# How does the average energy from equation (41.15) of Feynman's lectures approach $kT$ as omega goes to zero?

Feynman said the following equation should approach $$kT$$ as $$\omega$$ goes to zero, or $$T$$ goes to infinity.

$$\langle E \rangle = \frac{\hbar \omega}{e^{\hbar \omega/kT}-1}$$

Does anybody know how to prove this?

Get use of L'Hopital's rule where ' stands for the derivative

$$\lim_{\omega \to 0} \langle E\rangle= \lim_{\omega \to 0}\frac{(\hbar \omega)'}{(e^{\frac{\hbar \omega}{kT}}-1)'} = \lim_{\omega \to 0} kT e^{-{\frac{\hbar \omega}{kT}}}=kT$$

The expression for $$\langle E \rangle$$ is not defined at $$\omega = 0$$ as it would be $${0 \over 0}$$. So we need to take the limit $$\omega \to 0$$ i.e.

$$\lim_{\omega\to 0} \langle E \rangle = ?$$

To solve this, very simply, use the fact that the Taylor expansion of the exponential function is

$$e^x\approx 1+x+{1\over 2}x^2 + ...$$ which in your case is $$e^{\hbar\omega \over k_B T}\approx 1 + {\hbar\omega \over k_B T}+{1\over 2}\left({\hbar\omega \over k_B T}\right)^2+...$$

so $$\langle E \rangle \approx {\hbar\omega \over 1 + {\hbar\omega \over k_B T}+{1\over 2}\left({\hbar\omega \over k_B T}\right)^2+... -1}$$ which becomes

$$\langle E \rangle \approx {\hbar\omega \over {\hbar\omega \over k_B T}+{1\over 2}\left({\hbar\omega \over k_B T}\right)^2} = {\hbar\omega\over\hbar\omega} {k_BT \over 1+{1\over 2}{\hbar\omega \over k_B T}}$$ and finally

$$\langle E \rangle \approx {k_BT \over 1+{1\over 2}{\hbar\omega \over k_B T}}$$ This expression is valid if $$\omega\approx 0$$ and describes the behavior of $$\langle E \rangle$$ at low frequencies.

From this, just take the limit $$\omega\to 0$$, and it should be clear that $$\lim_{\omega\to 0} \langle E \rangle = \lim_{\omega\to 0} \langle E \rangle {k_BT \over 1+{1\over 2}{\hbar\omega \over k_B T}} \to k_B T$$

Use the Taylor series for the exponential function,

$$e^x=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\dots.$$

In addition to giving you the $$\omega\to 0$$ limit, it will give you the low-frequency corrections.