# Ludwig Boltzmann

AUSTRIAN PHYSICIST

1844–1906

Ludwig Edward Boltzmann is one of the foremost theoretical physicists of
the latter nineteenth century. A vigorous advocate for the existence of
atoms, he made monumental contributions to the
**
kinetic theory
**
of gases and established the statistical nature of the second law of
thermodynamics.

Boltzmann was born in Vienna, Austria, and graduated from high school in Linz. He entered the University of Vienna in 1863, and he received his doctorate in physics three years later. Then, for two years, he served as assistant professor at the university, where he was strongly influenced by the atomistic thinking of physicists Josef Loschmidt and Josef Stefan.

In 1869 Boltzmann became professor of mathematical physics at the
University of Graz, the first of his many academic appointments. After
leaving Graz in 1873, he held chairs in mathematics at Vienna
(1873–1876), experimental physics at Graz (1876–1890), and
**
theoretical physics
**
at Munich (1890–1893), Vienna (1893–1900), Leipzig
(1900–1902), and finally Vienna again for the remaining four years
of his life. According to many of his students, Boltzmann was an
outstanding teacher, and his lectures were often filled to overflowing. He
displayed a congenial attitude toward students and their learning,
something rather rare among Austrian and German professors at that time.

Boltzmann's first significant contribution to physics was the
generalization of James Clerk Maxwell's distribution of velocities
and energies for a sample of gaseous atoms. Although Maxwell had deduced
this distribution, he provided no physical basis for it. Boltzmann showed
that as atoms move toward
**
equilibrium
**
they assume the Maxwell distribution—later known as the
Maxwell-Boltzmann distribution—and further that this is the only
statistically possible distribution for a system at equilibrium.

Boltzmann connected his ideas with those of Rudolf Clausius, who had
introduced the concept of entropy in 1865. Somehow related to heat,
entropy was known to increase during irreversible processes, but its exact
nature was unknown. From the distribution of gas atoms, Boltzmann
described a quantity—later symbolized by the letter
*
H
*
—which is a minimum when atoms assume a Maxwell-Boltzmann
distribution. He recognized his
*
H
*
function as the negative of entropy, which is a maximum when the atoms
reach thermal equilibrium. Thus Boltzmann offered a kinetic explanation
for entropy and, more generally, a connection between the behavior of
atoms and thermodynamics.

One of the serious problems with Boltzmann's statistical treatment
arose from the reversibility of the laws of mechanics, which holds true
for a particle moving in one direction just as they do for a particle
moving in the opposite direction. How then could a given set of atomic
motions cause
*
H
*
to tend toward a minimum (or entropy toward a maximum), rather than away
from it? This was a vexing question for Boltzmann and a serious criticism
of kinetic theory.

In response, Boltzmann considered the number of different ways that a
sample of gaseous atoms could achieve a particular distribution. The more
ways the atoms can arrange themselves to achieve some distribution, the
more likely it becomes for that distribution to occur. This connection
between entropy (
*
S
*
) and the number of ways (
*
W
*
) that a given distribution can occur is embodied in the equation
*
S = k
*
ln
*
W
*
(
*
k
*
is now known as Boltzmann's constant, and ln is the natural
logarithm). This famous relationship, which is engraved on
Boltzmann's gravestone in Vienna, indicates that maximum entropy is
associated with the distribution that has the most ways of
occurring, that is, with the Maxwell-Boltzmann distribution. Although it
is possible for a system of atoms to move away from a Maxwell-Boltzmann
distribution, it is not probable, since it is statistically unlikely for
the system to move from a distribution with more ways of achieving it to
one with fewer ways. Much of Boltzmann's work in this area was
formalized somewhat differently under the name of statistical mechanics by
Josiah Willard Gibbs, an American physicist who was well known and well
respected in Europe, but not in his own country.

Boltzmann's achievements in theoretical physics are all the more remarkable in view of the considerable opposition to his ideas and in view of his own declining health. He had increasingly severe bouts of mental depression, and he tried to commit suicide several times during his life. In 1906 he succeeded in hanging himself while vacationing with his wife and family at Duino, near Trieste, on the Adriatic Sea.

**
SEE ALSO
**
Gibbs, Josiah Willard
;
Maxwell, James Clerk
.

*
Richard E. Rice
*

## Bibliography

Boltzmann, Ludwig (1905). "On the Trip of a German Professor into
El Dorado." In
*
Ludwig Boltzmann: His Later Life and Philosophy, 1900–1906
*
, ed. John Blackmore (1995). Boston: Kluwer Academic Publishers.

Cercignani, Carlo (1998).
*
Ludwig Boltzmann: The Man Who Trusted Atoms.
*
New York: Oxford University Press.

Cropper, William H. (2001).
*
Great Physicists: The Life and Times of Leading Physicists from Galileo
to Hawking.
*
New York: Oxford University Press.

Lindley, David (2001).
*
Boltzmann's Atom: The Great Debate That Launched a Revolution in
Physics.
*
New York: Free Press.

### Internet Resources

O'Connor, J. J., and Robertson, E. F. "Ludwig Boltzmann." September 1998. Available from http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Boltzmann.html .

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