An experiment to demonstrate the canonical distribution

Publisher Website
Google Scholar
Abstract
~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 energyE 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 diode4 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 inP(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(DE )o f a particle overcoming an energy barrier of height DE is pro- portional to * 0 ' g(e)e2(e1DE)/kTde, 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~DE!5 f ~ T!e2DE/kT, ~2!

This publication has 8 references indexed in Scilit: