Temperature and level temperatures!

  1. Shyan

    Shyan 1,809
    Gold Member

    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.
     
  2. jcsd
Know someone interested in this topic? Share a link to this question via email, Google+, Twitter, or Facebook

Have something to add?

0
Draft saved Draft deleted