The simulation can be stopped once it is running by striking the 's' key.

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.

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?

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• 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?

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• 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?

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• Watch the 'blue' particle, and speculate on what property of this particle is different from that of the other particles.

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.

• 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?

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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.

Now take some data. 

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.

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.  As an alternative, you could make marks on a piece of paper then type them out (you might even use simple vertical and horizontal dashes, like | and -, which you can then translate into y's and x's).

At whatever pace you prefer, write or type about 50 observations of x or y.  List them here.

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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. Values can be rounded to the nearest whole number.  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).

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What is the greatest KEx value you observed and what is the least?

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What is the greatest KEy value you observed and what is the least?

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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?

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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.

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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.

• 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?

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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.

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)

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• What do you think is the average speed of the red particle as a percent of the average speed of the green particles?

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• 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?

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• 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?

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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 following parameters can be controlled by electing to customize settings:

The number of particles (default setting is 30, maximum is 1000).

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.