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Continuous Path Brownian Trajectories for Diusion
Monte Carlo via First- and Last-Passage Distributions
Chi-Ok Hwang, Michael Mascagni and James A. Given
Department of Computer Science
Florida State University
Tallahassee, FL 32306-4530 USA
E-mail: chwang@csit.fsu.edu
URL: http://www.cs.fsu.edu/chwang/reserach.html
Research supported by DOE/ASCI, ARO, and NSF
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Table of Contents
linearized Poisson-Boltzmann Equation
First Passage (FP) Algorithms
\Walk on Spheres" (WOS) Algorithm
Green's Function First Passage (GFFP) Algorithm
{ Angle-averaging Method
Fluid Permeability
{ Simulation-Tabulation GF Method
Solc-Stockmayer Model
Mean Trapping Rate
Eective Conductivity
Feymann-Kac Formula
{
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Table of Contents
Last Passage Algorithms
Approach from Outside
Charge Density on a Circular Disk
Future Work
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Dirichlet problem for Laplace Equation
@
( )
X @ first−passage location
x; starting point
Ω
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First Passage Algorithms
Z
s
y
y
Z
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u(x) = q(x); x 2 (1)
u(x) = f (x);
x 2 @ :
(2)
@G(x; y )
u(x) =
f (y )dS +
G(x; y )q (y )dy
(3)
@n
Isomorphism: First passage probability is exactly the Laplacian
surface Green's function for the boundary value problem for
the given geometry.
Electrostatic charge density on a conducting rst-passage
surface induced by a charge to rst-passage probability density
on the absorbing rst-passage surface surrounding the diusing
particle.
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\Walk on Spheres" (WOS) and Green's
Function First Passage (GFFP)
Algorithms
WOS:
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Shell thickness
ε
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Timing with WOS:
3500
running time (secs)
3000
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2500
2000
1500
1000
500 −8
10
logarithmic regression
10
−7
10
−6
10
−5
−4
10
ε
10
−3
10
−2
10
−1
10
0
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Green's Functions
O
O
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(a) Putting back
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(b) Void space
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(c) Intersecting
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GFFP:
δ
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2000
1975
running time (secs)
1950
1925
1900
1875
1850
1825
ε = 0.0000
ε = 0.0003
ε = 0.0005
1800
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1775
1750
0
0.05
0.1
0.15
0.2
0.25
δ
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Darcy's Law
∆P : pressure difference
Fluid
flux
q
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Porous medium
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Fluid
thickness
L
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k ∆P
q = −−−−−−−
ηL
η : dynamic viscosity
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P
= kL
Darcy's Law
q
A slow steady ow of a single Newtonian uid through a
stationary, inert and isotropic porous medium
q :
L :
P :
k :
uid permeability
applied pressure dierence
thickness of the porous sample
: dynamic viscosity
volumetric ow per unit cross-sectional area (ux)
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(4)
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Hubbard-Douglas TFC for an Arbitrary
Nonskew Object
f
= 6C
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(5)
Using angle averaging, Hubbard and Douglas derived: vector or
tensor Laplace equations to scalar Laplace equations
f :
C :
translational friction coeÆcient
: dynamic viscosity
electrical capacitance
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2
rP = r V
0
2
k
= 6RG ()f1 + 23
V;
( )g
G0 1
(6)
(7)
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Felderhof's TFC for a Macromolecule
f
From the Debijf-Brinkman equation for porous ow under
pressure:
R :
translational friction coeÆcient
: dynamic viscosity
macromolecular radius
f :
we get:
{
{
{
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G0
= pR
() = 1 1 tanh k
(8)
(9)
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Felderhof's TFC for a Macromolecule
with:
uid permeability
dimensionless quantity:
k :
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Porous Media
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= pR
2
( )g
G0 1
Unit Capacitance Method
0
k
Combining two previous equations
C
3
=
G ( )f1 +
2
R
Getting C=R gives us From:
we get permeability
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(10)
(11)
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Sampling Methods
Sharp Boundary Method
sampling sphere
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intersected fragment of a sphere
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Polydispersed Spheres
10
2
10
1
10
0
Permeability
−1
10
−2
10
−3
10
−4
10
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explicit Stokes equation solution
cubic spline fit of explicit Stokes equation solution
angle−averaging estimate
−5
10
−6
10
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0
0.1
0.2
0.3
0.4
0.5 0.6
Porosity
0.7
0.8
0.9
1
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Summary and Conclusions for Permeability
porous media models
< 0:2).
Slide
Applying the new permeability estimation methods to more realistic
porous media.
The methods will work for general homogeneous and isotropic
simulation data.
spheres beds from N. S. Martys agrees well with the sharp boundary
The actual data set of mono-sized and polydispersed overlapping
one minute of 233 Mhz PII CPU time (very fast).
5
Each data point used 10 trajectories and consumed approximately
very low porosities (0
For overlapping sphere beds, the new estimate is good except at
each other except at very low porosities.
The two methods with sharp boundary sampling agree well with
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Solc-Stockmayer Model without Potential
Basic model for diffusion−limited protein−ligand binding
circular reactive patch (absorbing)
b
Θ
a
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diffusing small ligand
Ω
nonreactive (reflecting)
L
: launching
sphere
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Simulation-Tabulation Method
Green's function for the non-intersected surface of a sphere
located on the surface of a reecting sphere
Ω2
Absorbing Sphere
r2
Ω 1 O2
r1
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Reflecting Sphere
O1
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Solc-Stockmayer Model without Potential
1
κ/(4πDa)
0.8
0.6
0.4
6
Simulation (ntrj=10 )
Exact
0.2
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0
0
0.1
0.2
0.3
0.4
0.5
Θ/π
0.6
0.7
0.8
0.9
1
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Solc-Stockmayer Model without Potential
Timing with WOS :
3500
running time (secs)
3000
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2500
2000
1500
1000
500 −8
10
logarithmic regression
10
−7
10
−6
10
−5
−4
10
ε
10
−3
10
−2
10
−1
0
10
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Solc-Stockmayer Model without Potential
Timing with GFFPA :
1600
running time (secs)
1550
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1500
spline fit
1450
1400
1350
1300
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
δ
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Mean Trapping Rate
In a domain of nonoverlapping spherical traps :
δ
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Simulation-Tabulation Method
Average First-passage time
r3
r2
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...
r1
r 2i
τ = Σ τi = Σ −−−
6
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Mean Trapping Rate
In a domain of nonoverlapping spherical traps :
5
4
3
t
*
Zheng−Chiew simulation
our simulation
2
1
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0
0.05
0.1
0.15
0.2
φ2
0.25
0.3
0.35
0.4
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Mean Trapping Rate
In a domain of nonoverlapping spherical traps : Timing with WOS
2500
running time (secs)
2000
1500
1000
500
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logarithmic regression
0 −8
10
10
−7
10
−6
10
−5
−4
10
ε
10
−3
10
−2
10
−1
0
10
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Mean Trapping Rate
In a domain of nonoverlapping spherical traps : Timing with
GFFPA
1400
running time (secs)
1300
quadratic regression
1200
1100
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1000
0
0.05
0.1
δ
0.15
0.2
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Eective Conductivity
Perfectly insulating, nonoverlapping spherical inclusions :
σe =
X2
6 τe (X )
X
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Eective Conductivity
Perfectly insulating, nonoverlapping spherical inclusions :
1
0.9
Kim−Torquato
Three−point upper bound
Our simulation
0.8
0.7
σe/σ1
0.6
0.5
0.4
0.3
0.2
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0.1
0
0
0.1
0.2
0.3
0.4
0.5
φ2
0.6
0.7
0.8
0.9
1
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Eective Conductivity
Perfectly insulating, nonoverlapping spherical inclusions : Timing
with WOS
1000
running time (secs)
800
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600
400
200
logarithmic regression
0 −8
10
10
−7
10
−6
10
−5
−4
10
δ
10
−3
10
−2
10
−1
0
10
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Eective Conductivity
Perfectly insulating, nonoverlapping spherical inclusions : Timing
with GFFPA
190
running time (secs)
170
150
130
110
90
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quadratic regression
70
50
0
0.1
0.2
0.3
0.4
0.5
δ
0.6
0.7
0.8
0.9
1
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g]
dt ;
x2
x 2 @
D
Z x
0
2
Feynman-Kac Formula
2
0
x
2
d
E [exp( t)] =
sinh
d
(x) = E [0(XDx ) expf
= 0;
(x) = (x);
Linearized Poisson-Boltzmann Equation:
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(12)
(13)
(14)
(15)
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1.1
1.0
0.9
ψ/ψ0
0.8
0.7
0.6
0.5
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0.4
−1.5
−1.0
−0.5
0.0
0.5
−1
r (in units of κ )
1.0
1.5
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Last Passage Algorithms
Slide
Charge distribution on a conducting object with edges and
corners.
Problems in which a large fraction of the absorption takes
place on a very small fraction of the surface.
Problems in which more than one conducting object is present,
at close proximity, and at dierent voltages.
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Approach from Outside
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Approach from Outside
() =
P x
I
2
I
(
=0
(
) ( 1)
g x; y; 2
d yg x; y; p y;
d
)
1 d (x) = 1 d
(x) =
4
d
4
d
( ) = 41 d yg(x; y)p(y; 1);
x
( ) =
g x; y
=0
d =0
()
P x
probability of diusing to innity of a diusing particle
initiated at x -distant (very near) to the lower FP surface
where
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(16)
(17)
(18)
(19)
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Charge Density on a Circular Disk
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2
2
0
2
r2
@q
=x +y +z
3
I
b2
3
2
p
p
2
b
+ (r
b2
2
2
) + 4b x
2
A;
1
Charge Density on a Circular Disk
r2
g
= 3 cos 4
a
3 dS cos p(r; 1);
(x) =
16
a
where
(r; 1) = 1 2 arctan
p
where
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(20)
(21)
(22)
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Charge Density on a Circular Disk
charge density on a circular disk
10
9
analytic
simulation
charge density (σ/σ0)
8
7
6
5
4
3
2
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1
0
0
0.1
0.2
0.3
0.4
0.5
r
0.6
0.7
0.8
0.9
1
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Future Work
Extension of the Green's function library
{ Using analytic solutions
{ Sampling new Green's functions
Extension of these methods to other PDEs
{ Poisson equation
{ Linearized Poisson Boltzmann equation
{ Time-independent Schrodinger equation
More applications such as biological problems
Material Science
Biomolecular Science
{
{
Quasirandom walks
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