## Wednesday, February 26, 2014

### quantum statistics

can you write the Fermi-Dirac-distribution as a superposition of Bose-Einstein-distributions at different temperatures? is the inverse possible as well?

#### 1 comment:

1. this funny result can be explained by using the addition theorem of the hyperbolic tangent and cotangent,

\tanh(x) + \coth(x) = 2\coth(2x)

which follows directly from the addition theorem,

\tanh(x+y) = \frac{\tanh(x)+\tanh(y)}{1+\tanh(x)\tanh(y)}

by setting $x=y$. by writing the hyperbolic functions in terms of exponentials,

\tanh(x) = \frac{\exp(2x)-1}{\exp(2x)+1}

you can show that

\frac{1}{\exp(x)-1} - \frac{2}{\exp(2x)-1} = \frac{1}{\exp(x)+1}

such that the Fermi-Dirac-distribution is the superposition of two Bose-Einstein-distributions at two temperatures separated by a factor of two. because the addition theorem of the cotangent is equivalent to that of the tangent, you can't do the inverse.