120 Thermal History of the Universe
The±sign is ‘−’ for bosons and ‘+’ for fermions, and the name for these distributions
are theBose distributionand theFermi distribution, respectively. The Fermi distribu-
tion in the above form is actually a special case: it holds when the number of charged
fermions equals the number of corresponding neutral fermions (the ‘chemical poten-
tials’ vanish). In the following we shall need only that case.
The number density푁of nonrelativistic particles of mass 푚is given by the
Maxwell–Boltzmanndistribution for an ideal, nondegenerate gas. Starting from Equa-
tion (6.28) we note that for nonrelativistic particles the energykTis smaller than the
mass, so that the term±1 in can be neglected in comparison with the exponential.
Rewriting the Fermi distribution as a function of temperature rather than of momen-
tum we obtain the Maxwell–Boltzmann distribution
푁=푛spin
( 2 휋mkT)^3 ∕^2
(hc)^3e−퐸푖∕푘푇푖. (6.29)Note that because of the exponential term the number density falls exponentially as
temperature falls.James Clerk Maxwell(1831–1879) was a contemporary of Stefan
and Boltzmann.
Thermal Death. Suppose that the Universe starts out at some time with훾rays
at high energy and electrons at rest. This would be a highly ordered nonequilib-
rium system. The photons would obviously quickly distribute some of their energy
to the electrons via various scattering interactions. Thus the original order would
decrease, and the randomness or disorder would increase. The second law of ther-
modynamics states that any isolated system left by itself can only change towards
greater disorder. The measure of disorder is entropy; thus the law says that entropy
cannot decrease.
The counterexample which living organisms seem to furnish, since they build up
ordered systems, is not valid. This is because no living organism exists in isolation;
it consumes nutrients and produces waste. Thus, establishing that a living organism
indeed increases entropy would require measurement of a much larger system, cer-
tainly not smaller than the Solar System.
It now seems to follow from the second law of thermodynamics that all energy
would ultimately distribute itself evenly throughout the Universe, so that no further
temperature differences would exist. The discoverer of the law of conservation of
energy,Hermann von Helmholtz(1821–1894), came to the distressing conclusion in
1854 that ‘from this point on, the Universe will be falling into a state of eternal rest’.
This state was namedthermal death, and it preoccupied greatly both philosophers and
scientists during the nineteenth century.
Now we see that this pessimistic conclusion was premature. Since퐸scales as푎−^1
it follows that also the temperature of radiation푇rscales as푎−^1. Thus, from the time
when the temperatures of matter and radiation were equal,
푇m=푇r,we see from Equation (6.27) that the adiabatic expansion of the Universe causes mat-
ter to cool faster than radiation. Thus cold matter and hot radiation in an expanding