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Unformatted text preview: An experiment to demonstrate the canonical distribution M. D. Sturge and Song Bac Toh a) Department of Physics, Dartmouth College, Hanover, New Hampshire 03755 ~ Received 2 January 1999; accepted 2 June 1999 ! We describe a simple experiment, suitable for an undergraduate laboratory, in which the collector current in a transistor is measured as a function of the base–emitter voltage at various temperatures. The experiment gives a very convincing demonstration of the canonical distribution of statistical mechanics, in which the probability of occupancy of a state of energy E is proportional to e 2 E / kT . © 1999 American Association of Physics Teachers. Perhaps the single most important and useful result in sta tistical mechanics is the canonical distribution, in which the probability density P ( E ) that a system with a fixed number of particles, in equilibrium with a heat bath at temperature T , has energy E is proportional to the Boltzmann factor e 2 E / kT , weighted by the degeneracy g ( E ): P ~ E !} g ~ E ! e 2 E / kT , ~ 1 ! where k is Boltzmann’s constant. The result ~ 1 ! is also called the ‘‘Boltzmann distribution,’’ but students often confuse this distribution with the ‘‘Maxwell–Boltzmann distribu tion,’’ which applies only to an ideal gas. Instead we use the name ‘‘canonical distribution’’ because it refers to the distri bution in the canonical ensemble, and is the terminology used by Gibbs. 1 In spite of the central role of the canonical distribution, and its wide range of applications, it is not easy to find a simple experimental demonstration suitable for an under graduate physics laboratory. Although there are many chemi cal applications in which T is varied and the resulting change in chemical concentration or reaction rate is measured, 2 it is difficult to vary the energy E in such experiments and thus to demonstrate Eq. ~ 1 ! in its full generality. The same applies to physical experiments such as measurement of the density profile of a gas in a centrifuge. 3 The current–voltage charac teristic of a vacuum diode 4 depends on the canonical distri bution, but is complicated by space charge, electrode geom etry, and other confusing effects. This paper describes a simple undergraduate experiment in which E and T can both be varied, and the validity of Eq. ~ 1 ! confirmed over a range of six or more decades in P ( E ). The idea is to measure the collector current in a transistor as the base–emitter voltage is varied. Although such a measure ment of a transistor characteristic is a staple of electronics courses, it does not seem to be generally known that one can use such a measurement to demonstrate this fundamental re sult of statistical mechanics....
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This note was uploaded on 04/30/2008 for the course PHYSICS 2100 taught by Professor Livant during the Spring '08 term at Auburn University.
 Spring '08
 Livant
 Physics

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