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Date: 8-8-2016
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Date: 9-8-2016
880
Date: 16-3-2021
1388
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Blackbody Radiation and Early Universe
The entropy of the blackbody radiation in the early universe does not change if the expansion is so slow that the occupation of each photon mode remains constant (or the other way around). To illustrate this consider the following problem. A one-dimensional harmonic oscillator has an infinite series of equally spaced energy states, with εn = nhω where n is a positive integer or zero and ω is the classical frequency of the oscillator.
a) Show that for a harmonic oscillator the free energy is
b) Find the entropy S. Establish the connection between entropy and occupancy of the modes by showing that for one mode of frequency ω the entropy is a function of photon occupancy 〈n〉 only:
SOLUTION
a) By definition the free energy
(1)
b) The entropy is then
(2)
The energy of the system
(3)
Alternatively, the entropy can be found from
or
where τ can be expressed from (3) as
(4)
So,
(5)
where we let ξ ≡ hω. Performing the integral gives
(6)
Substituting the energy for this mode E = hω〈n〉, we recover the entropy in the form
(7)
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