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

Hit-or-Miss Random Sampling

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π ⋅12

22 =nhitnall

π =4nhitnall

Buffon’s Needle

Fracto-emission

Fracto-emission Measuring System

Zigzag Crack Profile Model

Fracto-emission particles bounce at the irregular surfaces.

Longtime Decay of theFracto-emission Intensity

Random Walk

Haphazrad paths on a lattice

A drop of ink through water.

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.

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)?

Diffusion

Screen shots of the trajectory of 500 random walkers, started together at the center.

Extension of Random WalkThis 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.

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

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

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

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Rg2 =AN2ν

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 DrivenSelf Assembly

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

Monte Carlo vs.Kinetic Monte Carlo

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pMC =exp−E1 −E2

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