When a continuous rather than a discrete distribution of energies is involved, g() is
replaced by g() d, the number of states with energies between and d.
We shall consider systems of three different kinds of particles:1 Identical particles that are sufficiently far apart to be distinguishable, for instance,
the molecules of a gas. In quantum terms, the wave functions of the particles overlap
to a negligible extent. The Maxwell-Boltzmann distribution functionholds for such
particles.
2 Identical particles of 0 or integral spin that cannot be distinguished one from another
because their wave functions overlap. Such particles, called bosonsin Chap. 7, do not
obey the exclusion principle, and the Bose-Einstein distribution functionholds for
them. Photons are in this category, and we shall use Bose-Einstein statistics to account
for the spectrum of radiation from a blackbody.
3 Identical particles with odd half-integral spin (^12 ,^32 , ^52 ,.. .) that also cannot be
distinguished one from another. Such particles, called fermions,obey the exclusion
principle, and the Fermi-Dirac distribution functionholds for them. Electrons are in
this category, and we shall use Fermi-Dirac statistics to study the behavior of the free
electrons in a metal that are responsible for its ability to conduct electric current.9.2 MAXWELL-BOLTZMANN STATISTICS
Classical particles such as gas molecules obey themThe Maxwell-Boltzmann distribution function states that the average number of parti-
cles fMB() in a state of energy in a system of particles at the absolute temperature Tis298 Chapter Nine
Ludwig Boltzmann(1844–1906)
was born in Vienna and attended
the university there. He then
taught and carried out both ex-
perimental and theoretical re-
search at a number of institutions
in Austria and Germany, moving
from one to another every few
years. Boltzmann was interested in
poetry, music, and travel as well as
in physics; he visited the United
States three times, something unusual in those days.
Of Boltzmann’s many contributions to physics, the most im-
portant were to the kinetic theory of gases, which he developed
independently of Maxwell, and to statistical mechanics, whose
foundations he established. The constant kin the formula ^32 kT
for the average energy of a gas molecule is named after him in
honor of his work on the distribution of molecular energies in
a gas. In 1884 Boltzmann derived from thermodynamic con-
siderations the Stefan-Boltzmann law RT^4 for the radiation
rate of a blackbody. Josef Stefan, who had been one of Boltzmann’steachers, had discovered this law experimentally 5 years earlier.
One of Boltzmann’s major achievements was the interpretation
of the second law of thermodynamics in terms of order and dis-
order. A monument to Boltzmann in Vienna is inscribed with
his formula Sklog W, which relates the entropy Sof a sys-
tem to its probability W.
Boltzmann was a champion of the atomic theory of matter,
still controversial in the late nineteenth century because there
was then only indirect evidence for the existence of atoms and
molecules. Battles with nonbelieving scientists deeply upset
Boltzmann, and in his later years asthma, headaches, and in-
creasingly poor eyesight further depressed his spirits. He com-
mitted suicide in 1906, not long after Albert Einstein published
a paper on brownian motion that was to convince the remain-
ing doubters of the atomic theory of its correctness. Boltzmann
had not been alone in his despair over doubters of the reality
of atoms. Planck was driven to an extreme of pessimism: “A
new scientific truth does not triumph by convincing its oppo-
nents and making them see the light, but rather because its
opponents eventually die and a new generation grows up that
is familiar with it.”bei48482_Ch09.qxd 1/22/02 8:45 PM Page 298