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course Phy 232
IntroductionThe program kinmodel_.EXE simulates in 2 dimensions the kinetic behavior of a user-specified number of spherical atoms with user-specified masses, colliding as hard elastic disks at a user-specified center-to-center distance. The initial positions and speeds of the particles are randomly generated by the computer and the simulation develops from the corresonding initial state. Information related to particle speeds, x- and y- kinetic energies, and energy distributions is provided in the form of unlabelled graphs on the screen.
The simulation can be stopped once it is running by striking the 's' key.
The default settings
The default settings are chosen to provide 43 particles or atoms, 32 colored green, 8 colored dark blue, 2 colored light blut and one colored red. One of the 'green' particles starts at rest and leaves a trail when it moves.
estimating the distribution of atomic speeds and mean free path
equipartition of energy and the tendency for an ordered system to move toward disorder
the improbability of 43 particles being segregated on one side of the viewing area (unlikelihood of an ordered configuration)
the probability that a particle's speed will occur in a given range
the connection between relative particle mass and average speed
the development of empirical frequency vs. speed and frequency vs. energy histograms (order and disorder, this time with statistical order emerging from the disordered system)
images of 2-dimensional collisions
appreciation of time scale of kinetic interactions in a gas at typical pressures and temperatures (at medium default speed the simulation represents many of the features of a thin slice approximately 10 nanometers on a side and, say, a nanometer thick, of a monatomic gas at room temperature and several atmospheres pressure, with 1 second of real-world time corresponding to a few thousand years of simulation time).
and others.
Experiments and Activities
Preliminary Observation
Run the program billiard simulation. Simply open the simulation and hit the 'Enter' key.
Watch the KEx and KEy values as they change with each collision, representing the total x and y kinetic energies of the particles.
One of the green particles traces out a path as it moves across the screen. This is the particle whose speed is indicated next to the word 'speed' (about halfway down the window, toward the right-hand side). Most of the time when this particle collides with another its speed changes. Watch for a minute or so and see if you can learn to estimate its speed before looking at the posted speed. How long does it take to move a distance equal to the height or width of the screen when its speed is 10? How long should it then take to move the same distance if its speed is 5? Is that about what you observe?
How frequently does that green particle collide with other particles? What percent of the time intervals between collisions do you think are less than a second? What percent are less than 2 seconds? What percent are less than 4 seconds? What percent are less than 10 seconds?
Watch the 'red' particle for a couple of minutes, estimating the average time between its collisions and its average speed. What percent of the time intervals between collisions do you think are less than a second? What percent are less than 2 seconds? What percent are less than 4 seconds? What percent are less than 10 seconds? At its average speed, how long do you think it would take to move a distance equal to the height or width of the screen? On the same scale you used for the speed of the green particle, what do you think is the average speed of the red particle?
Watch the 'blue' particle, and speculate on what property of this particle is different from that of the other particles.
****Preliminary Assignment
1. I estimate that the speed of the green particle is between 4 and 5 on average. It takes about a second for the green particle traveling at speed 10 to go from one end of the screen to the opposite side. Therefore, it takes approximately 5 seconds for the green particle to travel from one side of the screen to the opposite side at speed 5. I observed these approximations on my own.
2. It seems after viewing the simulation that the green particle collides with another particle about once every 1.0-1.5 seconds. This is a rough approximation. I believe that around 35% of the collisions occur in less than one second. About 9% occur in less than two seconds, around 99% in less than four seconds, and 100% of the collisions occur in less than ten seconds.
3. I estimate that the red particles average speed is 1 and that it collides with another particle around every 2 seconds. About 10% of the total collisions by other particles into the red particle occur in less than one second. About 40% occur in less than two seconds, about 80% in fewer than four seconds, and 100% of collisions into the red particle occur in less than ten seconds.
Experiment kinmodel_01: The Distribution of Atomic Speeds
When the speed of the simulation is moderate it is possible to watch a specific particle (the red particle or the blue particle in the default simulation) and obtain an intuitive feeling for the relative frequencies of various speeds.
Run the simulation billiard simulation at the default settings.
Observe the simulation long enough to get a feel for the maximum velocity you are likely to see. Then estimate how much time it spends at slow (less than 1/3 of max vel.), medium (between 1/3 and 2/3 of max. vel.) and fast (more than 2/3 of max. vel.) velocities.
Express your estimates in percents of the total time spent in the three different velocity ranges.
Draw a histogram (a bar graph) of your estimates. Describe your histogram in your writeup.
Now suppose you had estimated the percent of time spent in each of 10 velocity ranges (i.e., from 0 to .1 of max. vel., .1 to .2 of max. vel., etc, up to max. vel.). From your previous estimates, without further viewing the simulation, make a reasonably consistent estimate of the proportion of time spent in each of these ranges.
Sketch a histogram of your estimates and describe the graph in your writeup.
Sketch the smooth curve you think best represents the distribution, with the curve being highest at the most likely speed, near the horizontal axis for speeds you very seldom observe. According to your sketch, which speed is the most likely? What percent of the area under your curve corresponds to speeds within one unit of your most likely speed (e.g., if your most likely speed was 3, you would estimate the area under the curve between speed 3 - 1 = 2 and speed 3 + 1 = 4). For what speed(s) is the curve half as high as the maximum? For what speed(s) is it half this high?
Watch the green particle for long enough to estimate the percent of time it spends at speeds more than 2 units greater than the most likely speed, but not more than 4 units greater.
What percent of the time do you estimate that the green particle is moving at less than half its most likely speed?
Watch the number corresponding to the speed of the green particle.
Close your eyes for a few seconds at a time and open them suddenly, and each time write down the velocity of the particle as you see it immediately after your eyes open. Record about 100 velocities in this manner.
Tally your velocities to see how many of the 100 velocities were 0, how many were 1, how many were 2, etc.
Construct a histogram of your results and compare to the histograms you predicted earlier.
****Distribution of Atomic Speeds
1. I am focusing on the red particle for my observations in this experiment. The red particle seems to always stay in the lowest third of the maximum speed capable of the particles. The red particle remains at speeds less than one third of the maximum 10 speed capable 100% of the time. It never reaches speeds greater than 4.
2. The red particle spends 100% of the time in speeds less than 3.5, 0% in speeds between 3.5 and 6.7, and 0% in speeds greater than 6.7.
3. My histogram for the red particle would be only one bar in the speed interval from 0 to 3.5 because the red particle spends 100% of its time in this speed interval and never goes outside of it.
4. Speeds from 0 up to 3.5 the speed interval most likely to occur for the red particle. Speeds in this interval are 100% likely to occur the entire time for the red particle. Because the one speed interval is the only interval present, there are no other intervals that are half as much or a fourth as much.
5. The green particle seems to spend about 30% to 40% of its time at speeds greater than 2 units but no more than 4 unit speeds over the average. The green particle moves at less than half of its average speed about 20% of the time.
6. After recording 100 velocities of the green particle at random times, there were 0 occurrences at speed 0, 5 at speed 1, 6 6 at speed 2, 8 at speed 3, 24 at speed 4, 30 at speed 5, 9 at speed 6, 7 at speed 7, 6 at speed 8, 3 at speed 9, and 2 at speed 10.
Experiment kinmodel_02: Mean free path; mean time between collisions
It is possible to observe the mean free path of the green particle between collisions.
First observe the particle for a few minutes and try to get a feel for how the distances traveled between collisions with other particles are distributed. Make your best estimate of what percent of the time the particle travels less than 1 inch between collisions, the percent of the time the distance is between 1 and 2 inches, the percent of the time the distance is between 2 and 3 inches, etc.. When the particle collides with a 'wall', it doesn't count as a collision and distance keeps accumulating until it collides with another particle.
Sketch a histogram of your estimates, and also document the distance on your monitor between the 'walls' that confine the particles.
Now take some data.
Using the 'pause' and 'restart' buttons, stop and start the particle motion as required in order to observe the distances traveled by the green particle between collisions. Use a ruler to measure distances traveled. Don't leave any distances out, because this would bias the sample. Observe at least 100 distances.
Describe how you obtained your data and report your data as a frequency distribution (i.e., the number of observations for which the distance rounded to 0, 1, 2, 3, ..., inches).
Sketch a histogram of your results.
Sketch the histogram you would expect from a large number of observations.
Describe your histograms, and how they compare with your previous predictions.
****Mean Free Path; Mean Time Between Collisions
1. The green particle is within another particle around 80% of the time, excluding walls. This is because the green particle is moving at a high rate of speed and therefore approaches other particles frequently. The green particle is within about an inch of another particle 80% of the time, within 1 to 2 inches 15% of the time, and within 2 to 3 inches of another particle about 5% of the time.
2. The estimated dimensions of the box on my monitor are 6 inches by 5 inches.
3. I obtained my data through recording each distance as I paused the simulation and then continuing to run the simulation after recording the data. This was repeated until I obtained enough samples. My data in inches is: 1,2,0,3,2,2,1,0,0,4,3,4,2,2,2,1,1,4,1,1,2,0,0,1,1,0,2,3,3,1,2,2,1,1,2,0,0,0,2,4,5,3,2,1,2,2,1,3,2,1,1,0,2,1,3,4,4,5,3,4,1,2,1,0,2,4,3,2,1,1,1,0,1,2,1,1,3,2,4,2,3,2,1,1,2,2,3,2,2,3,3,2,1,0,0,3,0,1,2,0
4. I expected that the average number of occurrences would be around 1 inch between collisions and the data supports that claim. After analyzing the data, the mean was slightly above 1 inch between collisions. The histogram shows that 1 and 2 were the most popular distance intervals with 3 and 0 following close behind. 4 and 5 still occurred but not nearly as often as the others. Between 1 and 2 is the true estimation behind all of these data distance points.
Experiment kinmodel_03: Equipartition of energy and the direction of disorder to (increasing or decreasing)
NOTE: The program is not currently set up to run the experiment as given here. See the alternative, a few lines below.
Observe the first several seconds of the simulation at the 'slow' default speed. You will see how the particles initially are all moving in or very close to the x direction, with little or no y component. Note the x and y kinetic energies, displayed near the top of the screen.
Observe what happens to the directions of motion of the particles as they start colliding.
Observe what happens to the x and y kinetic energies.
Is the system more organized at the beginning of the simulation or after a couple of minutes?
If the x and y kinetic energies were averaged for 100 years, starting a few minutes after the simulation began, which do you think would be greater?
Run the simulation in this manner several times, and each time determine how long it takes before the total y kinetic energy is first greater than the total x kinetic energy. Report your results.
Now take some data:
Running at the fastest default speed, stop the simulation with the pause/break key every few seconds, keeping your eyes closed for at least 2 seconds before stopping the motion.
Write down the x and y kinetic energies each time.
Do this at least 30 times.
Find the average of all your x and all your y kinetic energies.
Do you believe the difference in the averages is significant, in that the direction that has the higher average will always tend to have the higher average every time the simulation is run?
ALTERNATIVE
Start the program using default values. Let it run for several seconds, then start observing the green particle. Keep track of whether it is moving more in the x or more in the y direction. Just say to yourself 'x x x y y y y y x x y x y y y ... ', according to what you see. Do this at a steady but comfortable pace. Continue this for a minute or so.
Then take a pencil and paper, or alternatively open a text editor in a separate window, and start writing down or typing your x and y observations. I just did this and in about a minute or two I got the following: xxyyyyxyyxxyxyyxxyxxxyyyxxyyxxyyxyxxyyyxyyyxyyxy. I haven't done this before and found this a little confusing. Every time the particle got hit I wanted to type a letter right away, but I hadn't had time to figure out in what direction it was headed. With practice I began to get over that. You will experience different glitches in the process, but with a few minutes of practice you'll be able to do a reasonably good job. I suspect I also had some tendency to type one of the letters in preference to the other (e.g., x in preference to y, or maybe y in preference to x). I don't recommend fighting this sort of tendency but just noticing it and gently trying to improve. I didn't do this with pencil and paper, and it would be interesting to see if the tendencies are the same when writing as opposed to typing. However that's not our purpose here.
At whatever pace you prefer, write or type about 50 observations of x or y. List them here.
Now notice the KEx and KEy values represented toward the right-hand part of the program's window, just a little ways below the middle of the screen. KEx represents the total x component of the kinetic energies of all the particles and KEy the total y component.
Using the Pause and Restart buttons, stop and start the program and with each stop record the KEx and KEy. After each observation quickly hit 'Restart' then 'Pause', and record another. Record about 50 observations.
Having recorded the 50 KEx and KEy values, write 'x' next to each pair for which the x value is greater, 'y' next to each pair for which the y value is greater. List your x's and y's in sequence here (don't list your values for the KE).
What is the greatest KEx value you observed and what is the least?
What is the greatest KEy value you observed and what is the least?
On a 50-trial sample of a normal distribution, the mean would be expected to occur about halfway between the least and greatest values observed, and the expected standard deviation would be very roughly 1/5 of the difference between the least and greatest values. According to this (very approximate) rule, what would be the mean and standard deviation of your KEx values, and what would be the mean and standard deviation of your KEy values?
Do you think the mean KEx value differs significantly from the mean KEy value? There is a difference. By 'significantly', we mean a difference that seems greater than what would naturally occur by chance statistical variations.
****Alternate
1. Yyxxyyyxxxyxyxyyxyyxyyyxxxxyyxyxyxyyxyyxyyyxyyxyyx 50 observations.
2. Xyxyxxyxxyxxyyyxyyxyyxyxyxxyxxyyxxxyyxyxyyxyxyxyxx 50 KEx and Key.
3. Greatest KEx value- 1733, least KEx value- 1267. Greatest Key value- 1655, least Key value- 1192.
4. Mean of KEx = 1537 and standard deviation for KEx = 101.4. Mean of Key = 1411 and standard deviation of KEy = 88.7.
5. I think that there is some variation in the means between the KEx and the Key but they are still relatively close to each other given the large span of possible number outcomes for the KEx and Key values.
Experiment kinmodel_04: The improbability of all particles being segregated on one side of the viewing area (order vs. disorder)
Any selected region of the screen can be selected for viewing by masking the rest of the screen. The viewer can estimate the probability of this region being vacated within an hour, within a day, within a year, ..., within the age of the universe. Results will differ with the size of the region, the number of particles and the speed of the simulation.
Cut out a 1-inch square and watch the simulation for 2 minutes on the middle default speed. Observe how many times the square becomes 'empty' of particles. Estimate what percent of the time this square is empty.
Enlarge the square to a 1-inch by 2-inch rectangle and repeat.
Enlarge to a 2-inch by 2-inch square and repeat.
Enlarge this square to a 2-inch by 4-inch rectangle and repeat.
Enlarge to a 4-inch by 4-inch square and repeat.
Mask all but 1/4 of the screen and repeat.
How long do you think it would take, on the average, for 1/4 of the screen to become completely empty of particles?
How long do you think it would take, on the average, for 1/2 of the screen to become completely empty of particles?
A typical closet is about 100 million times as far across as the distance represented by the screen. Ignoring for the moment that the closet is three-dimensional and hence contains many more air molecules than would be represented by a 2-dimensional simulation, how long do you think you would have to wait for all the molecules to move to one side of the closet?
****Improbability OF All Particles
1. The square becomes empty of particles for a second 14 times in a two minute span.
2. 1 inch by 2 inch square had 12 times where it was empty.
3. 2 inch by 2 inch square had 9 times where it was empty.
4. 2 inch by 4 inch square had 4 times where it was empty.
5. 4 inch by 4 inch square had 0 times where it was empty.
6. One fourth of the screen had 0 times where it was empty.
7. I believe that it would take a very long time for one fourth of the screen to be empty, if it could ever happen at all.
8. I believe that half of the screen will never be completely empty of particles because there are too many particles on the screen.
9. I believe that given the same size and number of particles here in the simulation, and translated them to space the size of a closet, it would be impossible for all the particles to be on one side. The probability of that happening can be calculated because the chance of one particle being on one side is (1/2) and there are 100 particles, therefore (1/2)^100 = 7*10^-31 = 0 percent chance of that happening, even in a larger space.
Experiment kinmodel_05: The probability that a particle's speed will occur in a given range
NOTE: This experiment is pretty much redundant with a previous one and is to be OMITTED.
The default settings will not work with this experiment. This time when you run the program you need to choose to customize the settings. For everything but the numbers of different particles and their masses, and the 'yellow' marker, you may use the defaults.
For the number and masses of particles:
When asked for the number of particles of type 1, enter 28. When asked for the mass of this type give 1.
When asked for the number of particles of type 2, enter 1. When asked for the mass of this type give 10.
When asked for the number of particles of type 3, enter 1. When asked for the mass of this type give 1.
Regarding the 'yellow' marker:
You will choose the minimum and maximum speeds which will result in the particle being 'painted' yellow. This will allow you to observe the proportions of the particles in different velocity ranges.
If you wish you may also adjust the speed factor, which has default value 3. If you want the simulation to slow down to 1/3 the pace, you can enter 1 for the speed factor. If you want the simulation to go as fast as practical for the other default setting, you could use a speed factor up to 5. Only the pace of the simulation is affected by the speed factor; the speeds displayed on the screen are not affected.
Now try to observe the numbers of particles in various ranges:
Run the simulation and use a 'yellow' range of 3 to 6 and attempt to observe the proportion of the particles falling within this range. You will be able to get a fairly good idea of the proportion, but it will be hard to get a really good estimate unless you repeatedly pause the program and count the 'yellow' particles.
Run the simulation using a 'yellow' range of 4 to 4, which will give mark only particles whose velocity is 4. Determine to reasonable accuracy the average percent of particles with this velocity.
Repeat for velocities 0, 1, 2, 3, 5, 6, 7, 8, 9 and 10.
What are the percentages corresponding to each of these velocities?
What therefore do you think is the average particle velocity?
Experiment kinmodel_06: The connection between relative particle mass and average speed; equality of average kinetic energies
Using default settings, answer the following:
What do you think is the average speed of the dark blue particles as a percent of the average speed of the green particles? (you might, for example, observe how long, on the average, it takes a particle of each color to move a distance equal to that across the screen)
What do you think is the average speed of the red particle as a percent of the average speed of the green particles?
A blue particle is 4 times more massive than a green particle. How do you think its average KE therefore compares with the average KE of the green particles?
A red particle is 64 times more massive than a green particle. How do you think its average KE therefore compares with the average KE of the green particles?
****Connection Between Relative Particle Mass
1. I believe that the average speed of the blue particles is 40% of the average speed of the green particles.
2. I believe that the average speed of the red particle is 10% of the average speed of the green particles.
3. The average KE of the blue particles is the same as the average KE of the green particles. This is because while the blue particle is more massive, it moves slower compensating for its heavier mass. The formula KE = (1/2)*mass*(velocity^2) shows this and shows how the faster green particle is evened out by its lesser mass.
4. The same principle applies here. The average KE of the red particle is the same as the average KE of the green particles. This is because while the red particle is way more massive, and therefore it moves slower, compensating for its heavier mass. The formula KE = (1/2)*mass*(velocity^2) shows this and shows how the faster green particle is evened out by its lesser mass.
Experiment kinmodel_07: The development of empirical frequency vs. speed and frequency vs. energy histograms (more about order vs. disorder, with statistical order emerging from the disordered system)
NOT CURRENTLY ASSIGNED: The frequency vs. speed, frequency vs. square root of energy, and frequency vs. energy histograms (it is left up to the student to determine which is which) are normalized to have a consistent total area. These distributions develop over time, eventually reaching a smooth distribution analogous to the Maxwell-Boltzmann distribution. This development occurs much more quickly if the settings are customized to encourage a maximal number of collisions.
Experiment kinmodel_08: Images of 2-dimensional collisions
It is easy to customize the settings to obtain two large relatively slow particles. Any student who has watched air hockey pucks or billiard balls colliding will recognize the validity of the simulation.
If the particles leave 'tracks' then a 'snapshot' in which a single collision between the particles occurs will provide data sufficient to validate conservation of momentum.
Customized settings
The following parameters can be controlled by electing to customize settings:
The number of particles (default setting is 30, maximum is 1000, which shouldn't be much of a restriction in the near future of PC's).
The number of different particle types (more than 9 different particle types is not recommended because of restricted number of colors).
The speed factor that determines how fast the particles move across the screen. If the speed factor is too great, particles may occasionally (or frequently, depending on how great) miss collisions. This is not a big problem unless data is being taken that assumes no 'misses'.
The radius of a particle (default radius is 1% the width of the square viewing area).
The proximity of the centers of the particles within which collision will occur (default is 5 particle radii). A greater value here will result in more collisions, other parameters being equal.
The minimum and maximum speeds defining a speed 'window'. Any particle whose speed is in this 'window' will be colored bright yellow. This range of speeds will be indicated by a yellow rectangle on one of the graphs.
Whether all the particles leave 'tracks' or not. The last two particles usually leave 'tracks'.
The number of iterations before the screen is cleared and the various graphs are updated. An iteration consists of the calculation and display of the position of every particle. A fairly small number allows the viewer to observe the evolution of the graphs, while a somewhat greater number permits observation of a significant number and variety of particle 'tracks'. If the number is too great the particle 'tracks' will be obscured.
The last two particles specified will have velocities indicated onscreen; the last of these particles will be sampled to obtain the velocity distribution shown at the right of the screen.
'Research' questions
For which particle speeds is the time between collisions likely to be greatest, and for which will it be least?
Initial particle speeds are uniformly distributed. After a short time a specific nonuniform distribution of speeds takes over. How long does it take before the contribution of the initial uniform distribution to the graphs and histograms displayed on the screen become indistinguishable? How will the shape of the graph differ from the ideal distribution during the transition?
For the default settings, what is the 'peak' particle energy? What is the 'peak' velocity of the sampled particle?
A narrower speed range near the peak of the speed distribution can result in more instances of 'yellow' particles than a wider speed range away from the peak. At each possible integer speed v, it is possible to define a speed range (v0, vf) with v at the midpoint of that range, such that the average number of 'yellow' particles will be the same as for the 'unit' range around the peak of the distribution. The 'unit' range is a velocity range of width 1 unit centered at the 'peak' velocity.
What does it take to get a massive molecule surrounded by low-mass particles moving fast?
Does the presence of an even more massive particle give a medium-mass particle, surrounded by a greater number of low-mass particles, an advantage in achieving greater speeds? Does the presence of a more massive particle affect the energy distribution of the medium-mass particle?
****Research Questions
1. The particle collision most likely to occur is a green particle on a green particle because they are moving at the fastest speed. The least likely collision to occur is between a red particle and a blue particle because those two speeds are the slowest available.
2. As soon as particles start colliding their speeds are no longer normal distributed because some particles speed up and slow down based on their collisions, while some continue at their normal speeds before colliding with another particle. The graph over a long period of time would come back to be normally distributed because the sample size is over 30 particles and the particles speeding up from collisions would balance with the particles that slowdown from the collisions.
3. The peak particle energy is approximately 2000 and the peak particle velocity is a speed of 10.
4. It takes time to get a large mass particle to be surrounded by many low mass and fast particles. This is because the large particle is moving so slowly that it is constantly being hit by the fast moving particles and is always being surrounded by them.
5. Yes because the more collisions that a particle has, the better chance that its speed will increase due to the collisions. The presence of a massive particle doesnt affect the energy distribution because the particles are randomly moving and the number of particles exceeds 30 particles.
At an advanced level: Derive Maxwell-Boltzmann distribution in 2 dimensions and compare the the empirical distribution.
More information on this model.
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&#Very good data and responses. Let me know if you have questions. &#