10.34, Numerical Methods Applied to Chemical Engineering
Professor William H. Green
Lecture #30: Modeling intrinsically Stochastic processes. Multiscale Modeling.
∫ ∫ " ∫ ∫ f (q)w(q)d
I=
∫ ∫ " ∫ ∫ w(q)d q
N
q
p(q) =
N
w(q )
∫ ∫ " ∫ ∫ w(q)d
N
q
I = ∫ ∫ " ∫ ∫ f (q ) p (q )d N q
N
lim
∑ f (q )
N →∞
i
i =1
N
= I where N is the number of sample points and qi is drawn from the partition
function p(q).
N
Will have statistical sampling error
dI ≈
1
N
δI MC ≈
f
2
− f
N
2
where
f
2
=
∑ f (q )
2
i
i =1
N
← does not converge very rapidly
~
f = f- < f > small
⎛f⎞
I = ∫ ∫ " ∫ ∫ ⎜⎜ ⎟⎟ pd N q
⎝ p⎠
→ best weight p ~ | f |
Suppose one samples 106 points. To obtain an additional significant figure requires 108
points, so it is difficult to acquire another significant figure. Often, only w(q) is known, not
p(q)
Metropolis Method
{q, …, qN}
randomly choose dq
if w(qN + δq) ≥ w(qN)
or if rand <
p(qN + δq)
w(qN + δq)
_________ = __________
p(q)
w(qN)
qN+1 = qN + δq
else
↓
qN+1 = qN
Cite as: William Green, Jr., course materials for 10.34 Numerical Methods Applied to Chemical Engineering, Fall
2006. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD
Month YYYY].
System in Thermal Equilibrium
p=
e − E ( q ) / k BT
Q(T )
w = e − E ( q ) / k BT
U = <E>
Ch. 10 of Tester and Modell 1
<E2> - <E>2 = kBT2
d<E>
B
/dT = kBT2CV
B
Real systems have fluctuations.
<N2> - <N>2 = kBT<N>2KT/V
B
KT ≡
− 1 ⎛ ∂V ⎞
⎜
⎟
V ⎝ ∂P ⎠ T
If the number of particles N is small, the above difference is noticeable.
N ± N i.e. 10 6 ± 10 3
The reason fluctuations do not have to be considered in many systems is N = 1020 or
greater. Thus, 1020+ 1010 shows the fluctuations do not affect greatly the significant
figures. In nanoscale systems, biological systems, and ignition events, the volumes are
small so then fluctuations are noticeable.
Bayesian Experiments vs. Models
Markov Chain Monte Carlo
I = ∫ ∫ " ∫ ∫ f (Ymodel (θ )) p (θ ) d N θ
Molecular Dynamics (MD)
Time-Dependent Case, Not at Equilibrium, and Probability Density. Although the equations
below are used, they are not correct, because atoms do not move according to Newtonian
Mechanics.
Assume in Thermal Equilibrium
df − 1 ∂E (q )
=
+ F thermal (rand )
dt
m ∂q
The second term on the right hand side represents a ‘kick’ to the system.
dq
dt
= P. / m MatLab Notation
IR “Spectroscopy”
100 cm-1 Æ 3000 cm-1
x–H
10.34, Numerical Methods Applied to Chemical Engineering
Prof. William Green
Lecture 30
Page 2 of 3
Cite as: William Green, Jr., course materials for 10.34 Numerical Methods Applied to Chemical Engineering, Fall
2006. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD
Month YYYY].
τeffect ~ 100 s
τfast ~ 10-14 s “we have a stiffness problem”
Double precision explicit solver Æ Verlet algorithm
(10-14)(106) ~ 10-8 s = 10 nanoseconds
Stiff solvers (need Jacobian), number of variables less than 104. Cannot solve biological
problems.
Many problems use combined atoms (no hydrogens), remove bond stretches, keep bond
torsions and bends.
Be careful of error bars when using these methods.
Multiscale modeling
Unsolved Problem.
10.34, Numerical Methods Applied to Chemical Engineering
Prof. William Green
Lecture 30
Page 3 of 3
Cite as: William Green, Jr., course materials for 10.34 Numerical Methods Applied to Chemical Engineering, Fall
2006. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD
Month YYYY].