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)).
GdR CHANT, Interactions EDPs/Probas, Grenoble, 23-25 novembre 2011 – p. 2
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.
GdR CHANT, Interactions EDPs/Probas, Grenoble, 23-25 novembre 2011 – p. 2
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.
GdR CHANT, Interactions EDPs/Probas, Grenoble, 23-25 novembre 2011 – p. 3
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.
GdR CHANT, Interactions EDPs/Probas, Grenoble, 23-25 novembre 2011 – p. 3
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.
GdR CHANT, Interactions EDPs/Probas, Grenoble, 23-25 novembre 2011 – p. 4
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).
GdR CHANT, Interactions EDPs/Probas, Grenoble, 23-25 novembre 2011 – p. 5
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).
GdR CHANT, Interactions EDPs/Probas, Grenoble, 23-25 novembre 2011 – p. 5
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).
GdR CHANT, Interactions EDPs/Probas, Grenoble, 23-25 novembre 2011 – p. 7
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, . . . ).
GdR CHANT, Interactions EDPs/Probas, Grenoble, 23-25 novembre 2011 – p. 8
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.
GdR CHANT, Interactions EDPs/Probas, Grenoble, 23-25 novembre 2011 – p. 9
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.
GdR CHANT, Interactions EDPs/Probas, Grenoble, 23-25 novembre 2011 – p. 10
Fast and slow degrees of freedomin molecular simulation
GdR CHANT, Interactions EDPs/Probas, Grenoble, 23-25 novembre 2011 – p. 11
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.
GdR CHANT, Interactions EDPs/Probas, Grenoble, 23-25 novembre 2011 – p. 12
Typical potential energy surfaces
Typical potential energy surfaces (material science and biomolecules)
There are many local minima.
GdR CHANT, Interactions EDPs/Probas, Grenoble, 23-25 novembre 2011 – p. 13
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.
GdR CHANT, Interactions EDPs/Probas, Grenoble, 23-25 novembre 2011 – p. 14
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?
GdR CHANT, Interactions EDPs/Probas, Grenoble, 23-25 novembre 2011 – p. 15
Reference dynamics
We are interested in dynamical properties. Two possible choices for thereference dynamics of the system:
overdamped Langevin equation (our main choice):
dXt = −∇V (Xt) dt+√
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
dXt = Pt dt, dPt = −∇V (Xt) dt− γPt dt+√
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
dXt = −∇V (Xt) dt+√
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
GdR CHANT, Interactions EDPs/Probas, Grenoble, 23-25 novembre 2011 – p. 17
Metastability and reaction coordinate
dXt = −∇V (Xt) dt+√
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
dXt = −∇V (Xt) dt+√
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)
GdR CHANT, Interactions EDPs/Probas, Grenoble, 23-25 novembre 2011 – p. 19
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,
dXt = −∇V (Xt) dt+√
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)).
GdR CHANT, Interactions EDPs/Probas, Grenoble, 23-25 novembre 2011 – p. 19
Effective dynamics in theoverdamped Langevin case
GdR CHANT, Interactions EDPs/Probas, Grenoble, 23-25 novembre 2011 – p. 20
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.
GdR CHANT, Interactions EDPs/Probas, Grenoble, 23-25 novembre 2011 – p. 21
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
GdR CHANT, Interactions EDPs/Probas, Grenoble, 23-25 novembre 2011 – p. 21
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:
GdR CHANT, Interactions EDPs/Probas, Grenoble, 23-25 novembre 2011 – p. 22
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.
GdR CHANT, Interactions EDPs/Probas, Grenoble, 23-25 novembre 2011 – p. 22
The general case: X ∈ Rn and arbitrary ξ - 1
dXt = −∇V (Xt) dt+√
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.
GdR CHANT, Interactions EDPs/Probas, Grenoble, 23-25 novembre 2011 – p. 23
The general case: X ∈ Rn and arbitrary ξ - 1
dXt = −∇V (Xt) dt+√
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.
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.
GdR CHANT, Interactions EDPs/Probas, Grenoble, 23-25 novembre 2011 – p. 23
The general case: X ∈ Rn and arbitrary ξ - 2
dXt = −∇V (Xt) dt+√
2β−1 dWt, ξ : Rn → R
yields
d [ξ(Xt)] =(−∇V · ∇ξ + β−1∆ξ
)(Xt) dt+
√2β−1 |∇ξ(Xt)| dBt
where Bt is a 1D brownian motion.
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
GdR CHANT, Interactions EDPs/Probas, Grenoble, 23-25 novembre 2011 – p. 25
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).
GdR CHANT, Interactions EDPs/Probas, Grenoble, 23-25 novembre 2011 – p. 27
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
GdR CHANT, Interactions EDPs/Probas, Grenoble, 23-25 novembre 2011 – p. 28
Effective dynamics and free energy
GdR CHANT, Interactions EDPs/Probas, Grenoble, 23-25 novembre 2011 – p. 29
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
).
GdR CHANT, Interactions EDPs/Probas, Grenoble, 23-25 novembre 2011 – p. 30
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).
GdR CHANT, Interactions EDPs/Probas, Grenoble, 23-25 novembre 2011 – p. 31
Questionning this natural candidate - 1
Consider the fine scale dynamics
dXt = −∇V (Xt) dt+√
2β−1 dWt
and
dXt = Pt dt, dPt = −∇V (Xt) dt− γPt dt+√
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
dut = −A′(ut) dt+√
2β−1 dBt
would always be a good approximation?
GdR CHANT, Interactions EDPs/Probas, Grenoble, 23-25 novembre 2011 – p. 32
Questionning this natural candidate - 2
dut = −A′(ut) dt+√
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?
GdR CHANT, Interactions EDPs/Probas, Grenoble, 23-25 novembre 2011 – p. 33
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
dut = −A′(ut) dt+√
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′.
GdR CHANT, Interactions EDPs/Probas, Grenoble, 23-25 novembre 2011 – p. 34
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).
GdR CHANT, Interactions EDPs/Probas, Grenoble, 23-25 novembre 2011 – p. 35
Numerical results
GdR CHANT, Interactions EDPs/Probas, Grenoble, 23-25 novembre 2011 – p. 36
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).
GdR CHANT, Interactions EDPs/Probas, Grenoble, 23-25 novembre 2011 – p. 37
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
GdR CHANT, Interactions EDPs/Probas, Grenoble, 23-25 novembre 2011 – p. 39
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
GdR CHANT, Interactions EDPs/Probas, Grenoble, 23-25 novembre 2011 – p. 41
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)
GdR CHANT, Interactions EDPs/Probas, Grenoble, 23-25 novembre 2011 – p. 42
Error estimator: estimates on the lawof the random variables ξ(Xt) and zt
GdR CHANT, Interactions EDPs/Probas, Grenoble, 23-25 novembre 2011 – p. 43
Some background materials: relative entropy
dXt = −∇V (Xt) dt+√
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)).
GdR CHANT, Interactions EDPs/Probas, Grenoble, 23-25 novembre 2011 – p. 44
Some background materials: relative entropy
dXt = −∇V (Xt) dt+√
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∞).
GdR CHANT, Interactions EDPs/Probas, Grenoble, 23-25 novembre 2011 – p. 44
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.
GdR CHANT, Interactions EDPs/Probas, Grenoble, 23-25 novembre 2011 – p. 45
Some background materials: exponential convergence
dXt = −∇V (Xt) dt+√
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.
GdR CHANT, Interactions EDPs/Probas, Grenoble, 23-25 novembre 2011 – p. 46
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∞
GdR CHANT, Interactions EDPs/Probas, Grenoble, 23-25 novembre 2011 – p. 47
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.
GdR CHANT, Interactions EDPs/Probas, Grenoble, 23-25 novembre 2011 – p. 48
Back to the effective dynamics: a convergence result
dXt = −∇V (Xt) dt+√
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 . . .
GdR CHANT, Interactions EDPs/Probas, Grenoble, 23-25 novembre 2011 – p. 49
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)
GdR CHANT, Interactions EDPs/Probas, Grenoble, 23-25 novembre 2011 – p. 50
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.
GdR CHANT, Interactions EDPs/Probas, Grenoble, 23-25 novembre 2011 – p. 50
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 |∇ξ| is close to a constant on each Σz, e.g.
λ = maxX
∣∣∣∣|∇ξ|2(X) − σ2(ξ(X))
σ2(ξ(X))
∣∣∣∣ is small
GdR CHANT, Interactions EDPs/Probas, Grenoble, 23-25 novembre 2011 – p. 50
Error estimate
E(t) = error =
∫
R
ψexact(t, ·) lnψexact(t, ·)φeff(t, ·)
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.
GdR CHANT, Interactions EDPs/Probas, Grenoble, 23-25 novembre 2011 – p. 51
Rough estimation in a particular case
Standard expression in MD: Vε(X) = V0(X) +1
εq2(X) : ∇q ≡ fast direction
E(t) = error =
∫
R
ψexact(t, ·) lnψexact(t, ·)φeff(t, ·)
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
).
GdR CHANT, Interactions EDPs/Probas, Grenoble, 23-25 novembre 2011 – p. 52
Rough estimation in a particular case
Standard expression in MD: Vε(X) = V0(X) +1
εq2(X) : ∇q ≡ fast direction
E(t) = error =
∫
R
ψexact(t, ·) lnψexact(t, ·)φeff(t, ·)
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).
GdR CHANT, Interactions EDPs/Probas, Grenoble, 23-25 novembre 2011 – p. 52
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.
GdR CHANT, Interactions EDPs/Probas, Grenoble, 23-25 novembre 2011 – p. 53
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
GdR CHANT, Interactions EDPs/Probas, Grenoble, 23-25 novembre 2011 – p. 54
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.
GdR CHANT, Interactions EDPs/Probas, Grenoble, 23-25 novembre 2011 – p. 55
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)
]+ β−1∂xx [ψexact]
where b is the ideal/exact drift: b(t, x) = −∫ψ(t, x, y)∂xV (x, y) dy.
GdR CHANT, Interactions EDPs/Probas, Grenoble, 23-25 novembre 2011 – p. 56
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)
]+ β−1∂xx [ψexact]
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 .
GdR CHANT, Interactions EDPs/Probas, Grenoble, 23-25 novembre 2011 – p. 56
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
Sketch of the proof - 3
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).
GdR CHANT, Interactions EDPs/Probas, Grenoble, 23-25 novembre 2011 – p. 58
Sketch of the proof - 3
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∞)
GdR CHANT, Interactions EDPs/Probas, Grenoble, 23-25 novembre 2011 – p. 58
Sketch of the proof - 4
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).
GdR CHANT, Interactions EDPs/Probas, Grenoble, 23-25 novembre 2011 – p. 59
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)
GdR CHANT, Interactions EDPs/Probas, Grenoble, 23-25 novembre 2011 – p. 60
Effective dynamics in the Langevincase
GdR CHANT, Interactions EDPs/Probas, Grenoble, 23-25 novembre 2011 – p. 61
Extension to the Langevin equation
We have considered until now the overdamped Langevin equation:
dXt = −∇V (Xt) dt+√
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.
GdR CHANT, Interactions EDPs/Probas, Grenoble, 23-25 novembre 2011 – p. 62
Building the effective dynamics - 1
dXt = Pt dt, dPt = −∇V (Xt) dt− γPt dt+√
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
GdR CHANT, Interactions EDPs/Probas, Grenoble, 23-25 novembre 2011 – p. 64
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.
GdR CHANT, Interactions EDPs/Probas, Grenoble, 23-25 novembre 2011 – p. 65
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.
GdR CHANT, Interactions EDPs/Probas, Grenoble, 23-25 novembre 2011 – p. 66
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.
dXt = −∇V (Xt) dt+√
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.
GdR CHANT, Interactions EDPs/Probas, Grenoble, 23-25 novembre 2011 – p. 68