Energie libre et dynamique réduite en dynamique moléculaire · Energie libre et dynamique réduite en dynamique moléculaire Fred´ eric Legoll´ Ecole des Ponts ParisTech et INRIA

Energie libre et dynamique réduite

en dynamique moléculaire

Frederic Legoll

Ecole des Ponts ParisTech et INRIA

http://cermics.enpc.fr/∼legoll

Travaux en collaboration avec T. Lelievre (Ecole des Ponts et INRIA)

Interactions EDPs/Probas: modeles probabilistes pour la simulation moleculaire

GdR CHANT, Interactions EDPs/Probas, Grenoble, 23-25 novembre 2011 – p. 1

Effective dynamics

In these lectures, we are interested in deriving effective dynamics, in thecontext of molecular simulation.

This question (absent the specific context) is very general.

Considering a dynamics

X = f(X), X ∈ Rn,

and a quantity of interest ξ(X) ∈ R, we wish to obtain a dynamics thatapproximates t 7→ ξ(X(t)).


Effective dynamics

In these lectures, we are interested in deriving effective dynamics, in thecontext of molecular simulation.

This question (absent the specific context) is very general.

Considering a dynamics

X = f(X), X ∈ Rn,

and a quantity of interest ξ(X) ∈ R, we wish to obtain a dynamics thatapproximates t 7→ ξ(X(t)).

To achieve such a goal, a time scale separation needs to be present, and ξ(X)

be a slow variable.Many results in the literature use the fact that:

a small parameter (encoding the time scale separation) appears in anexplicit way in the problem.

X = (x, y) where x is slow and y is fast: natural splitting of X into a slowand a fast component.


A first example

To fix the idea, consider the Ordinary Differential Equation

dxε

dt= f(xε, yε),

dyε

dt=h(xε, yε)

ε=g(xε) − yε

ε

In the fast time scale τ = t/ε, we have

dxε

dτ= εf(xε, yε),

dyε

dτ= g(xε) − yε

which converges, in the limit ε→ 0, on bounded time intervals, to

dx0

dτ= 0,

dy0

dτ= g(x0) − y0

We thus see that xε is a slow variable and yε a fast one.


A first example

To fix the idea, consider the Ordinary Differential Equation

dxε

dt= f(xε, yε),

dyε

dt=h(xε, yε)

ε=g(xε) − yε

ε

In the fast time scale τ = t/ε, we have

dxε

dτ= εf(xε, yε),

dyε

dτ= g(xε) − yε

which converges, in the limit ε→ 0, on bounded time intervals, to

dx0

dτ= 0,

dy0

dτ= g(x0) − y0

We thus see that xε is a slow variable and yε a fast one.

For a fixed xε = x, the dynamicsdyε

dt=h(x, yε)

ε=g(x) − yε

εhas a specific

structure: for any ε > 0, whatever the initial condition yε(0), we havelimt→∞

yε(t) = g(x). The presence of ε only affects the characteristic time of

evolution of yε to g(x), which is proportional to ε.


Tikhonov result

dxε

dt= f(xε, yε),

dyε

dt=h(xε, yε)

ε=g(xε) − yε

ε

Classical result [Tikhonov] : assume xε(0) = x0 independent of ε. Then, for anyfinite T , we have

supt∈[0,T ]

|xε(t) − x0(t)| →ε→0 0

where x0 solvesdx0

dt= f

(x0, g

(x0

))

We have eliminated the fast variable yε, and obtained a reduced (effective)dynamics x0(t) that approximates xε(t).

Remark: a similar result holds for more general functions h, that may benonlinear wrt y, . . . We took here the simplest example for the sake ofillustration.


Fast equations that are ergodic

Similar results hold when the dynamics of the fast variable (for a frozen slowvariable) is ergodic: consider

dxε

dt= f(xε, yε),

dyε

dt=h(xε, yε)

ε

Assume that the fast dynamics (for slow variable frozen at x) is ergodic for

µx(dy): the solution y tody

dt= h(x, y) satisfies, for any test function φ

sufficiently smooth, limT→∞

1

T

∫ T

0

φ(y(t)) dt =

∫

R

φ(y)µx(dy).


Fast equations that are ergodic

Similar results hold when the dynamics of the fast variable (for a frozen slowvariable) is ergodic: consider

dxε

dt= f(xε, yε),

dyε

dt=h(xε, yε)

ε

Assume that the fast dynamics (for slow variable frozen at x) is ergodic for

µx(dy): the solution y tody

dt= h(x, y) satisfies, for any test function φ

sufficiently smooth, limT→∞

1

T

∫ T

0

φ(y(t)) dt =

∫

R

φ(y)µx(dy).

Then we can again obtain an effective dynamics for xε, in the sense

supt∈[0,T ]

|xε(t) − x0(t)| →ε→0 0 withdx0

dt= F (x0),

where F (x) :=

∫

R

f(x, y)µx(dy). [Artstein, Arnold, Verhulst, . . . ]

Remark: the previous case corresponds to µx(dy) = δg(x): conv. to g(x).


A Hamiltonian case (q1 ∈ Rn, q2 ∈ R)

Consider

Hε(q1, q2, p1, p2) =1

2(p2

1 + p22) + U(q1, q2) +

ω2(q1)q22

2ε2

with ω(q1) > 1 and V (q1, q2) ≥ 0. The dynamics is

q =∂Hε

∂p(q, p), p = −∂Hε

∂q(q, p), q = (q1, q2), p = (p1, p2).

Consider initial conditions depending on ε such that Hε(t = 0) = H0

independent of ε. By energy preservation, for all t, |q2(t)| ≤ Cε. When ε→ 0,this looks like constraining at q2 = 0.

Actually, this is NOT equivalent to constraining:

p1 = −∂Hε

∂q1= − ∂U

∂q1(q1, q2)−

ω(q1) q22

ε2∂ω

∂q1(q1) 6= − ∂U

∂q1(q1, 0)

because the second term is of order 1, even if ε is small.GdR CHANT, Interactions EDPs/Probas, Grenoble, 23-25 novembre 2011 – p. 6

Effective Hamiltonian dynamics

Hε(q1, q2, p1, p2) =1

2(p2

1 + p22) + U(q1, q2) +

ω2(q1)q22

2ε2

When ε→ 0, the dynamics of the slow variables (q1, p1) converges to theHamiltonian dynamics associated to

Heff(q1, p1) =1

2p21 + U(q1, 0) + C ω(q1)

where the constant C depends on the initial conditions:

C = limε→0

[1

2ω(q1)

(p22 +

ω2(q1)q22

ε2

)∣∣∣∣t=0

]

The unexpected term is known as the Van Kampen (or Fixman) potential.

Remark: if q2 is not scalar, resonances may occur (Takens, Bornemann andSchuette 1997).


A stochastic case

Consider now the stochastic differential equation

dxε = f(xε, yε) dt, dyε =h(xε, yε)

εdt+

1√εdWt

and assume that the fast equation, when the slow variable is frozen at x,

dyε =h(x, yε)

εdt+

1√εdWt

is ergodic for some measure µx(dy). Observe that, due to the scaling, theinvariant measure (if it exists) is independent from ε.

Introduce x0 solution of

dx0

dt= F (x0), F (x) :=

∫

R

f(x, y)µx(dy).

Then xε(t) converges to x0(t) (Papanicolaou, book of Pavliotis and Stuart andreferences therein, . . . ).


Our aim

In the above works:the state variable naturally splits into a fast component y and a slowcomponent x, that are cartesian coordinates of the state variable.

Note though that this assumption has recently been relaxed: [Arstein,

Kevrekidis, Slemrod, Titi, 2007] and [Tao, Owhadi, Marsden, 2010] have

considered the problemdx

dt= G(x) +

1

εF (x)

there is an identified small parameter ε in the problem.

In molecular simulation, these two assumptions are not always satisfied. Ouraim is to

consider a stochastic differential equation on X , and assume that ξ(X) isa slow variable (in a sense to be made precise)

build an effective dynamics for ξ(X)

The slow component ξ(X) does not have to be some cartesian coordinate ofX , and it is slow in a broader sense that those considered in the above slides.


Outline

Motivation: molecular simulation problems

The overdamped case:

A strategy to build an effective dynamics for ξ(X) ∈ R, when X ∈ Rn

evolves according to

dXt = −∇V (Xt) dt+√

2β−1 dWt

Numerical results

Error estimation: estimates in law, pathwise estimates

The Langevin case: the reference dynamics is

dXt = Pt dt, dPt = −∇V (Xt) dt− γPt dt+√

2γβ−1 dWt,

and we now look at ξ(X) and the associated momentum.


Fast and slow degrees of freedomin molecular simulation


Molecular simulation

We consider systems at the atomistic scale:

the state of the system is given by the positions xi (and the momenta pi)of each atom, considered as a classical point particle.

largest systems simulated ∼ 1010 atoms

space scale ∼ 10−10 metre, time scale ∼ 10−15 second

We assume that we are given an interaction energy V (X) = V (x1, x2, . . . , xn),that depends on the position xi of each atom. All the physics is contained inthe precise expression of V .

A simple example: V (X) =∑

i<j

Vpair(|xi − xj |), with, for instance,

Vpair(r) = 4εLJ

[(σLJ

r

)12

− 2(σLJ

r

)6], the Lennard-Jones potential.

Building a good V for a given materials is a research topic by itself.


Typical potential energy surfaces

Typical potential energy surfaces (material science and biomolecules)

There are many local minima.


Quantities of interest - 1

Following statistical physics, equilibrium quantities are given bythermodynamical averages with respect to the Boltzmann-Gibbs measure:

〈Φ〉 =

∫

Rn

Φ(X) dµ, dµ = Z−1 exp(−βV (X)) dX, X ∈ Rn

with β = 1/(kBT ) and Z =

∫

Rn

exp(−βV (X)) dX

The precise choice for Φ depends on what macroscopic quantities we areinterested in. For example, the (macroscopic) pressure is obtained with

Φ(X) = ρT − 1

3|Ω|

n∑

i=1

xi ·∂V

∂xi(X), ρ ≡ density, Ω ≡ occupied domain

If the atoms are non-interacting, then V ≡ 0 and we recover Mariotte’s lawfor ideal gas.


Quantities of interest - 2

Dynamical quantities are also of interest:

diffusion coefficients (e.g. of defects in materials)

rate constants (of chemical reactions)

residence times in metastable basins

In practice, quantities of interest often depend on a few variables.

Reduced description of the system,that still includes some dynamical information?


Reference dynamics

We are interested in dynamical properties. Two possible choices for thereference dynamics of the system:

overdamped Langevin equation (our main choice):


2β−1 dWt, Xt ∈ Rn

Langevin equation (with masses set to 1), that we will consider at the endof the lectures: Xt ∈ R

n and Pt ∈ Rn with


2γβ−1 dWt

For both dynamics,

1

T

∫ T

0

Φ(Xt) dt −→∫

Rn

Φ(X) dµ, dµ = Z−1 exp(−βV (X)) dX.

Remark: the dynamics dyε =h(xε, yε)

εdt+

1√εdWt (with a large coefficient in

front of the noise) does not enter this framework.GdR CHANT, Interactions EDPs/Probas, Grenoble, 23-25 novembre 2011 – p. 16

Metastability and reaction coordinate


2β−1 dWt, Xt ≡ position of all atoms

in practice, the dynamics is metastable: the system stays a long time in awell of V before jumping to another well:

25002000150010005000

1

0

-1

1

T

∫ T

0

φ(Xt) dt =

∫φ(X) dµ+O

(1√T

)with a large constant in the O


Metastability and reaction coordinate


2β−1 dWt, Xt ≡ position of all atoms

in practice, the dynamics is metastable: the system stays a long time in awell of V before jumping to another well:

25002000150010005000

1

0

-1

1

T

∫ T

0

φ(Xt) dt =

∫φ(X) dµ+O

(1√T

)with a large constant in the O

we assume that wells are fully described through a well-chosen reactioncoordinate ξ : R

n 7→ R

ξ(x) may e.g. be a particular angle in the molecule.

Quantity of interest: path t 7→ ξ(Xt).GdR CHANT, Interactions EDPs/Probas, Grenoble, 23-25 novembre 2011 – p. 17

Our aim


2β−1 dWt in Rn, ergodic for dµ = Z−1 exp(−βV (X)) dX

Given a reaction coordinate ξ : Rn 7→ R,

propose a dynamics zt that approximates ξ(Xt).

preservation of equilibrium properties:when X ∼ dµ, then ξ(X) is distributed according to exp(−βA(z)) dz,where A is the free energy: for any smooth Φ,∫

Rn

Φ(ξ(X)) dµ = Z−1

∫

Rn

Φ(ξ(X)) exp(−βV (X)) dX =

∫

R

Φ(z) exp(−βA(z)) dz

The dynamics zt should be ergodic wrt exp(−βA(z)) dz.

recover in zt some dynamical information included in ξ(Xt).

Related approaches: Mori-Zwanzig formalism, averaging principle for SDE(Pavliotis and Stuart, Hartmann, . . . ), effective dynamics using Markov statemodels (Schuette and Sarich), asymptotic expansion of the generator(Papanicolaou, . . . ), . . . GdR CHANT, Interactions EDPs/Probas, Grenoble, 23-25 novembre 2011 – p. 18

Fokker-Planck equations

consider the SDE

dXt = b(t,Xt) dt+ σ(t,Xt) dWt, Xt ∈ Rn, b ∈ R

n, σ ∈ R

Let ψ(t, x) be the density of Xt: for any B ⊂ Rn, P(Xt ∈ B) =

∫

B

ψ(t, x)dx.

Then∂ψ

∂t=

1

2∆

(σ2ψ

)− div (bψ)


Fokker-Planck equations

consider the SDE

dXt = b(t,Xt) dt+ σ(t,Xt) dWt, Xt ∈ Rn, b ∈ R

n, σ ∈ R

Let ψ(t, x) be the density of Xt: for any B ⊂ Rn, P(Xt ∈ B) =

∫

B

ψ(t, x)dx.

Then∂ψ

∂t=

1

2∆

(σ2ψ

)− div (bψ)

in the particular case of the overdamped Langevin equation,


2β−1 dWt,

the Fokker-Planck equation reads

∂ψ

∂t=

1

β∆ψ + div (∇V ψ) =

1

βdiv

[ψ∞∇

(ψ

ψ∞

)]

with ψ∞(x) = Z−1 exp(−βV (x)).


Effective dynamics in theoverdamped Langevin case


A super-simple case: ξ(x, y) = x

Consider the dynamics in two dimensions: X = (x, y) ∈ R2,

dxt = −∂xV (xt, yt) dt+√

2β−1 dW xt ,

dyt = −∂yV (xt, yt) dt+√

2β−1 dW yt ,

and assume that ξ(x, y) = x.


A super-simple case: ξ(x, y) = x

Consider the dynamics in two dimensions: X = (x, y) ∈ R2,

dxt = −∂xV (xt, yt) dt+√

2β−1 dW xt ,

dyt = −∂yV (xt, yt) dt+√

2β−1 dW yt ,

and assume that ξ(x, y) = x. Let ψ(t, x, y) be the density of Xt = (xt, yt).

Introduce the mean of the drift over all configurations satisfying ξ(X) = z:

b(t, z) := −

∫

R

∂xV (z, y) ψ(t, z, y) dy∫

R

ψ(t, z, y) dy

= − E [∂xV (X) | ξ(Xt) = z]

and consider dzt = b(t, zt) dt+√

2β−1 dBt

Then, for any t, the law of zt is equal to the law of xt (Gyongy 1986).

proof


Making the approach practical

b(t, z) = − −∫

R

∂xV (z, y) ψ(t, z, y) dy = − E [∂xV (X) | ξ(Xt) = z]

b(t, z) is extremely difficult to compute . . . Need for approximation:


Making the approach practical

b(t, z) = − −∫

R

∂xV (z, y) ψ(t, z, y) dy = − E [∂xV (X) | ξ(Xt) = z]

b(t, z) is extremely difficult to compute . . . Need for approximation:

b(z) := − −∫

R

∂xV (z, y) ψ∞(z, y) dy = − Eµ [∂xV (X) | ξ(X) = z]

with ψ∞(x, y) = Z−1 exp(−βV (x, y)).

Effective dynamics:

dzt = b(zt) dt+√

2β−1 dBt

Idea: b(t, x) ≈ b(x) if the equilibrium in each manifold

Σx = (x, y), y ∈ Ris quickly reached: xt is much slower than yt.


The general case: X ∈ Rn and arbitrary ξ - 1


2β−1 dWt, ξ : Rn → R

From the dynamics on Xt, we obtain (chain rule)

d [ξ(Xt)] = ∇ξ(Xt) · dXt + β−1∆ξ(Xt) dt

=(−∇V · ∇ξ + β−1∆ξ

)(Xt) dt+

√2β−1 ∇ξ(Xt) · dWt

=(−∇V · ∇ξ + β−1∆ξ

)(Xt) dt+

√2β−1 |∇ξ(Xt)| dBt

where Bt is a 1D brownian motion.





From the dynamics on Xt, we obtain (chain rule)

d [ξ(Xt)] = ∇ξ(Xt) · dXt + β−1∆ξ(Xt) dt

=(−∇V · ∇ξ + β−1∆ξ

)(Xt) dt+

√2β−1 ∇ξ(Xt) · dWt

=(−∇V · ∇ξ + β−1∆ξ

)(Xt) dt+

√2β−1 |∇ξ(Xt)| dBt


Introduce averages:

b(z) := −∫ (

−∇V · ∇ξ + β−1∆ξ)(X) ψ∞(X) δξ(X)−z dX

σ2(z) := −∫

|∇ξ(X)|2 ψ∞(X) δξ(X)−z dX

Recall that Fokker-Planck for dXt = b(Xt) dt+ σ(Xt) dWt reads∂ψ

∂t=

1

2∆

(σ2ψ

)− div (bψ) . It is natural that σ2 (and not σ) is an average.





yields

d [ξ(Xt)] =(−∇V · ∇ξ + β−1∆ξ

)(Xt) dt+

√2β−1 |∇ξ(Xt)| dBt


b(z) := −∫ (

−∇V · ∇ξ + β−1∆ξ)(X) ψ∞(X) δξ(X)−z dX

σ2(z) := −∫

|∇ξ(X)|2 ψ∞(X) δξ(X)−z dX

Eff. dyn.: dzt = b(zt) dt+√

2β−1 σ(zt) dBt

The approximation makes sense if, in the manifold

Σz = X ∈ Rn, ξ(X) = z ,

Xt quickly reaches equilibrium. ξ(Xt) much slower than evolution of Xt in Σz.GdR CHANT, Interactions EDPs/Probas, Grenoble, 23-25 novembre 2011 – p. 24

Definition of conditional averages

b(z) := −∫B(X) ψ∞(X) δξ(X)−z dX =

∫B(X) ψ∞(X) δξ(X)−z dX∫

ψ∞(X) δξ(X)−z dX

with B(X) = −∇V (X) · ∇ξ(X) + β−1∆ξ(X).

By definition of ψ∞(X) δξ(X)−z, we have, for any B(X) and any ϕ(z),

∫

R

ϕ(z)

[∫B(X) ψ∞(X) δξ(X)−z dX

]dz =

∫

Rn

ϕ(ξ(X))B(X) ψ∞(X) dX

This is the co-area formula (Evans and Gariepy).

If ξ(x, y) = x, then ψ∞(x, y) δξ(x,y)−zdxdy = ψ∞(z, y) δz(dx)dy, since, byFubini,

∫

R

ϕ(z)

[∫B(z, y) ψ∞(z, y) dy

]dz =

∫

Rn

ϕ(x)B(x, y) ψ∞(x, y) dxdy


Computation of conditional averages - 1

b(z) = −∫B(X) ψ∞(X) δξ(X)−z dX

To compute b(z), follow [Ciccotti, Leli evre, Vanden-Eijnden, 2008] :

introduce the projection operator P = Id − ∇ξ ⊗∇ξ|∇ξ|2 .

the dynamics

dXt = −∇V (Xt) +√

2β−1 dWt + dYt

where Yt is such that ξ(Xt) = z and P (Xt) · dYt = 0, is ergodic forexp(−βV (X)) |∇ξ(X)| δξ(X)−zdX

choose V such that

exp(−βV (X)) |∇ξ(X)| = exp(−βV (X)) = ψ∞(X)

that is,

V (X) := V (X) +1

βln |∇ξ(X)|GdR CHANT, Interactions EDPs/Probas, Grenoble, 23-25 novembre 2011 – p. 26

Computation of conditional averages - 2

b(z) = −∫B(X) ψ∞(X) δξ(X)−z dX

we want to consider

dXt = −∇V (Xt) +√

2β−1 dWt + dYt

where Yt is such that ξ(Xt) = z and P (Xt) · dYt = 0.

in practice, run the numerical scheme

Xm+1 = Xm − ∆t∇V (Xm) +√

2β−1∆tGm + λ∇ξ(Xm)

with the Lagrange multiplier λ such that ξ(Xm+1) = z.

up to time step errors (expected to be of the order of O(∆t)),

b(z) = limM→∞

1

M

M∑

m=1

B(Xm).


Some remarks

Effective dynamics:

dzt = b(zt) dt+√

2β−1 σ(zt) dBt

OK from the statistical viewpoint: the dynamics is ergodic wrtexp(−βA(z))dz.

Using different arguments, this dynamics has been obtained by [E and

Vanden-Eijnden, 2004] , and [Maragliano, Fischer, Vanden-Eijnden and Ciccotti,

2006].

In the following, we will

compare the obtained dynamics with a natural candidate

numerically assess its accuracy

derive error bounds


Effective dynamics and free energy


Useful remarks on the free energy

For any smooth Φ,

Z−1

∫

Rn

Φ(ξ(X)) exp(−βV (X)) dX =

∫

R

Φ(z) exp(−βA(z)) dz

in the case ξ(x, y) = x, we have

A′(x) =

∫R∂xV (x, y) exp(−βV (x, y)) dy∫

Rexp(−βV (x, y)) dy

=

∫

R

∂xV (x, y)dµx

where dµx =exp(−βV (x, y)) dy∫R

exp(−βV (x, y)) dy≡ Gibbs measure conditioned at x.

in the general case, we have A′(z) =

∫

Rn

F (X)dµz, where dµz is the

Gibbs measure conditioned on the manifold Σz = X ∈ Rn, ξ(X) = z,

dµz =ψ∞(X) δξ(X)−z dX∫ψ∞(X) δξ(X)−z dX

and F =∇V · ∇ξ|∇ξ|2 − β−1 div

( ∇ξ|∇ξ|2

).


A natural candidate

Residence times in wells are often associated with free energy barriers:

assume that ut := ξ(Xt) follows the dynamics

dut = −A′(ut) dt+√

2β−1 dBt. (1)

this dynamics is ergodic wrt exp(−βA(z)) dz.

in the limit of small temperature, the residence time of ut solution to (1) ina well W1 before going to the well W2 is given by

τW1→W2∝ exp (β∆AW1→W2

)

where ∆AW1→W2is the free energy barrier (large deviation theory,

Transition State Theory, Arrhenius law, . . . ).

For these reasons, the dynamics (1) may be considered as a natural candidatefor the effective dynamics of ξ(Xt).


Questionning this natural candidate - 1

Consider the fine scale dynamics


2β−1 dWt

and


2γβ−1 dWt.

They are all ergodic for the same Gibbs measure. The equilibrium densityof ξ(X) is always exp(−βA(z)) dz. However, the dynamics of ξ(Xt)

certainly depends on γ, is different for Langevin and overdampedLangevin, . . .

Why


2β−1 dBt

would always be a good approximation?


Questionning this natural candidate - 2


2β−1 dBt. (2)

dynamics (2) is not invariant by a reparametrization: consider aone-to-one function h, and consider ζ(X) := h(ξ(X)):

level sets associated to ξ = level sets associated to ζ

Dynamics (2) for RC ξ and next one-to-one change of coordinate ξ → ζ

6=Dynamics (1) for RC ζ.

the free energy A is defined in terms of equilibrium properties. Whyshould it also have a dynamical content?


Effective dynamics and free energy

Is the effective dynamics we found,

dzt = b(zt) dt+√

2β−1 σ(zt) dBt, σ2(z) = 〈|∇ξ|2〉Σz,

with Σz = X ∈ Rn, ξ(X) = z, the same as


2β−1 dBt, A free energy associated with ξ(X)

If |∇ξ(X)| = 1, then σ(z) = 1 and the effective dynamics is ergodic forexp(βB(z)) dz, with B′ = b. Since the effective dynamics preservesexp(−βA(z)) dz, we deduce that B = −A, thus b(z) = −A′(z). Werecover the natural candidate. An example:

ξ(X) = Xj , (e.g. ξ(x, y) = x)

In general, σ(z) is not a constant, and b(z) 6= −A′(z). Dynamics aredifferent, and yield very different numerical results.

Our dynamics is invariant through reparametrization ζ(X) = h(ξ(X)), incontrast to the dynamics associated with A′.


A particular example

Vε(x, y) = U(x, y) +ω2(x)y2

2ε2

Choose ξ(x, y) = x: since ∇ξ is constant, the effective dynamics is

dzt = −A′ε(zt) dt+

√2β−1 dBt

withA′ε(z) =

∫R∂xVε(z, y) exp(−βVε(z, y)) dy∫

Rexp(−βVε(z, y)) dy

.

We calculateA′

0(z) := limε→0

A′ε(z) = ∂xU(z, 0) +

ω′(z)

βω(z).

The effective dynamics (when ε→ 0) is the overdamped Langevin equationassociated to the potential (see also [Reich 2000] )

A0(x) = U(x, 0) +1

βlnω(x).

Again, the limit ε→ 0 differs from constraining at y = 0 by a Fixman correction(which is different from the Fixman correction in the Hamiltonian case).


Numerical results


Accuracy assessment: residence times

In practice:

we consider a finite interval for z

we pre-compute b(z) and σ(z) for values on a grid (remember z is scalar)

we linearly interpolate between these values

Residence times computations: for a given reaction coordinate ξ,

we define the starting well as x ∈ Rn; ξ(x) > ξth

generate a set of initial conditions in the well distributed according to theGibbs measure

run the full dynamics from each xi until the other well has been reached→ reference residence time

run the effective dynamics

dzt = b(zt) dt+√

2β−1 σ(zt) dBt, zt=0 = ξ(xi).


Residence times for the butane molecule - 1

V (X) =∑

i

V2

(‖Xi+1 −Xi‖

)+ V3(θ1) + V3(θ2) + V4(φ)

φ

V2(ℓ) =k2

2(ℓ−ℓeq)2, V3(θ) =

k3

2(θ−θeq)2

V4(φ)

1000-100

Reaction coordinate: ξ(X) = φ, the dihedral angle.

Residence time in primary well before going to one of the two secondary wells:

complete model (reference): 31.937 ± 0.561

effective dynamics prediction: 32.037 ± 0.558

free energy dynamics prediction: 37.122 ± 0.644GdR CHANT, Interactions EDPs/Probas, Grenoble, 23-25 novembre 2011 – p. 38

Residence times for the butane molecule - 2

Temperature Reference Eff. dyn. Dyn. based on A

β−1 = 1 31.9 ± 0.56 32.0 ± 0.56 37.1 ± 0.64

β−1 = 0.5 7624 ± 113 7794 ± 115 9046 ± 133

σ is close to one: the effective dynamics agrees well with that based on A.

b(ξ), β = 2b(ξ), β = 1

ξ

43210-1-2-3-4

10

5

0

-5

-10

-15 σ(ξ), β = 2σ(ξ), β = 1

ξ

43210-1-2-3-4

1.088

1.086

1.084

1.082

1.08

1.078


Dimer in solution: comparison of residence times

solvent-solvent, solvent-monomer: truncated LJ on r = ‖xi − xj‖:

VWCA(r) = 4ε

(σ12

r12− 2

σ6

r6

)if r ≤ σ, 0 otherwise (repulsive potential)

monomer-monomer: double well on r = ‖x1 − x2‖

Reaction coordinate: the distance between the two monomers

β Reference Eff. dyn. Dyn. based on A

0.5 262 ± 6 245 ± 5 504 ± 11

0.25 1.81 ± 0.04 1.68 ± 0.04 3.47 ± 0.08GdR CHANT, Interactions EDPs/Probas, Grenoble, 23-25 novembre 2011 – p. 40

Error estimators


Measuring the error

We wish to measure the “distance” between the reference evolution ξ(Xt) andthe proposed effective dynamics zt.Several ways to do so:

for any t, compare the densities of the random variables ξ(Xt) and zt.

⊖ time correlations are lost

⊕ this will already tell us a lot

compare the law of the paths (ξ(Xt))0≤t≤T and (zt)0≤t≤T

difficult from the analytical viewpoint

residence times previously shown are an example of functionalsdepending on the law of the paths

observation: predicted good accuracy in the sense of the first criterionabove ⇐⇒ good accuracy in term of residence times

pathwise estimates:

E

[sup

0≤t≤T|ξ(Xt) − zt|2

]≤ ? (ongoing work with T. Lelièvre and S. Olla)


Error estimator: estimates on the lawof the random variables ξ(Xt) and zt


Some background materials: relative entropy


2β−1 dWt

Let ψ(t, x) be the probability distribution function of Xt:

P (Xt ∈ B) =

∫

B

ψ(t, x) dx

Under mild assumptions, ψ(t, x) converges to ψ∞(x) = Z−1 exp(−βV (x)).


Some background materials: relative entropy


2β−1 dWt

Let ψ(t, x) be the probability distribution function of Xt:

P (Xt ∈ B) =

∫

B

ψ(t, x) dx

Under mild assumptions, ψ(t, x) converges to ψ∞(x) = Z−1 exp(−βV (x)).

To measure the difference between ψ(t, x) and ψ∞(x), we introduce therelative entropy

H (ψ(t, ·)|ψ∞) :=

∫ψ(t, ·) ln

ψ(t, ·)ψ∞

Since u lnu− u+ 1 ≥ 0 for all u > 0, we have, taking u =ψ(t, ·)ψ∞

, and

multiplying by ψ∞, that

ψ(t, ·) lnψ(t, ·)ψ∞

− ψ(t, ·) + ψ∞ ≥ 0

hence H (ψ(t, ·)|ψ∞) ≥ 0 (and equal to 0 if and only if ψ(t, ·) ≡ ψ∞).


Some background materials: logarithmic Sobolev inequality

H (ψ(t, ·)|ψ∞) :=

∫ψ(t, ·) ln

ψ(t, ·)ψ∞

We also have (Csiszar-Kullback inequality)

‖ψ(t, ·) − ψ∞‖L1 ≤√

2H (ψ(t, ·)|ψ∞)

(control on H hence implies control on the difference of averages ofobservables wrt to ψ(t, ·) and ψ∞).

Definition A probability measure ψ∞ satisfies a logarithmic Sobolev inequalitywith a constant ρ > 0 if, for any probability measure ϕ,

H(ϕ|ψ∞) ≤ 1

2ρI(ϕ|ψ∞)

where the Fisher information I(ϕ|ψ∞) is defined by

I(ϕ|ψ∞) =

∫ ∣∣∣∣∇ ln

(ϕ

ψ∞

)∣∣∣∣2

ϕ.


Some background materials: exponential convergence


2β−1 dWt (3)

Theorem Consider Xt solution to (3), and assume the stationary measureψ∞(x) dx = Z−1 exp(−βV (x)) dx satisfies a logarithmic Sobolev inequalitywith a constant ρ > 0. Then the probability distribution ψ(t, ·) of Xt convergesto ψ∞ exponentially fast, in the sense:

∀t ≥ 0, H(ψ(t, ·)|ψ∞) ≤ H(ψ(0, ·)|ψ∞) exp(−2ρβ−1t). (4)

Conversely, if (4) holds for any initial condition ψ(0, ·), then the stationarymeasure ψ∞(x) dx satisfies a logarithmic Sobolev inequality with theconstant ρ.

proof

Keep in mind that, the larger the logarithmic-Sobolev constant ρ is, the fasterthe convergence to equilibrium.


Other entropies - 1

We have considered the relative entropy

H (ψ(t, ·)|ψ∞) =

∫ψ(t, ·) ln

ψ(t, ·)ψ∞

Alternative:

E2(t) := H2 (ψ(t, ·)|ψ∞) :=

∫ψ∞

(ψ(t, ·)ψ∞

− 1

)2

=

∥∥∥∥ψ(t, ·)ψ∞

− 1

∥∥∥∥2

L2(ψ∞)

Using∂ψ

∂t=

1

βdiv

[ψ∞∇

(ψ

ψ∞

)]

we obtaindE2

dt= − 2

β

∫ ∣∣∣∣∇(ψ(t, ·)ψ∞

)∣∣∣∣2

ψ∞


Other entropies - 2

Assume that the following Poincaré-Wirtinger type inequality holds:

∥∥∥∥u−−∫u

∥∥∥∥L2(ψ∞)

≤ C ‖∇u‖L2(ψ∞) for u =ψ(t, ·)ψ∞

Since −∫u =

∫uψ∞ =

∫ψ(t, ·) = 1, we deduce

∥∥∥∥ψ(t, ·)ψ∞

− 1

∥∥∥∥L2(ψ∞)

≤ C

∥∥∥∥∇(ψ(t, ·)ψ∞

)∥∥∥∥L2(ψ∞)

which exactly reads

E2(t) ≤ −C2 β

2

dE2

dt.

Using the Gronwall lemma, we deduce the exponential convergence ofE2(t) = H2 (ψ(t, ·)|ψ∞) to 0.


Back to the effective dynamics: a convergence result


2β−1dWt, consider ξ(Xt)

Let ψexact(t, z) be the probability distribution function of ξ(Xt):

P (ξ(Xt) ∈ I) =

∫

I

ψexact(t, z) dz

On the other hand, we have introduced the effective dynamics

dzt = b(zt) dt+√

2β−1 σ(zt) dBt

Let φeff(t, z) be the probability distribution function of zt.

Introduce the error

E(t) :=

∫

R

ψexact(t, ·) lnψexact(t, ·)φeff(t, ·)

We would like ψexact ≈ φeff , e.g. E small . . .


Decoupling assumptions

Σz = X ∈ Rn, ξ(X) = z , dµz ∝ exp(−βV (X)) δξ(X)−z

assume that dµz, the Gibbs measure restricted to Σz, can be easilysampled: they satisfy a logarithmic Sobolev inequality with a largeconstant ρ≫ 1 (no metastability)





assume that the coupling between the dynamics of ξ(Xt) and thedynamics in Σz is weak:If ξ(x, y) = x, we request ∂xyV to be small.In the general case, recall that the free energy derivative reads

A′(z) =

∫

Σz

F (X)dµz

We assume that max |∇ΣzF | ≤ κ.





assume that the coupling between the dynamics of ξ(Xt) and thedynamics in Σz is weak:If ξ(x, y) = x, we request ∂xyV to be small.In the general case, recall that the free energy derivative reads

A′(z) =

∫

Σz

F (X)dµz

We assume that max |∇ΣzF | ≤ κ.

assume that |∇ξ| is close to a constant on each Σz, e.g.

λ = maxX

∣∣∣∣|∇ξ|2(X) − σ2(ξ(X))

σ2(ξ(X))

∣∣∣∣ is small


Error estimate

E(t) = error =

∫

R


Under the above assumptions, for all t ≥ 0,

E(t) ≤ C(ξ, Initial Cond.)(λ+

β2κ2

ρ2

)

Hence, if the coarse variable ξ is such that

ρ is large (no metastability in Σz),

κ is small (small coupling between dynamics in Σz and on zt),

λ is small (|∇ξ| is close to a constant on each Σz),

then the effective dynamics is accurate:

at any time, law of ξ(Xt) ≈ law of zt.

Remark: this is not an asymptotic result, and this holds for any ξ.


Rough estimation in a particular case

Standard expression in MD: Vε(X) = V0(X) +1

εq2(X) : ∇q ≡ fast direction

E(t) = error =

∫

R


The measure dµz concentrates around X s.t. ξ(X) = 0, and locally looks like aGaussian measure of variance O(ε), hence ρ = O(1/ε).

Estimate κ: A′(z) =

∫

Rn

F (X)dµz, with F =∇Vε · ∇ξ|∇ξ|2 − β−1 div

( ∇ξ|∇ξ|2

).


Rough estimation in a particular case

Standard expression in MD: Vε(X) = V0(X) +1

εq2(X) : ∇q ≡ fast direction

E(t) = error =

∫

R


The measure dµz concentrates around X s.t. ξ(X) = 0, and locally looks like aGaussian measure of variance O(ε), hence ρ = O(1/ε).

Estimate κ: A′(z) =

∫

Rn

F (X)dµz, with F =∇Vε · ∇ξ|∇ξ|2 − β−1 div

( ∇ξ|∇ξ|2

).

If ∇ξ · ∇q = 0, then the direction ∇ξ is decoupled from the fast direction∇q, hence ξ is indeed a slow variable, and it turns out that E(t) = O(ε).

If ∇ξ · ∇q 6= 0, then the variable ξ does not contain all the slow motion,and bad scale separation: E(t) = O(1), hence the laws of ξ(Xt) and of ztare not close one to each other.

The condition ∇ξ · ∇q = 0 seems important to obtain good accuracy (in termsof laws of random variables).


Tri-atomic molecule

B

CA

B

C

A

V (X) =1

2k2 (rAB − ℓeq)

2+

1

2k2 (rBC − ℓeq)

2+ k3 WDW(θABC), k2 ≫ k3,

where WDW is a double well potential.

Two possible reaction coordinates:

the angle ξ1(X) = θABC

the distance ξ2(X) = r2AC

ξ1(X) = θABC satisfies the orthogonality condition, whereas ξ2(X) = r2ACdoes not.


Residence times as a function of β

-1

0

1

2

3

4

5

6

7

8

9

1 1.5 2 2.5 3 3.5 4 4.5 5

β

Dyn. driven by A′1

ReferenceEff. dyn. with ξ1

-2

0

2

4

6

8

10

1 1.5 2 2.5 3 3.5 4 4.5 5

β

Dyn. driven by A′2

ReferenceEff. dyn. with ξ2

ξ1 = θABC ξ2 = r2AC

ln(residence time) as a function of β


Accuracy of the effective dynamics

We observe that

if ξ(X) = ξ1(X) = θABC , then orthogonality condition is ok, and wepredict a good accuracy on the distributions of the random variables(theoretical error bound).We numerically observe good accuracy on the residence times (notcovered by our estimator, depends on the law of the paths and not only onthe law of the random variables).

if ξ(X) = ξ2(X) = r2AC , then orthogonality condition is NOT ok. We do notexpect a good accuracy on the distributions of the random variables. Wenumerically observe a bad accuracy on the residence times.


Sketch of the proof in the case X = (x, y), ξ(X) = x

Fokker-Planck (reference dynamics): ∂tψ = div [ψ∇V ] + β−1∆ψ

Let ψexact(t, x) be the law of ξ(Xt) = xt, that is ψexact(t, x) =

∫ψ(t, x, y) dy:

∂tψexact = ∂x

[∫ψ∂xV dy

]+ β−1∂xx [ψexact]

= ∂x

[b(t, x) ψexact(x)


where b is the ideal/exact drift: b(t, x) = −∫ψ(t, x, y)∂xV (x, y) dy.


Sketch of the proof in the case X = (x, y), ξ(X) = x

Fokker-Planck (reference dynamics): ∂tψ = div [ψ∇V ] + β−1∆ψ

Let ψexact(t, x) be the law of ξ(Xt) = xt, that is ψexact(t, x) =

∫ψ(t, x, y) dy:

∂tψexact = ∂x

[∫ψ∂xV dy


= ∂x

[b(t, x) ψexact(x)


where b is the ideal/exact drift: b(t, x) = −∫ψ(t, x, y)∂xV (x, y) dy.

The effective dynamics reads dzt = −A′(zt) dt+√

2/β dBt, with

A′(z) = Eµ [∂xV (X) | ξ(X) = z] = −∫ψ∞(x, y)∂xV (x, y) dy.

The law φeff of zt satisfies ∂tφeff = [A′φeff ]′+ β−1φ′′eff .


Sketch of the proof - 2

E(t) = H(ψexact|φeff) =

∫

R

ln

(ψexact(t, x)

φeff(t, x)

)ψexact(t, x) dx

Time derivative of the error:

dE

dt= −β−1I(ψexact|φeff) +

∫

R

ψexact ∂x

(lnψexact

φeff

)(A′(x) − b(t, x)

)

≤ −β−1I(ψexact|φeff) +1

2α

∫ψexact

(∂x

(lnψexact

φeff

))2

+α

2

∫ψexact

(A′(x) − b(t, x)

)2

=

(1

2α− β−1

)I(ψexact|φeff) +

α

2(∗) (arbitrary α)

Note that

b(t, x) = −∫

R

∂xV (x, y)ψ(t, x, y) dy

A′(x) = −∫

R

∂xV (x, y)ψ∞(x, y) dyGdR CHANT, Interactions EDPs/Probas, Grenoble, 23-25 novembre 2011 – p. 57


A′(x) − b(t, x) =

∫∂xV (x, y) νx1 (y) dy −

∫∂xV (x, y) νt,x2 (y) dy

=

∫(∂xV (x, y1) − ∂xV (x, y2)) k

t,x(y1, y2) dy1 dy2

∣∣∣A′(x) − b(t, x)∣∣∣ ≤ ‖∂xyV ‖L∞

∫|y1 − y2| kt,x(y1, y2) dy1 dy2

with∫kt,x(y1, y2) dy2 = νx1 (y1) and

∫kt,x(y1, y2) dy1 = νt,x2 (y2).



A′(x) − b(t, x) =

∫∂xV (x, y) νx1 (y) dy −

∫∂xV (x, y) νt,x2 (y) dy

=

∫(∂xV (x, y1) − ∂xV (x, y2)) k

t,x(y1, y2) dy1 dy2

∣∣∣A′(x) − b(t, x)∣∣∣ ≤ ‖∂xyV ‖L∞

∫|y1 − y2| kt,x(y1, y2) dy1 dy2

with∫kt,x(y1, y2) dy2 = νx1 (y1) and

∫kt,x(y1, y2) dy1 = νt,x2 (y2). Optimize on

kt,x:∣∣∣A′(x) − b(t, x)

∣∣∣ ≤ ‖∂xyV ‖L∞ W1(νx1 , ν

t,x2 ) [Wasserstein distance]

≤ ‖∂xyV ‖L∞ρ

√I(νt,x2 |νx1 ) [Talagrand + ISL on νx1 ]

Hence,

(∗) =

∫ψexact

(A′(x) − b(t, x)

)2

≤ ‖∂xyV ‖2L∞

ρ2

∫ψexactI(ν

t,x2 |νx1 ) ≤ ‖∂xyV ‖2

L∞

ρ2I(ψ|ψ∞)



dE

dt≤

(1

2α− β−1

)I(ψexact|φeff) +

α

2

‖∂xyV ‖2L∞

ρ2I(ψ|ψ∞)

=

(1

2α− β−1

)I(ψexact|φeff) − αβ‖∂xyV ‖2

L∞

2ρ2∂tH(ψ|ψ∞)

Take 2α = β to cancel the first term:

dE

dt≤ −β

2‖∂xyV ‖2L∞

4ρ2∂tH(ψ|ψ∞)

Integrate in time, with E(0) = 0, and H(ψ(t)|ψ∞) ≥ 0:

E(t) ≤ β2‖∂xyV ‖2L∞

4ρ2H(ψ(t = 0)|ψ∞)

We hence need ‖∂xyV ‖L∞ = ‖∇ΣzF‖L∞ < +∞ and ISL on

νx1 (dy) = ψ∞(x, y) dy, e.g. marginals of dµ at fixed ξ(X).


Remark

We have shown that

‖ψexact(t, ·) − ψeff(t, ·)‖L1 ≤√

2H (ψexact|φeff) ≤ C

Long-time behavior:

we also know that the effective dynamics is ergodic forexp(−βA(z))dz, thus φeff(t, ·) converges exponentially fast toexp(−βA(z))

the reference dynamics is also ergodic, hence ψexact(t, ·) alsoconverges exponentially fast to exp(−βA(z))

we thus expect (and can indeed prove) that

‖ψexact(t, ·) − ψeff(t, ·)‖L1 ≤ C exp(−Rβ−1t)


Effective dynamics in the Langevincase


Extension to the Langevin equation

We have considered until now the overdamped Langevin equation:


2β−1 dWt

Consider now the Langevin equation:

dXt = Pt dt

dPt = −∇V (Xt) dt− γPt dt+√

2γβ−1 dWt

Aim:

build a (tractable) effective dynamics

show equilibrium consistency

numerical results

Joint work with F. Galante and T. Lelièvre.


Building the effective dynamics - 1


2γβ−1 dWt

We compute

d [ξ(Xt)] = ∇ξ(Xt) · Pt dt

We introduce the coarse-grained velocity

v(X,P ) = ∇ξ(X) · P ∈ R

and have (chain rule)

d [v(Xt, Pt)] =[P Tt ∇2ξ(Xt)Pt −∇ξ(Xt)

T∇V (Xt)]dt

− γv(Xt, Pt) dt+√

2γβ−1∣∣∇ξ(Xt)

∣∣ dBt

where Bt is a 1D Brownian motion.

We wish to write a closed equation on ξt = ξ(Xt) and vt = v(Xt, Pt).

Introduce D(X,P ) = P T∇2ξ(X)P −∇ξ(X)T∇V (X).GdR CHANT, Interactions EDPs/Probas, Grenoble, 23-25 novembre 2011 – p. 63

Building the effective dynamics - 2

Without any approximation, we have obtained

dξt = vt dt,

dvt = D(Xt, Pt) dt− γvt dt+√

2γβ−1 |∇ξ(Xt)| dBt

To close the system, we introduce the conditional expectations with respect tothe equilibrium measure µ(X,P ) = Z−1 exp[−β

(V (X) + P TP/2

)]:

Deff(ξ0, v0) = Eµ

(D(X,P ) | ξ(X) = ξ0, v(X,P ) = v0

)

σ2(ξ0, v0) = Eµ

(|∇ξ|2(X) | ξ(X) = ξ0, v(X,P ) = v0

)

Effective dynamics:

dξt = vt dt

dvt = Deff(ξt, vt) dt− γvt dt+√

2γβ−1 σ(ξt, vt) dBt


Equilibrium properties

The equilibrium measure on (ξ(X), v(X,P )) is

exp [−βA(ξ0, v0)] =

∫exp

[−β

(V (X) + P TP/2

)]δξ(X)−ξ0, v(X,P )=v0

which is such that, for any smooth test function Φ, we have

∫

Rn×Rn

Φ(ξ(X), v(X,P )) dµ(X,P ) =

∫

R×R

Φ(ξ0, v0) exp(−βA(ξ0, v0)) dξ0 dv0

Proposition: The measure exp [−βA(ξ0, v0)] dξ0 dv0 is an invariant measureof the effective dynamics.

Under classical assumptions, our effective dynamics is hence ergodic for thecorrect equilibrium measure.


Practical implementation

We need to compute conditionned averages, such as

Deff(ξ0, v0) = Eµ

(D(X,P ) | ξ(X) = ξ0, v(X,P ) = v0

)

One possibility is to use the constrained dynamics

dXt = Pt dt,

dPt = −∇V (Xt) dt− γ Pt dt+√

2γβ−1 dWt + ∇ξ(Xt) dλt,

ξ(Xt) = ξ0,

as proposed by [Leli evre, Rousset and Stoltz, 2011] , to sample the Gibbs measureconditionned at ξ(X) = ξ0, v(X,P ) = 0.


Numerical results: the tri-atomic molecule

B

CA

B

C

A

V (X) =1

2k2 (rAB − ℓeq)

2+

1

2k2 (rBC − ℓeq)

2+ k3 WDW(θABC), k2 ≫ k3,

where WDW is a double well potential.

Reaction coordinate: ξ(X) = θABC .

Inverse temp. Reference Eff. dyn.

β = 1 9.808 ± 0.166 9.905 ± 0.164

β = 2 77.37 ± 1.23 79.1 ± 1.25

Excellent agreement on the residence times in the well.GdR CHANT, Interactions EDPs/Probas, Grenoble, 23-25 novembre 2011 – p. 67

Conclusions

In the framework of SDE, e.g.


2β−1 dWt

we have proposed a “natural” way to obtain a closed equation on ξ(Xt).

encouraging numerical results and rigorous error bounds.

FL, T. Lelièvre, Nonlinearity 23, 2010.FL, T. Lelièvre, Springer LN Comput. Sci. Eng., vol. 82, 2011, arXiv 1008.3792FL, T. Lelièvre, S. Olla, in preparationF. Galante, FL, T. Lelièvre, in preparation

Similar derivations can be performed in the framework of kinetic Monte Carlomodels: S. Lahbabi, F.L., S. Olla, in preparation.


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Energie libre et dynamique réduite en dynamique moléculaire · Energie libre et dynamique réduite en dynamique moléculaire Fred´ eric Legoll´ Ecole des Ponts ParisTech et INRIA