Temperature and level temperatures!

by Shyan
Tags: temperature, temperatures
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Jan12-14, 12:37 AM
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I remember a day in the last year thinking about Fermi temperature([itex] T_f=k \varepsilon_f [/itex]) 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 [itex] \frac{1}{T}=\frac{\partial S}{\partial E} [/itex] which can also be written in the form [itex] \frac{1}{T}=\sum \frac{\partial S}{\partial N_i}\frac{\partial N_i}{\partial E} [/itex].
Taking into account [itex] E=\sum N_i \varepsilon_i[/itex] we can write for a constant volume system [itex] dE=\sum \varepsilon_i dN_i [/itex] and assuming [itex] \frac{\partial x}{\partial y}=\frac{1}{\frac{\partial y}{\partial x}} [/itex],we have [itex] \frac{\partial N_i}{\partial E}=\frac{1}{\varepsilon_i} [/itex].
Now considering the Boltzmann statistics,we have [itex] \frac{\partial S}{\partial N_i}=k\ln{\frac{g_i}{N_i}} [/itex] so at last we have [itex] \frac{1}{T}=\sum \frac{1}{T_i}\ln{\frac{g_i}{N_i}} [/itex] where [itex] T_i=\frac{\varepsilon_i}{k} [/itex].
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.
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