ENTROPY AND PROBABILITY 69the context of statistical considerations. It seems that he had already been brooding
for some time about the mysterious formula Eq. 4.2. Much later he wrote,
'Already soon after 1900, i.e., shortly after Planck's trailblazing work, it became
clear to me that neither mechanics nor thermodynamics could (except in limiting
cases) claim exact validity' [E52].
His statement that thermodynamics is not exact refers, of course, to the phe-
nomena of fluctuations. Einstein turned to fluctuations for the first time in 1904,
when he considered a system with variable energy E in thermal equilibrium with
a very large second system at temperature T. The equilibrium energy {E) of the
first system is given by
where u(E) is the density of states with energy E. In 1904 Einstein deduced a
formula for the mean square energy fluctuation
of the first system. Differentiating Eq. 4.11 with respect to /3, he obtained
The quantity («^2 ) (Einstein noted) is a measure for the thermal stability of the
system. The larger the fluctuations, the smaller the system's degree of stability.
'Thus the absolute constant* [k] determines the thermal stability of the system.
[Equation 4.13] is of interest since it does not contain any quantities which remind
one of the assumptions on which the theory is based' [El2].
Next, Einstein introduced a criterion for fluctuations to be large:
This relation is not satisfied by a classical ideal gas under normal conditions, since
then (E) = nkT/2 (n is the number of particles) so that £ = 0(n~'), indepen-
dent of the volume. He went on to note that £ can be of order unity only for one
kind of system: blackbody radiation. In that case, (E) = aVT^4 , by the Stefan-
Boltzmann law (V is volume, a is a constant), and hence £ = 4k/aVTi. The
temperature T is proportional to the inverse of Xmax, the wavelength at which the
spectral distribution reaches its maximum. He therefore concluded that volume
dependence is important: for fixed T, £ can become large if X^ax/ V is large, i.e.,
'Einstein used a symbol other than k.