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Approximation of Percolation Thresholds William Rogers

Approximation of Percolation Thresholds William Rogers. What is Percolation Theory?. Theory that deals with fluid flow (or a similar process) through a randomly arranged media Developed to mathematically deal with disordered media. What is Percolation Theory?.

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Approximation of Percolation Thresholds William Rogers

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  1. Hirophysics.com Approximation of Percolation Thresholds William Rogers

  2. Hirophysics.com What is Percolation Theory? • Theory that deals with fluid flow (or a similar process) through a randomly arranged media • Developed to mathematically deal with disordered media

  3. Hirophysics.com What is Percolation Theory? • Deals with “sites” of arbitrary dimensions which consist of randomly placed and interconnected sub units • The theory is concerned with the existence of a percolation threshold • States that there is a minimum threshold where a “path” can be found from one end to another

  4. Hirophysics.com Why Study Percolation Theory? Can be used to assist in prediction of: • Propagation of fires through a forest • The distribution of gas inside porous rocks and reservoirs • Electrical resistance in a mixture of two materials

  5. Hirophysics.com Problems • Probabilistic Model • More accuracy = More trials • Experiments can be computationally expensive • Early tests required >36 hours of continuous processing time Solutions • Develop a means of threshold approximation • This experiment was developed to determine if small trials with small sites could be used to approximate the threshold of larger sites

  6. Hirophysics.com Simulations • Two batches • 1,000,000 trial to determine thresholds • 1,000 to approximate • Square Sites • Site Sizes: 5x5, 25x25, 50x50, 100x100 • Rectangular Sites • Two ratios: 2x1 and 1x2 • 1x2 Site Sizes: 5x10, 25x50, 50x100, 100x200 • 2x1 Site Sizes: 10x5, 50x25, 100x50, 200x100

  7. Hirophysics.com Testing Method 1. For each designated bin, make certain amount of trials. (1,000,000 and 1,000 are used.) 2. Calculate the probability of the successful connection for each bin. 3. Plot the all of the probabilities for each bin.

  8. Hirophysics.com Results

  9. Hirophysics.com Results

  10. Hirophysics.com Results

  11. Hirophysics.com Results

  12. Hirophysics.com Results

  13. Hirophysics.com Results

  14. Hirophysics.com Conclusions • Behavior of square sites was accurately approximated by all square site sizes tested • Behavior of rectangular sites varied by size too much for a “one size fits all” approximation • However, small trials approximated the behavior of large trials for the same sized rectangular sites • Efficient approximation • Small trial time, in most cases, was ~0.1% of the large trial time • One exceptional case: 5x5 – 0.003%

  15. Hirophysics.com Further Research Possibilities • Refinement of results – Attempt better accuracy • Further development of approximation techniques • Further testing of rectangular sites • Other site shapes, node shapes and node patterns

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