102 STATISTICAL PHYSICSthrough a gas increases strongly in a neighborhood O(1°C) of the critical point.
In 1908 Smoluchowski became the first to ascribe this phenomenon to large den-
sity fluctuations [S7]. He derived the following equation for the mean square par-
ticle number fluctuations S^5 :valid up to terms O((d}p/dV^)T). For an ideal gas, W ~ I/TV, but near the critical
point, where (dp/dV)T = (8ip/dV^2 )T = 0, the rhs of Eq. 5.22 blows up. 'These
agglomerations and rarefactions must give rise to corresponding local density fluc-
tuations of the index of refraction from its mean value and thus the coarse-
grainedness of the substance must reveal itself by Tyndall's phenomenon, with a
very pronounced maximal value at the critical point. In this way, the critical
opalescence explains itself very simply as the result of a phenomenon the existence
of which cannot be denied by anybody accepting the principles of kinetic theory'
[S8].
Thus, Smoluchowski had seen not only the true cause of critical opalescence
but also the connection of this phenomenon with the blueness of the midday sky
and the redness at sunset. Already in 1869 John Tyndall had explained the blue
color of the sky in terms of the scattering of light by dust particles or droplets, the
'Tyndall phenomenon' [T5]. Rayleigh, who worked on this problem off and on
for nearly half a century, had concluded that the inhomogeneities needed to
explain this phenomenon were the air molecules themselves. Smoluchowski
believed that the link between critical opalescence and Rayleigh scattering was a
qualitative one. He did not produce a detailed scattering calculation: 'A precise
calculation ... would necessitate far-reaching modifications of Rayleigh's calcu-
lations' [S7].
Along comes Einstein in 1910 and computes the scattering in a weakly inho-
mogeneous nonabsorptive medium and finds [E10] (for monochromatic polarized
light)
where r is the ratio of the scattered to the primary intensity, n the index of refrac-
tion, v the specific volume, A the incident wavelength, $ the irradiated gas volume,
A the distance of observation, and t? the scattering angle. For an ideal gas (n «
1),'[Equation 5.24] can also be obtained by summing the radiations off the individual
molecules as long as these are taken to be randomly distributed'. Thus Einstein