where we utilize the average frequency 𝜔 ̅ as the characteristic phonon frequency of the mobile ion and 𝑛 c is the average critical phonon occupation number needed for the jump to occur. Since phonon occupation is also frequency dependent (e.g. Bose-Einstein statistics), it is not clear at first glance if this theory provides any further insights than those of Rice and Roth 21 or Mahan. Still, the phenomenological implementation of the multi-excitation theory has been used to explain Meyer-Neldel behavior in diverse physical systems, including ionic conductors. Because 𝑛 c cannot be treated as an independent variable, i.e. its value cannot change independent of the frequency, the Taylor expansion in frequency (Eq. 2) must make use of the chain rule, giving