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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 Spring '08
 Livant
 Physics, Statistical Mechanics, J. Phys., Am. J. Phys., canonical distribution, M. D. Sturge

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