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Exercise_9

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

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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.

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