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# GetPDFServlet - An experiment to demonstrate the canonical...

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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. It follows from Eq. ~ 1 ! that the probability P ( D E ) of a particle overcoming an energy barrier of height D E is pro- portional to * 0 g ( e ) e 2 ( e 1 D E )/ kT d e , where e is the energy measured from the top of the barrier and g ( e ) is the density of states in the barrier region. This relation can be integrated to give P ~ D E ! 5 f ~

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