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Stochastic Simulations Monday, 9/9/2002 Monte Carlo simulations are generally concerned with large series of computer experiments using uncorrelated random numbers. Explore order out of randomness Random sampling Fractoemission Diffusion Polymer Growth model

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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 the Fracto-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 http://polymer.bu.edu/java/java/1drw/1drwapplet.html Wandering ant Try and extract an equation from the plot relating the mean squared distance to the step number.

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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 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. Schematic model for polyethylene

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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 Driven Self Assembly How to grow desired fine nanoscale structures by pre-patterning some coarse structures.

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Monte Carlo vs. Kinetic Monte Carlo

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The probability of a particle escaping a given cell so that it can it can move either to the cell’s right or left can be calculated as: Where is defined.

The probability of a particle escaping a given cell so that it can it can move either to the cell’s right or left can be calculated as: Where is defined.

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