Stochastic Simulations

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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
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Hit-or-Miss Random Sampling
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Buffons Needle
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Fracto-emission
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Fracto-emission Measuring System
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Zigzag Crack Profile Model
Fracto-emission particles bounce at the irregular
surfaces.
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Longtime Decay of theFracto-emission Intensity
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Random Walk
Haphazrad paths on a lattice
A drop of ink through water.
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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.ht
ml
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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 reverse-biased motion)?
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Diffusion
Screen shots of the trajectory of 500 random
walkers, started together at the center.
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Extension of Random Walk
This model is a two-dimensional 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.
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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.
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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.
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Polymers as Long Molecular Chains
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Self-Avoiding Random Walk
Mean square distance of gyration of a linear
polymer molecule consists of N monomer unites has
the leading asymptotic behavior
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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.
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DLA Growth Model
http//apricot.polyu.edu.hk/lam/dla/dla.html
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Thermodynamic Force DrivenSelf Assembly
How to grow desired fine nanoscale structures by
pre-patterning some coarse structures.
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Monte Carlo vs.Kinetic Monte Carlo