Transcript Stress Matrix and MATLAB basics

Stochastic Simulations
Monday, 9/9/2002
•Random sampling
•Fractoemission
•Diffusion
•Polymer
•Growth model
Monte Carlo simulations are generally concerned with large
series of computer experiments using uncorrelated random
numbers.
Explore order out of randomness
Hit-or-Miss Random Sampling
 12
2
2
nhit

nall
nhit
 4
nall
Buffon’s Needle
Fracto-emission
Fracto-emission Measuring System
Zigzag Crack Profile Model
Fracto-emission particles bounce at the irregular surfaces.
Longtime Decay of the
Fracto-emission Intensity
Random Walk
Haphazrad paths on a lattice
A drop of ink through water.
One Dimensional Random Walk
Wandering ant
Try and extract an equation from the plot relating
the mean squared distance to the step number.
http://polymer.bu.edu/java/java/1drw/1drwapplet.html
Question
How do the answers change is the probability is p (!= 1/2)
to move right and 1-p to move left (a forward- or reversebiased motion)?
Diffusion
Screen shots of the trajectory of 500 random walkers, started
together at the center.
Extension of Random Walk
This model is a twodimensional extension of
a random walk.
Displayed is the territory
covered by 500 random
walkers. As the number
of walkers increases the
resulting interface
becomes more smooth.
Different kinds of random walks on a square lattice
Random Walk (RW): the walker may cross the walk in an infinite number of
times with no cost.
Self-Avoiding Walk (SAW): the walker dies when attempting to intersect a
portion of the already completed walk.
Growing Self-Avoiding Walk (GSAW): the process proceeds at first as for
SAWs, but a walker senses a ‘trap’ and chooses instead between the remaining
‘safe’ directions so that it can cancontinue to grow.
Polymer Model
Schematic model for
polyethylene
Bond lengths of polymers tend to be rather fixed as do bond angles.
Thus, as a more computationally friendly model we may construct a
polymer which is made up of bonds which connect nearest neighbor
sites (monomers) on a lattice.
Polymers
as Long
Molecular
Chains
Self-Avoiding Random Walk
Mean square distance of gyration
of a linear polymer molecule
consists of N monomer unites
has the leading asymptotic
behavior
Rg2  AN 2
Diffusion Limited Aggregation (DLA)
A seed is placed at the center of the box. A point is chosen at
random in the box, excluding a zone around the cluster. A particle
then random walks from this point until it either sticks to the cluster
or is lost from the box.
DLA Growth Model
http://apricot.polyu.edu.hk/~lam/dla/dla.html
Thermodynamic Force Driven
Self Assembly
How to grow desired fine nanoscale structures by pre-patterning
some coarse structures.
Monte Carlo vs.
Kinetic Monte Carlo
pMC
 E1  E 2 

 exp
 k B T 
pkMC
 E1  E 2   E A 
exp 

 exp 
 k B T   kB T 