1. Approximation of Percolation Thresholds
William Rogers
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2. 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
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3. 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
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4. 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
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5. Problems Solutions 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
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6. Simulations Two batches Square Sites Rectangular Sites
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
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7. 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.
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8. Results
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9. Results
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10. Results
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11. Results
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12. Results
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13. Results
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14. 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%
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15. 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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