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In my text (Oxford Solid State Basics by Steven Simon, page 11), it was stated that Debye wrote the following expression

⟨E⟩=3∑→kℏω(→k) [nB(βℏω(→k))+12]

What was not stated was the meaning of this expression. The only mention was that it was completely analogous to Einstein's expression for the averaged energy of a quantum harmonic oscillator in 1D.

⟨E⟩=∑kℏω [nB(βℏω)+12]

However, I can't seem to draw the link between the 2 expressions. Could someone explain to me

1) the interpretation of Debye's expression

2) how Debye's expression arises from a partition function (and how the partition function comes about),

3) and also the link between the 2 equations?

Many thanks in advance!