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Determine the Einstein relations by upward and downward transition.

While the system is in equilibrium, the upward and downward transition rates should be equal.

N₁ρ(ν) B₁₂= N₂ ρ(ν) B₂₁+ N₂ A₂₁ ........... (1)

Hence ρ(ν) = N₂ A₂₁ / N₁B₁₂- N₂ B₂₁ ........... (2)

The Einstein relations are specified by the equations 3 and 4.

g₁B₁₂= g₂B₂₁ ........ (3)

A₂₁/B₂₁= 8πhν³/c³........ (4)

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