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--- dev:quiz_ulam [2015/06/15 02:37] – xychen
+++ dev:quiz_ulam [2015/06/28 17:52] (current) – xychen
@@ Line 46: / Line 46: @@
 The number of pairs is actually 1/2 of the number of ''Pbox''. Then we get some pictures of the time for ''Boxes'' transform to ''Pbox'' and finally one one or two pairs are left walking.
 {{:dev:1c1_pairs_plot.png?210 |}}   {{:dev:2c2_pairs_plot.png?215 |}}
+==== Boxes Experiments ====
+Here are some more experiments on the dancing of ''Boxes'' and ''Pairs''. We assume that a dancing ''Pair'' can leap further if it encounters some ''Boxes'' on its trail. This is possible because a standalone ''Box'' may replace one //dancer// of the ''Pair'' to form a new pair. And the visual effect seems that the ''Pair'' leaps from its origin position to the ''Box'''s position.
+=== Base Case ===
+We first observed how a single ''Pair'' moves without interacting with other ''Boxes''. We put a ''Pair'' in the center of a square stage of 4 by 4 tiles (128*128=16384 sites). We let the ''Pair'' dance for 50 kAEPS. This size of the MFM simulator and the time length is set so because we hope the pair will not hit the boundary too much. We want to observe the natural position of the ''Pair'' as if it is dancing on a infinite large stage. However, 10 out of 51 experiments still got the ''Pair'' hit the boundary. We also tried bigger stage of 6 by 6 tiles. In this case, the AER decreased from 60 to 30. Although the ''Pair'' hit the boundary less, but this may be the result of the less event it acquired and the decreased distance it walked. So we finally used 4 by 4 stage in the following experiments.
+We plot the 41 //end-positions// of the ''Pair'' as the picture.  The left graph shows this base case of the ''Pair'''s random walks. The average distance that the ''Pair'' traverses is 46.9 with a standard deviation of 20.7 (The distance is measured as the mahattan distance in 41 runs).
+**Looks like we need a lot more data there, right?** Yes, we need more data points.
+=== Shot Case ===
+We put a line of ''Boxes'', east of the center of the 4 by 4 stage. The population of this line is relative dense because each ''Box'' in the line is only 3 sites away from its horizontal neighbor.
+How this single trail can interact with the ''Pair''? After 50 kAEPS. We found the //end-positions// are not scattered randomly as they are in the base case.They are interfered and over 42% of them are in the first quadrant. We have more situations of //hit-boundary// than the base case. We suppose this is because of the interactions of the ''Pair'' and the ''Boxes'', 12 Out of 40 experiments had this result. Another interesting thing we observed is sometimes the interaction produced more ''Pairs''. There are 2 experiments end with more than one ''Pair''s. We need more experiments for the //Shot Case// because the much higher unexpected results of //hit boundary// and //multiple Pairs//. The middle graph shows this //Shotgun// case of the ''Pair'''s interfered walks. In the 26 experiments, the average distance that the ''Pair'' traversed is 41.8 and the standard deviation is 19.3. Almost the same average distance only the direction or the //end-positions// are affected because of the line of ''Boxes''. **We already ran 10 more experiments. We can merge the results soon.**
+The following pictures show the layout of our experiments. The left graph shows part of the layout of //Shot Case// experiment. Each ''Box'' in the line is 3 sites away from its neighbor. The middle graph shows part of the layout of //Grid Case// experiment. Each ''Box'' in the grid is 3 sites away from its vertical neighbor or horizontal neighbor. This dense population will create a chain reaction. The right graph shows part of the layout of //Grid Case// experiment. Each ''Box'' in the grid is 7 sites away from its vertical neighbor or horizontal neighbor.
+{{:dev:4shot_layout.png?200|}} {{:dev:4grid_layout.png?200|}} {{:dev:8grid_layout.png?200|}}
+=== Grid Case ===
+This is the real party here. We put a lot of ''Boxes'' into the stage. They are dancing alone on a 8 by 8 grid
+(Each ''Box'' is horizontally and vertically 7 sites away from its neighbor.). There are more interactions between ''Pair'' and ''Boxes''. We had to expand the distance of ''Boxes'' from 3 to 7. The short distance brought a //chain reaction//. More and more ''Pairs'' are produced and we cannot see how the origin ''Pair'' walks. The left graph is the //chain reaction// of a 4 by 4 grid (3 sites distance) after about 30kAEPS.
+{{:dev:chainrec.png?300|}}
+The intensive interactions let the ''Pair'' hit the boundary more often. Also the chance of produce more ''Pairs'' is greater. In 11 of the 32 runs, the ''Pair'' hit boundaries. In 4 of the 32 runs, more ''Pairs'' were created. So we have only 17 runs that produced the results we can use. The right graph shows this //Grid// case of the ''Pair'''s interfered walks. In the 17 experiments, the average distance that the ''Pair'' traversed is 48.0 and the standard deviation is 18.8. So we can at least say that the ''Boxes'' grid helped to extend the average mahattan distance that a ''Pair'' can traverse from around 40 to 50 (in 50kAEPS).
+ {{:dev:2base_500.png?300|}} {{:dev:2shot_600.png?280|}} {{:dev:2grid_400.png?280|}}
+The left graph is 526 //Base Case// runs on 2 by 2 simulator after 12 kAEPS. Average distance for //Shot Case// is 18.8. Standard deviation is 9.9. The middle graph is 642 //Shot Case// runs for the same size and time. Average distance for //Shot Case// is 19.2. Standard deviation is 10.2.(This time ''Pairs'' do not seem to favor the 1st quadrant. We see better with more data points.)  The right graph is 466 //Grid Case// runs. Average distance for //Grid Case// is 19.4. Standard deviation is 10.0. (This time the three layouts seemed to produce similar end-positions. )
+{{:dev:4base_500.png?280|}} {{:dev:4shot_600.png?280|}} {{:dev:4grid_500.png?280|}}
+nearly 1/4 data in the ''{2C2}'' experiments are invalid because of they hit the boundary. To rule out the influence of th missing data points(those hit the boundary), we ran the experiments on bigger simulator again. The left graph is 516 //Base Case// runs on ''{4C4}'' simulator after 10 kAEPS. Average distance for //Base Case// is 21.1. Standard deviation is 11.8. The middle graph is 604 //Shot Case// runs on ''{4C4}'' simulator. Average distance for //Shot Case// is 21.8. Standard deviation is 11.9.  The right graph is 521 //Grid Case// runs on ''{4C4}''. Average distance for //Grid Case// is 23.1. Standard deviation is 13.0.
+In the bigger simulator and less time length, we had no data hitting the boundary. The ''{4C4}'' average distance we got seems greater than the ''{2C2}'' average distance. However, nearly 1/4 of the ''{2C2}'' data fell out of the edge. So the actual average distance in ''{2C2}'' should be greater than the data shows. In both experiments, we can see the grid helped to extend the range a ''Pair'' traversed. The extension in ''{2C2}'' seems to be less than 5% (from 18.8 to 19.4) because more data points fell out of the edge. In ''{4C4}'' experiments, the average distances are extended nearly by 9% (from 21.1 to 23.1). This shows that the some //pulses// that travel faster than //walks// are produced in the grid.
+The end-positions of the experiments didn't show obvious bias. That's fair because of the initial layout of our experiments. When ''Boxes'' are evenly placed in grid pattern, they //attract// the ''Pairs'' evenly, which will produce no bias. When ''Boxes'' are placed in a line, they are too sparse to //trap// ''Pairs''. This did not produce bias either. In the following experiment, we will place a bunch of ''Boxes'' in the 1st quadrant. This cluster of ''Boxes'' will //attract// the first ''Pair'' and produce more ''Pairs''. Then, the end-positions will show obvious favor to the 1st quadrant.
+{{:dev:4overlay_500.png?350|}}
+The above graph is what we saw if we put the three pictures together. If the end-positions of ''Pairs'' are not surprising, how about the ''Boxes''? Did they still remain in line or in grid? How they were messed up by the ''Pair''?
+Excluding runs that hit the boundary messes up the data.  Should use a bigger grid, so that hitting the boundary is very rare.  It will run slower but just use more machines.  And we don't care if more pairs are created, right?  We just wanted the distance to the farthest pair.
+Q:20-Jun-2015 09:11:31PM-0600:Yes, we can use bigger simulator to find out the relative different outcomes of these three layout. I still have this question. The farthest pair maybe does not travel to this location. It's possible that this pair is formed somewhere and then travels to the farthest location. Do we count this new formed pairs?
+A:20-Jun-2015 10:55:40PM-0600: When a pulse of electricity flows through a wire, do you think it's literally the same electrons coming out one end that went in the other end?  The electrical pulse travels //much// faster than do individual electrons in the wire.  The same thing here.  There is **no differerence** between the first pair that starts the dynamics, and some later pair that the first pair takes part in creating by "cutting in" and so forth.  For purposes of this experiment we only cared about how fast 'pboxness' can travel.
+-Jun-2015 11:58:06PM-0600 Thank you! That's much clear! We care about the pulse more than the individual one here. I am running a 6 by 6 //Shot Case// this night. I hope this experiment can show us more obvious **bias** because of the preset ''Boxes'' lines. In the previous experiments, we observed the line of ''Boxes'' may let the ''Pair'' favor the first quadrant. I don't know if this preference comes from the //clockwise// swap order of each ''Box''. This time I put more ''Boxes'' in the first quadrant. I am hoping that the ''Pair'' will fall more often into the first quadrant than the other three areas. We can see the result tomorrow morning.
+The new //Shot Case// layout helped to create a great bias. The result is as what we had expected. Almost 75% of the ''Pairs'' ended up at the first quadrant. Only this 6 by 6 tiles simulator runs quite slow on my 2 cores computer. It took near 50 minutes for a 50kAEPS run. When more ''Pairs'' were created, we simply measured the farthest ''Pair's'' location. Although we chose a much bigger simulator this time, The ''Pair'' hit the boundary in 8 experiments.
+{{:dev:6shot_layout.png?300|}}    {{:dev:shot6.png?360|}}