Consider, for instance, a simple system that has only two energy levels \( \epsilon_0 = 0 \) and \( \epsilon_1 = \Delta E \).
For a system governed by the Boltzmann distribution we find (the partition function has been taken out)
$$ W(0\rightarrow 1)\exp{-(\epsilon_0/kT)} = W(1\rightarrow 0)\exp{-(\epsilon_1/kT)}. $$We get then
$$ \frac{W(1\rightarrow 0)}{W(0 \rightarrow 1)}=\exp{-(\Delta E/kT)}, $$which goes to zero when \( T \) tends to zero.