10.34, Numerical Methods Applied to Chemical Engineering
Professor William H. Green
Lecture #31: Fluctuation – Dissipation Theorem.
Fluctuation – Dissipation Theorem
Nobel Prize: Einstein 1905
Drag ~ Brownian motion Æ Diffusion
We say “flux” – But each particle is moving on its own. Only when we consider their
movement over time can we consider there to be flux in a room.
This applies in lots of other physical situations
•
D=
coupling between work and heat
k BT
F drag = −ζ v
ζ
< (δx) 2 >= 2 Dt =
2k B T
ζ
< v > ter min al =
F ext
ζ
Dζ = k B T
t
v = velocity, ζ = drag coefficient
Johnson Noise
< (δV ) 2 >= 2k B TR (ω )
Δω
π
< (δI ) 2 > R 2 = 2k B TR (ω )
< (δI ) 2 >=
2k B T Δω
R (ω ) π
Δω
V = voltage, R = resistance,
π
{V = IR}
Δω = bandwidth of voltmeter
Exxon Example
Find best lubricant
shear on computer
not a good way to
To find viscosity:
1/shear rate < τsimulator
calculate viscosity
because τ is small so
shear rate is high,
Figure 1. Fluid
under shear.
which results in
More practical way
shear thinning and
to do experiment:
an irrelevant
viscosity calculation
Figure 2. Fluctuation-dissipation on static system.
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].
Use fluctuation – dissipation on static system to calculate diffusivity and then use Stokes –
Einstein to obtain viscosity.
Use molecular dynamics for 10 ns or faster. Diffusion of orientation faster than diffusion of
center of mass.
How to get time-dependent quantity? M. D. for 10 nanoseconds
d
/dt p( ) = Ô[p( )]
Master Equation
d
p i ( E ) = − Z collision p i ( E ) − ∑ k i → j ( E ) p i ( E ) + Z collision ∫ PE '→ E p i ( E ' )dE '+ k i → j ( E ) p j ( E )
dt
j
C+D
k(E)
k(E)
B
A
Figure 3. Energy diagram.
k(T,P) since Zcollision depends on P
as P Æ ∞
Æ Boltzmann p(ε) Æ k(T)
10.34, Numerical Methods Applied to Chemical Engineering
Prof. William Green
Lecture 31
Page 2 of 4
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].
Alaska
Alaska
Canada
Canada
radio-labeled
collars tracking
migration of a few
animals
track herd of
caribou from
airplane
INDIVIDUAL
MOLECULE
APPROACH
CONTINUUM
APPROACH
PROBABILISTIC
VIEW
Figure 4. Two methods for tracking caribou as a way to compare the individual molecule to
the continuum approach.
Individual Molecule Approach (Stochastic)
Gillespie Algorithm for Kinetic Monte Carlo
1
= k dissociate (ε ) + k isomerization (ε ) + Z collision
τ total
τ change = τ total (− ln(rand ))
if rand #2 <
if
k dissoc
Æ molecule dissociated
k dissoc + k isom + Z
k dissoc + k isom
> rand #2 >
k dissoc + k isom + Z
k dissoc
Æ molecule isomerized
k dissoc + k isom + Z
else molecule underwent an energy-changing collision
Variance
10.34, Numerical Methods Applied to Chemical Engineering
Prof. William Green
Lecture 31
Page 3 of 4
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].
σ
f
~
1− f
N traj f
if low-probability event (f is small), variance is BIG Æ no good
Alternative: solve master equation deterministically by discretizing.
Xin = pi(En)
discrete {En}
∫ δ (E − E
d
p i ( E )dE = ......
dt
n
)
10.34, Numerical Methods Applied to Chemical Engineering
Prof. William Green
Lecture 31
Page 4 of 4
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].