Part 2 - Squiggle Ball Experiment

Objective:  

A Boltzmann machine is used to illustrate the Ergodic Hypothesis.  In one case, there is an energy difference between the two states.  In the second case, there is a phase space difference between the two states.

Background: 

Squiggle Balls function as a constant temperature bath, or constant source of energy for a test ball (ping pong ball) in these two experiments.  In the first investigation, the energy of the two platforms was varied.  The right platform was raised in small increments.  As it's energy grew higher, the likelihood of finding the test ball in the rightmost section decreased.  The phase space (that is, number of available states for the test ball, or more concretely, the available area on each side) was invariant.  

In the second Squiggle Ball investigation, the energy of the states was invariant, but the phase space was subject to change.  A large area is more likely to hold the test ball than is a smaller area.  In this case the number of Squiggle Balls is altered to adjust the amount of area available to the test ball.  A large number of balls takes up considerable area, or decreases the phase space.

 

Part A - Energy States

Analysis and Results:

(t Right)/(t left) = [(Area Right)/(Area Left)]*e^E/kT

The ratio of phase spaces is 1:1, so that term drops out.  E is mgh in this case.

(t Right)/(t Left) = e^mgh/kT

Ln [(t Right)/(t Left)] = (1/kT)*mgh

kT = mgh/Ln[(t Right)/(t Left)]

kT represents the Kinetic energy of the test ball.  So, kT=(1/2)mv^2.  Solve for v to find the average velocity of the test ball.

 

Change in E = mgh (t Right)/(t Left) Avg v (m/s)
0 1.0857 ?
1.5435*10^-4 0.3626 0.38
2.44902*10^-4 0.2162 0.39
3.9102*10^-4 0.1328 0.42

The slope of the graph below is 1/kT, the activation energy.  The average velocity of the test ball can also be found in this manner.

The value of kT is 185 microJoules.  If we assume that the activation energy is equivalent to the kinetic energy ((1/2)mv^2) of the test ball, then the average velocity of the test ball is 1.33 m/s.  So why don't these two values agree??  

Part B - Phase Space

Analysis and Results:

(t Right)/(t left) = [(Area Right)/(Area Left)]*e^E/kT

There is no energy difference between the two platforms, so the exponential term goes to 1.

(t Right)/(t left) = (Area Right)/(Area Left)

In theory, the time ratio ought to be equivalent to the phase space ratio.

Phase Space Ratio Time Ratio
2.17 / 1 1275 / 525 = 2.41 / 1
2.03 / 1 839 / 961 = 0.87 / 1
2.10 / 1 1018 / 782 = 1.30 / 1
2.33 / 1 1229 / 751 = 1.64 / 1

These ratios do not match as one would expect.  It may be that there are other factors contributing to the probability that have been neglected in this experiment.

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