Achieving Equilibrium:

Average Properties Become Predictable

The properties of individual parts of a random system, like an individual coin flip, are quite often unpredictable. (See random systems for other examples.) However, if enough coin flips are tallied, approximately half of the flips will show heads while about half show tails.

This principle, that the average behavior of a random system is predictable, is quite general. If enough leaves are measured from the same species of tree, the average leaf length will be extremely similar from one tree to another, even though individual leaf lengths vary widely.

This is the also the basis for opinion polls. If one person is asked a question, not much is learned about the nation's opinion. Now suppose that a news station asks 1000 people the same question, and suppose that 700 respond "yes" while 300 respond "no". This sample is likely large enough that any other group of 1000 people would also split roughly 70/30 on the question, and it can be assumed that this reflects the opinions of the whole nation. (Whenever live subjects are involved, various sources of bias could enter; these have been neglected in this example.)

Exercises to Illustrate Equilibrium

  1. Run the JAVA applet for 10 atoms at 1000K. The histogram of speeds tallies the observed speeds of the atoms over a long period of time. After a long run, what is the most probable speed (position of the highest peak in the histogram)?
  2. Conduct two more long runs with the same parameters. Compare the most probable speeds for the three runs.
  3. Run the applet for 10 atoms and a range of temperatures. For each temperature, how long does it take for the system to converge to a stable value of most probable speed?
  4. Thought question: Once an ideal gas system converges to a stable value of most probable speed, can that value change significantly? How can your answer be rationalized with the fact that an ideal gas is a random system?

The Big Picture

As you may have observed, the ideal gas system arrives at equilibrium differently than the other examples mentioned (like coin flips). The different atoms actually collide with each other and transfer energy. This way of approaching equilibrium is very important for many real-world systems.

© Andrew M. Rappe