Temperature and level temperatures

[ShayanJ]

I remember a day in the last year thinking about Fermi temperature($T_f=k \varepsilon_f$) and its interpretation.People were saying its just a notation and has no meaning but I doubted that.So I did some calculations and had some interpretations.Then I forgot it until recently that I remembered it again.Now I want to know others' ideas about it.I know maybe some of you say personal theories aren't allowed here but I don't think its that big to be called a personal theory.

Consider the formula $\frac{1}{T}=\frac{\partial S}{\partial E}$ which can also be written in the form $\frac{1}{T}=\sum \frac{\partial S}{\partial N_i}\frac{\partial N_i}{\partial E}$.
Taking into account $E=\sum N_i \varepsilon_i$ we can write for a constant volume system $dE=\sum \varepsilon_i dN_i$ and assuming $\frac{\partial x}{\partial y}=\frac{1}{\frac{\partial y}{\partial x}}$,we have $\frac{\partial N_i}{\partial E}=\frac{1}{\varepsilon_i}$.
Now considering the Boltzmann statistics,we have $\frac{\partial S}{\partial N_i}=k\ln{\frac{g_i}{N_i}}$ so at last we have $\frac{1}{T}=\sum \frac{1}{T_i}\ln{\frac{g_i}{N_i}}$ where $T_i=\frac{\varepsilon_i}{k}$.
There can be similar formulas for the other two statistics.
I know it can't be used for true calculations but I think an interpretation of it is that we can assign a temperature to each energy level and then each of these "level temperatures" contribute to the real temperature by a factor which is determined by the properties of that level and also the statistics we're assigning to the particles.
Any idea is welcome.

 

[Greg Bernhardt]

@ShayanJ have you come to more understanding on this?