60 STATISTICAL PHYSICSconcerning the foundations of statistical mechanics preceded the appearance of the
first papers in which it was noted that all was not well with Boltzmann's ergodic
hypothesis. In what follows, I shall therefore have no occasion to make reference
to ergodic theory.4b. Maxwell and Boltzmann*
Boltzmann's grave, in the Central Cemetery in Vienna, is marked by a monument
on which the formulais carved. 'It is immaterial that Boltzmann never wrote down the equation in this
form. This was first done by Planck. ... The constant k was also first introduced
by Planck and not by Boltzmann' [S4]. Indeed, k is a twentieth century symbol
which was used for the first time in the formulaproposed on December 14, 1900, by Planck [PI] for the thermal equilibrium dis-
tribution of blackbody radiation.** The quantity p(v, T)dv is the radiative energy
per unit volume in the frequency interval v to v + dv at temperature T. Equation
4.1, or rather (and better)
is also found for the first time in a paper by Planck, one completed a few weeks
later [P3]. Lorentz referred to k as Planck's constant as late as 1911 [L3]. Nor
was he the only one to do so at that time [Jl].
The essence of Eq. 4.3, the insight that the second law of thermodynamics can
be understood only in terms of a connection between entropy and probability, is
one of the great advances of the nineteenth century.f It appears that Maxwell was
*In writing this section, M. Klein's studies of the work of Maxwell and Boltzmann have served me
as an indispensable guide.
"Planck's discovery will be treated in Chapter 19. An equation equivalent to Eq. 4.2 but in which
h and k do not yet occur explicitly had been proposed by Planck on the preceding October 19 [P2].
•(•Recall that the period of discovery of the first law of thermodynamics (the impossibility of a per-
petuum mobile of the first kind) is approximately 1830 to 1850. Many scientists, from engineers to
physiologists, made this discovery independently [Kl]. The law of conservation of energy for purely
mechanical systems is, of course, much older. The second law was discovered in 1850 [Cl] by Rudolf
Julius Emmanuel Clausius while he was pondering the work of Sadi Carnot. In its original form
(Clausius's principle), the second law said in essence that heat cannot go from a colder to a warmer
body without some other accompanying change. The term entropy was also introduced by Clausius,
in 1865, at which time he stated the two laws as follows: 'The energy of the world is constant, its
entropy strives toward a maximum,' and commented that 'the second law of thermodynamics is much
harder for the mind to grasp than the first' [C2].