Stochastic Simulations Monday 992002 Random sampling Fractoemission Diffusion

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

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/1 drwapplet. 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 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

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. 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