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Quantum Statistics

Bose-Einstein distribution

1. Abstarct:

In 1924, with the concept of photon, Sateyndra Bose rethink the study of the blackbody radiations. In a cavity at a temperature T, the thermal ray is constituted by monochromatic and independent electromagnetic waves, with all possible wavelengths. We can say then each radiation is a photon carrying enrgy hν. The all radiations inside the cavity form a set of photons. Inside the blackbody cavity, we have a gas of photons. Bose assumed that the photons are individually indistinguishable, and set a new statistical method to study the thermal ray. In addition to the indistinguishability, if the internal (total) energy of the blackbody cavity is well defined, It is not the case for the number N of photons. The number of photons of the system (vacuum of the cavity) fluctuates. Indeed, we can add any number of photons in the cavity with null energy (frequency null), without affecting the actual photons distribution. This number total number N of photons inside the cavity is then not delimited. The constraint N = ∑ni is removed. Bose used these two new particularities to derive, for photons, the spectral energy density of the blackbody cavity. Einstein extended the Bose theory to the molecules gas. The related distribution is then called the Bose-Einstein distribution. That is the purpose of this chapter.

2. Canonical ensembles

Let's consider any thermodynamic system. Its internal energy U can vary in the three foolowing ways:

- Supplied work dW = - PdV. P is the exerted pressure responsible to the change dV in the volume.

- Supplied heat dQ = TdS: isentropique process. T is the temperature brought from the exterior (surrounding) to the system, and cause a change in th entropy.

- Supplied matter: dM = μdN . μ is the chemical potential responsible to a matter supply to the system.

All gives to the system the variation: dU = -PdV + TdS + μdN. We consider the surrounding for the system as a reservoir that feeds this system. Once the equilibriun is reached, the temperature of the system and the temperature of the reservoir are equal.

2.1 Microcanonical description

If the system is islated (cut from the reservoir), we can write the entropy of the system as S = k lnΩ K is the Boltzmann constant and Ω is the number of the macrostates for the system, and deduce the temerature of the system by 1/T = ∂S/∂U. Such description is called microcanonical.

2.2. Canonical description

If the system exchanges energy with the reservoir, we say that they are in thermal contact. The reservoir imposes its temperature and becomes a thermostat. If the system is in a state "i" with the energy Ei, and the energy of the reservoir is E, then the energy of the reservoir is E - Ei. The probability to have the system this situation Pi can be written as: Pi = constant Ω(E - Ei); whwere Ω(E - Ei) is the number of macrostates for the reservoir. Using the Boltzman formula S(E - Ei) = k ln Ω(E - Ei) leads to Pi = exp {- Ei/kT}/Z. Z is the partition function for the system = ∑ exp {- Ei/kT} ( sum over i, that is all the states of the system). Having the probability Pi for each state, we can deduce the average of many physical quantites (X = ∑PiXi), such as internal energy, free energy. Such description is called canonical.

2.3. Grand canonical ensemble

In addition the supply of energy, now the reservoir feeds the system with particles. Here we develop S(E - Ei, N - Ni), where N is the total number of particles. Using the definition of the chemical potential μ = - T ∂S/∂N and β = 1/kT, we will derive Pi = exp {- β (Ei - μNi)}/Ξ, where Ξ = ∑ exp {- β (Ei - μNi)} ( sum over i, that is all the states of the system). Ξ is called the parition function of the grand canonical.

For the system, the number of particles fluctuates. Let, at one moment, Ni the number of the particles for the system, and Ei its energy. All these Ni particles, asw ell as the energy Ei, are distributed among the nj levels (states) of the system. If the state "j" of the system, with nj particles and energy ej, we can write:

Ni = &sum nj and Ei = &sum nj ej Each configuration {nj} corresponds to a current state "i" of the system with energy Ei and particles Ni.

Of course, the number of particles for (the reservoir + the system) is constant, but the number of particles for the system changes.

Now, we can rewrite the grand canonical partition function : Ξ = ∑ exp {- β (&sum nj ej - μnj)} = ∑ exp {- β [&sum (ej - μ)nj]}
The sum is over nj, that is over all the number nj of the configuration {i} state of the system. In other words, the sum is set over the set {nj}.

Now, if the particles are distinguishable, the multiplicity of the system for the configuration {i} is Ω(i) = ΠΩ(j) = Ni!/Πnj!, then the grand canonical partition function becomes:
Ξ = ∑ ΠΩ(j) x exp {- β &sum (ej - μ)nj} = ∑ Ni!/Πnj! x exp {- β &sum (ej - μ)nj} = ∑ Ni!/Πnj! x exp {- β &sum ejnj} x exp {+ βμ ∑nj} = ∑ Ni!/Πnj! x Π exp {- βejnj} x exp {+ βμ Ni} =
∑ {exp βμ} Ni Ni! /Πnj! x Π exp {- βejnj}.

Ξ = ∑ (expβμ)Ni Ni! /Πnj! x Π exp {- βEi}
The sum is over all the states "i" of the system.

But the photons are indistinguishable. If Ni from the reservoir fluctuates then nj fluctuates as well. Then the number of particles nj over the levels "j" of the system are independents and fed directly from the reservoir. The constraint ∑nj = Ni is released.

Then at a certain configuration "j", we have: ξ as the partition function for the system. ξ = ∑ exp {- β (ej - μ)nj} (sum over independents nj) Then the total partition function for all the levels "j", we have:

Ξ = Π ξ (product over j); that is: Ξ = Π ∑ exp {- β (ej - μ)nj} This description is called grand canonical. For more details, see Partition functions chapter.

3. Chemical potential of the photon gas

From the expression of ξ we can derive the average number of photon within the level "j". This number is nj = ∑ Probability(nj) nj/∑nj Probability(nj) = ∑nj x exp {- β (ej - μ) nj}/Ξ
Thus: nj = ∑ exp {- β (ej - μ) nj}/Ξ nj But - (1/β) ∂ ln(ξ)/∂ej = - (1/β) [∂ξ/∂ej]/ξ = - (1/β) x ( - β nj exp {- β (ej - μ) nj})/ξ =
nj exp {- β (ej - μ) nj})/ξ = nj Hence:
nj = (1/β) ∂ ln(ξ)/∂ej.
Whith the bosons nothing limits the number to fill in the states nj, that is a state "j" with the numer of bosons nj and energy ej can contains any number of photons. Then ξ = ∑ exp {- β (ej - μ)nj}, the sum is over j from 0 to intinity. We can write then:
ξ = ∑ exp {- β (ej - μ)nj} [nj: 0 → ∞] = 1/(1 - exp {- β (ej - μ)})

The derivative gives:
nj = 1/ (exp {β (ej - μ) - 1})

Within a photon gas, the average number of bosons with the energy ε is:
n(ε) = 1/ (exp {β(ε - μ)}- 1)
μ is the chemical potential of the system.

4. photon gas in thermal equilibrium

The free energy F of the gas of photons is F = U - TS; then dF = dU -TdS - SdT . With dU = -PdV + TdS + μdN , we have: dF = -PdV + TdS + μdN -TdS - SdT = -PdV + μdN - SdT, then μ = [∂F/∂N](V,T constant). That is at the equilibrium between the wall of the cavity and the photons inside, F must be stationary; then dF = 0 and μ = 0

The condition for the thermal equilibrium of the system is mu; = 0

The photons have a spin equal to 1; then they are bosons. Finally, the distribution of the phtons inside the cavity is:

n (ε = hν) = 1/ (exp {β(hν - μ)}- 1) That is the Planck law.

1. In the above results we have assumed that the photons are independents and we didn't take account of the pair production (e -, e +) that can be happen with the collision between photons.
2. The advantage of the bose theory (grand canoncal) is that we consider the real existent facts which are the indistinguishability of photons and fluctuation (the wall of the blackbody cavity emits and absorbs photons continuously, the mechanism necessary to maintain the equilibrium. But the Planck law that used the Boltzmann formula Nn = N0 = exp{(En - Eo)/kT}, didn't consider the indistinguishability nor the fluctuation of photons. We have calculated an average number n. Let's say that the grand canonical methode is more adequate.

5. Some specific quantities of the photon gas

We have seen that the number of vibrational modes in the range ν and ν + dν is:
in the volume V: dg = (8πV/c32dν. With ε = hν, we have: dg = (8πV/(hc)32

The density of states is: ρ(ε) = dg/dε = (8πV/(hc)32

The average number of photons in the range: ε and ε + ε is :
dn = ρ(ε)n(ε) dε = (8πV/(hc)32/(exp{ε/kT} - 1)
Their energy is:
dE = ρ(ε)nε = (8πV/(hc)33 dε/(exp{ε/kT} - 1)
with ε = hν

The total numbe of particles in the cavity can be calculated as: N = ∫ dn [0 → ∞] = ∫ (8πV/(hc)32 dε/(exp{ε/kT} - 1)
Let x = ε/kT, then: N = (8πV/(hc)3) k3T3 ∫ x2 dx /(ex - 1) ∫ x2 dx /(ex - 1) = Γ(3) x ζ(3) = 2 x 1.202 = 2.404 Γ is the gamma function, and ζ is the Riemann zeta function.
N = 0.244 V (kT/ℏc)3

For example at T = 300 oK , kT ≈ 1 eV/40
ℏc = 197.32 Mev.Fermi;, we have within 1 m3, about 1 x 1020> photons.

E = ∫ dE [0 → ∞] = ∫ (8πV/(hc)33 dε/(exp{ε/kT} - 1) with x = ε/kT,
we have: E = (8πV (kT)4/(hc)3) ∫ x3 dx/(ex - 1) ∫ x3 dx/(ex - 1) [0 → ∞] = π4/15 = 6.494 Hence: E = E/V = (8π5k4/15(hc)3) x T4 ( Stefan law)

The total energy is U = EV = aV T4
E/N = 6.494 kT/2.404. Then E = 0.569 (kT)4/(ℏc )3
E = 2.70 N kT . At T = 300oK 1 kT ≈ 1 eV/40; then
E = 2.70 x 1020 x 1.6 10 -19/40 Joules

In 1 m3 at 300oK, we have 1 x 1020 photons and an available energy equal to 1 Joule; the equivalent of lifting one meter a mass of 100 grams.

The entropy of the photons at a constant volune V is: dS = dQ/T CvdT/T, where Cv = V cv; and cv is the specific heat capacity for the photons. We have cv = (1/V) dE/dT = 4a T4 at V = constan, then S = 4a ∫ T2 dT [from 0 to T]. According to Nerst principle (S = 0 at T = 0), we have:
S = (4/3)aV T3.

The entropy of the photons in a volume V, at the temperature T is: S = (4/3)aV T3

Using the relation E = aVT4 = 2.70 NkT, the ratio S/N has the following value: S/N = 2.70 kT x (4/3)aV T3/aVT4 = k x (4/3)/ 2.70 = 3.60 k.

The average entropy per photon is: s = 3.60 k ( universal constant independent of the temperature).

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