Exercise_9 - Exercise 9 A&EP 264 Spring,07 The Boltzmann...

Exercise 9, A&EP 264, Spring ‘07
The Boltzmann Machine
In this exercise a simple dynamical “Boltzmann Machine” [Prentis, J., “Experiments in
statistical mechanics”, Am. J. Phys. 68 (2000) 1073-1083] designed to simulate
classical Boltzmann statistics is used to illustrate the Boltzmann distribution (canonical
ensemble), dynamical equilibrium and transition rates (the law of detailed balance), and
state occupation numbers (degeneracy). This experiment is intended to provide the
student with a visual demonstration of the approach of a simple system to statistical
equilibrium. Before beginning this experiment, be sure to carefully read the document
“The Boltzmann Distribution Law” located in the AEP 264 folder and the course
blackboard.
The apparatus
A motorized “molecule”, known commercially as a Squiggle Ball TM [1] consists of a
plastic spherical shell enclosing a battery operated motor mounted on an axis extending
between poles of the spherical shell. The motor rotates the shell around the axis at
approximately 180 rpm. When placed on a horizontal surface bounded by rubber band
walls to form a “boxing ring”, the ball rolls continuously in random directions with a
distribution of speeds. Over time the ball will eventually visit every square centimeter of
the surface. The kinetic energy of the ball varies with time, but is constant when
averaged over time durations ca. 1 minute. The ball has a mass of 120 g and a
diameter of 8 cm. The maximum velocity of the center of mass is approximately 1 m/s.
The apparatus consists of two rectangular horizontal surfaces of equal area (ca. 1 ft 2 )
placed side-by-side and covered with a rubber sheet. Provision is made to elevate one
surface with respect to the other by heights of 0, 0.25, 0.5, or 0.75 inches (the wooden
elevators in the box at the front of the room). The rubber sheet provides a smooth
transition between surfaces. The surfaces are surrounded by rubber band walls that
confine a single squiggle ball to each surface, but allow ping pong balls to roll freely
from one surface to the other when propelled by collisions with a squiggle ball.
The apparatus models a two-level system in which the ping pong balls may reside either
on the lower surface of zero potential energy or the upper surface of potential energy
mgh, where m is the mass of the ping pong ball, g is the acceleration of gravity (980 cm
s -2 ) and h is the difference in height between upper and lower surfaces. The squiggle
ball has a mass of 120 g, approximately 50 times larger than the mass of a ping pong
ball (ca. 2.3 g). The average energy of the squiggle ball is much larger than that of the
ping pong balls and acts as a large energy reservoir from which a ping pong ball may
“borrow” sufficient energy to surmount the potential energy barrier separating the upper
and lower levels.