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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

$$\langle E \rangle = 3\sum_{\vec{k}} \hbar \omega (\vec{k}) \ [ n_B (\beta \hbar \omega (\vec{k})) + \frac{1}{2}]$$

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.

$$\langle E \rangle = \sum_k \hbar \omega \ [ n_B (\beta \hbar \omega ) + \frac{1}{2}]$$

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!