Weinan E. Principles of Multiscale Modeling
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9.3. COARSE-GRAINED MOLECULAR DYNAMICS 417
As µ → ∞,
¯ρ
µ
→ ¯ρ. (9.3.22)
To captur e the marginal density of X, we create an artificial separation of the time
scales associated with x and X by making γ small. In this case, the variables X(·) evolves
much more slowly than x(·) and only feel the average effect of the latter. This puts us
in a situation for which averaging theorems for stochastic ODEs can be applied (see the
beginning of this chapter or Section 2.3). The effective equation governing the dynamics
of X is obtained by averaging the right hand-side of the second equation in (9.3.19) with
respect to the conditional probability density:
ρ
µ
(x|X) = Z
−1
µ
(X)e
−βV (x)−
1
2
µβ|X−q(x)|
2
, (9.3.23)
where
Z
µ
(X) =
Z
R
3N
e
−βV (x)−
1
2
µβ|X−q(x)|
2
dx, (9.3.24)
is a normalization factor. S ince
− Z
−1
µ
(X)
Z
R
n
µ(X − q(x))e
−βV (x)−
1
2
µβ|X−q(x)|
2
dx
= β
−1
Z
−1
µ
(X)∇
X
Z
µ
(X)
= β
−1
∇
X
log Z
µ
(X),
(9.3.25)
the limiting equations for X(·) (9.3.19) as γ → 0 can be written as
˙
X = −∇
X
F
µ
+
p
2β
−1
˙
W, (9.3.26)
where
F
µ
(X) = −β
−1
log Z
µ
(X). (9.3.27)
From (9.3.24) and (9.3.27), we have, upto a constant,
F
µ
(X) → F (X) = −β
−1
log ¯ρ(X) (9.3.28)
as µ → ∞. Therefore the limiting equation for X(t) as γ → 0 and then µ → ∞ is
˙
X = −∇
X
F +
p
2β
−1
˙
W, (9.3.29)
This shows that (9.3.19) can be used in this parameter regime to capture the desired free
energy density for X.
418 CHAPTER 9. PROBLEMS WITH MULTIPLE TIME SCALES
A similar argument shows that (9.3.21) is also the reduced density for the extended
Hamiltonian system
m
¨
x = −∇V (x) + µ(X − q(x))∇q(x)
M
¨
X = −µ(X − q(x)),
(9.3.30)
corresponding to the extended Lagrangian
L(x, X) =
1
2
m|
˙
x|
2
− V (x) +
1
2
M|
˙
X|
2
−
1
2
µ|X − q(x)|
2
. ( 9.3.31)
Here M is an artificial mass parameter associated with the variable X. Therefore the
extended Hamiltonian system (9.3.30) can also be used to compute the free energy asso-
ciated with X. This approach is often called the extended Lagrangian method.
As one can see, the justification given here is only for the equilibrium situ ation.
Therefore at this point, it is only safe to use th ese techniques as sampling techniques
for the equilibrium distribution, even though the formulation might also contain useful
information for dynamics.
9.4 Notes
Numerous relevant topics have been omitted here. Some, such as bo osting and pro-
jective integration, were discussed in Chapter 6 and are not repeated here. An interesting
perspective on HMM, seamless HMM, boosting and projective integrators was presented
in [43]. Also relevant are the exponential integrators, the techniques of using integrating
factors based on Duhammel’s principle, integrators with adaptive time step selection,
and so on. These topics have already been covered in many excellent textbooks or review
articles.
We have focused on coarse-grained molecular dynamics, but there is also a parallel
issue and parallel development in Monte Carlo algorithms. In particular, coarse-grained
Monte Carlo methods have been proposed in [26]. We refer to the review article [27] for
details and more recent development. The basic philosophy is quite similar to that of
CGMD. For this reason, it is also expected to have the same problems discussed above,
namely that the memory effects may not be accur ately modeled. These difficulties can
in principle be overcome using more sophisticated renormalization group techniques in
the Mori-Zwanzig formalism [8].
9.4. NOTES 419
Another interesting idea is the “parareal” algorithm [33]. This was intro du ced as a
strategy for applying domain decomposition methods in time. Obviously the domain
decomposition structure is naturally suited for developing multiscale integrators.
420 CHAPTER 9. PROBLEMS WITH MULTIPLE TIME SCALES
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