I have a phononic system with its distribution

$$N_B(E,T) = \frac{1}{e^{E/k_BT}-1}$$

where $E$ is the energy and $T$ the temperature of the system.

I'd like to know how to make an expansion of this at low temperatures, that is $$E>>k_B T$$

I already googled but I haven't found anything useful

  • 3
    $\begingroup$ This is a basic calculus question, is it not? $\endgroup$ – Jon Custer Feb 10 '17 at 3:32
  • $\begingroup$ So basic that I couldn't find a convincing answer $\endgroup$ – Daniel Feb 10 '17 at 11:42
  • 1
    $\begingroup$ Low temperature compared to what? An expansion with respect to what? What is the context? $\endgroup$ – Mark Mitchison Feb 10 '17 at 12:44

Hint: Multiply both the numerator and the denominator by $e^{-E/k_BT}$ to get

$$N_B(E,T) = \frac{1}{e^{E/k_BT}-1}=e^{-E/k_BT}\cdot \frac{1}{1-e^{-E/k_BT}}$$

This gives you something that looks like $\dfrac{1}{1-x}$ which can be expanded for $|x|<1$.


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