3. The Kinetic Definition of Entropy. Another
definition of entropy, which we owe to Ludwig Boltz-
mann, has been provided by the kinetic theory of gases.
Elaborating on J. C. Maxwell's famous statistical deri-
vation of the velocity distribution of gas molecules
under equilibrium conditions, Boltzmann studied the
change of this distribution
f(
v) under equilibrium ap-
proach and showed that
f(
v) always tends toward the
Maxwellian form (Boltzmann, 1872). Boltzmann ob-
tained this result by introducing a certain one-valued
function of the instantaneous state distribution of the
molecules, which he called the
E-function and later
the
H-function (Burbury, 1890), and of which he could
show, apparently on the basis of pure mechanics alone,
that it decreases until
f(
v) reaches the Maxwellian
form. His proof relied on the simple fact that the
expression (
x - y) (log
y - log
x) is always negative
for positive numbers
x and
y. Since under equilibrium
conditions
E turned out to be proportional to the
thermodynamic entropy, Boltzmann realized that his
E-function (or
H-function) provides an extension of the
definition of entropy to nonequilibrium states not
covered by the thermodynamic definition.
