthe introduction of a single parameter, the correlation time τ governing the dissipa-tion. This is to be considered either an adjustable parameter which can be obtainedphenomenologically or, better, derived, although non straightforwardly, from the the-ory of dynamical phenomena. The second problem arises in the definition of thethermal random noise terms, which require to compute the elements of the squareroot of the friction matrix. Because this needs to be done at every time step, theprocedure would be computationally quite heavy. We have shown, however, that inthe limit of stiff springs the friction matrix reduces to a very simple dyadic productthat renders the computation of the stochastic terms a straightforward task.The magnitude of the correlation time τ is expected to be of the order of the

period of the stiff bond and, therefore, small. An important result that is manifestin the exponential approximation is that the friction matrix scales as τε2 and, there-fore, the friction matrix is proportional to κ−2 which is, in general, very small. Webelieve that although this is obtained under the exponential approximation, the order


of magnitude is nevertheless correct. This suggests that dissipation (and noise) arenegligible in the dimer problem as compared with the reversible dynamics generatedby the CG Hamiltonian (which is of order κ0).We have not considered yet other model systems with more complex stiff potentials

(on angles, for example) and at present it remains a conjecture whether dissipationand noise may always be neglected in the CG elimination of stiff bonds. If this wasthe case, the resulting scheme would be certainly much simpler. Note that in thisparticular case of a solvated dimer when the bond is infinitely stiff the CG dynamicsreduces to the one defined by a constrained Hamiltonian. This is a feature of thepresent system and needs not to be true in general for other type of bond potentials,acting, for example, on angles, where the Fixman potential can explicitly depend onthe slow degrees of freedom. In any case, we have given an explicit simple expres-sion for the CG Hamiltonian in terms of Gauss-Hermite quadratures, that allows toexplore the effects of finite stiffness of the bond.The use of generalized coordinates may refrain from using the present approach

in practice, because it may be inconvenient to implement in a general purpose sim-ulation package, and care must be taken to avoid singular points in the mapping ofcoordinates that could lead to numerical inconsistencies [15]. Of course it is alwayspossible to go back to a stochastic dynamics equation written in cartesian coordinateswith holonomic constraints. Appropriate numerical algorithms are already available[16], but then the entire Mori-Zwanzig derivation has to be carried out again afresh.

A special thank to Pablo Echenique who organized in Zaragoza the interesting discussionwhich brought us to explore the subject of the present paper. We acknowledge financialsupport from SFI Grant No. 08- IN.1-I1869. GC acknowledges also support from the Isti-tuto Italiano di Tecnologia under the SEED project grant No. 259 SIMBEDD – AdvancedComputational Methods for Biophysics, Drug Design and Energy Research. Financial sup-port from the MICINN under project FIS2010-22047-C05-03 and BIFI is acknowledged. P.E.acknowledges the kind hospitality of the Freiburg Institute for Advanced Studies where thiswork was initiated.

Appendix

The purpose of this appendix is to compute the Taylor expansion up to second orderin ε of conditional averages defined in Eq. (48), this is,

〈A〉x = A(x, ρ0) + ε ddε〈A〉x

∣∣∣∣ε=0

+ε2

2

d2

dε2〈A〉x

∣∣∣∣ε=0

. (72)

The first derivative of the conditional average (48) with respect to ε is

d

dε〈A〉x =

∑i

d

dεWi(x)A(x, ρi) +

∑i

Wi(x)d

dεA(x, ρi) (73)

where we note that the weight function (49) also depends on ρi and, therefore, on ε.The second derivative is

d2

dε2〈A〉x =

∑i

d2

dε2Wi(x)A(x, ρi) + 2

∑i

d

dεWi(x)

d

dεA(x, ρi) +

∑i

Wi(x)d2

dε2A(x, ρi).

(74)


To start we need to compute the first derivative of the weight function

d

dεWi(x) =

wieg(x,ρi)∑n

j wjeg(x,ρj)

dg

dε(x, ρi)− wie

g(x,ρi)

(∑nj wje

g(x,ρj))2

n∑j

wjeg(x,ρj)

dg

dε(x, ρj)

= Wi(x)

[dg

dε(x, ρi)−

⟨dg

dε

⟩x]. (75)

The second derivative of the weight function is

d2

dε2Wi(x) =

(d

dεWi(x)

)[dg

dε(x, ρi)−

⟨dg

dε

⟩x]+Wi(x)

(d

dε

[dg

dε(x, ρi)−

⟨dg

dε

⟩x])

= Wi(x)

[dg

dε(x, ρi)−

⟨dg

dε

⟩x]2+Wi(x)

d2g

dε2(x, ρi)−Wi(x)

⟨d2g

dε2

⟩x

−Wi(x)n∑j

d

dεWj(x)

dg

dε(x, ρj)

= Wi(x)

[dg

dε(x, ρi)−

⟨dg

dε

⟩x]2+Wi(x)

d2g

dε2(x, ρi)−Wi(x)

⟨d2g

dε2

⟩x

−Wi(x)n∑j

[Wj(x)

[dg

dε(x, ρj)−

⟨dg

dε

⟩x]dg

dε(x, ρj)

]

= Wi(x)

[d2g

dε2(x, ρi)−

⟨d2g

dε2

⟩x]+Wi(x)

[(dg

dε(x, ρi)

)2−

⟨(dg

dε

)2⟩x]

−2Wi(x)[(dg

dε(x, ρi)−

⟨dg

dε

⟩x)⟨dg

dε

⟩x]. (76)

Now we have to evaluate these expressions at ε = 0. By using Eq. (45) and the chainrule, the derivatives with respect to ε are

d

dεA(x, ρi) = ∂ρA(x, ρi)

d

dερi = ρ0∂ρA(x, ρi)si

(77)

d2

dε2A(x, ρi) = ρ

20∂2ρA(x, ρi)s

2i

which, when evaluated at ε = 0 become

d

dεA(x, ρi)

∣∣∣∣ε=0

= ρ0∂ρA(x, ρ0)si

(78)

d2

dε2A(x, ρi)

∣∣∣∣ε=0

= ρ20∂2ρA(x, ρ0)s

2i .

Therefore, the derivatives of the weights at ε = 0 are

d

dεWi(x)

∣∣∣∣ε=0

= ρ0∂ρg(x, ρ0)Wi(x) [si − 〈s〉x]d2

dε2Wi(x)

∣∣∣∣ε=0

= ρ20∂2ρg(x, ρ0)Wi(x)

[s2i −

⟨s2⟩x]+ (ρ0∂ρg(x, ρ0))

2Wi(x)[s2i −

⟨s2⟩x]

−2Wi(x)(ρ0∂ρg(x, ρ0))2 [(si − 〈s〉x) 〈s〉x] . (79)


Note that the averages have the following form at ε = 0

〈s〉x =n∑i

W (x, ρ0)si =

∑ni wisi∑ni wi

≈∫∞−∞ e

−s2sds∫∞−∞ e

−s2ds= 0

(80)

⟨s2⟩x=

n∑i

W (x, ρ0)s2i =

∑ni wis

2i∑n

i wi≈

∫∞−∞ e

−s2s2ds∫∞−∞ e

−s2ds=1

2

where we have assumed that the above Gauss-Hermite quadratures are sufficientlygood approximations. The derivatives of the weights at ε = 0 are then

d

dεWi(x)

∣∣∣∣ε=0

= ρ0∂ρg(x, ρ0)wi∑nj wjsi

(81)

d2

dε2Wi(x)

∣∣∣∣ε=0

=(ρ20∂

2ρg(x, ρ0) + (ρ0∂ρg(x, ρ0))

2) wi∑n

j wj

[s2i −

1

2

].

Therefore, the first derivative of the conditional average (73) vanishes at ε = 0 becauseit is proportional to 〈s〉x. The second derivative is

d2

dε2〈A〉x = ρ0∂ρg(x, ρ0)ρ0∂ρA(x, ρ0) + 1

2ρ20∂

2ρA(x, ρ0). (82)

Finally, the Taylor expansion (72) to second order in ε of the conditional average is

〈A〉x = A(x, ρ0) + ε2

2

[ρ0∂ρg(x, ρ0)ρ0∂ρA(x, ρ0) +

1

2ρ20∂

2ρA(x, ρ0)

]. (83)

We may now use this result to compute the conditional covariances

〈AB〉x − 〈A〉x〈B〉x = A(x, ρ0)B(x, ρ0)+ε2

2

[ρ0∂ρg(x, ρ0)ρ0∂ρ [A(x, ρ0)B(x, ρ0)]

+1

2ρ20∂

2ρA(x, ρ0)B(x, ρ0)

]

−[A(x, ρ0) +

ε2

2

[ρ0∂ρg(x, ρ0)ρ0∂ρA(x, ρ0) +

1

2ρ20∂

2ρA(x, ρ0)

]]

×[B(x, ρ0) +

ε2

2

[ρ0∂ρg(x, ρ0)ρ0∂ρB(x, ρ0) +

1

2ρ20∂

2ρB(x, ρ0)

]]

=ε2

2ρ20 (∂ρA(x, ρ0)) (∂ρB(x, ρ0)) . (84)

Therefore, conditional covariances turn out to be just the product of two functions.This leads to the dyadic structure of the friction matrix in the stiff limit.

References

1. M.E. Tuckerman, G.J. Martyna, B.J. Berne, J. Chem. Phys. 97, 1990 (1992)2. T.C. Bishop, R.D. Skeel, K. Schulten, J. Comp. Chem. 18, 1785 (1997)


3. J.P. Ryckaert, G. Ciccotti, H.J.C. Berendsen, J. Comp. Phys. 23, 277 (1977)4. J.P. Ryckaert, G. Ciccotti, J. Chem. Phys. 78, 7368 (1983)5. M. Fixman, Proceedings of the National Academy of Sciences of the United States ofAmerica 71, 3050 (1974)

6. E. Helfand, J. Chem. Phys. 71, 5000 (1979)7. N.G. Van Kampen, Applied Scientific Research 37, 67 (1981)8. S. Reich, Num. Algorithm 19, 213 (1998)9. H. Mori, Prog. Theor. Phys. 33, 423 (1965)10. R. Zwanzig, Phys. Rev. 124, 983 (1961)11. H. Grabert, Projection Operator Techniques in Nonequilibrium Statistical Mechanics(Springer Verlag, Berlin, 1982)

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Coarse-graining stiff bonds

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