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Mean energy behavior as a function of T.

  1. Nov 14, 2006 #1

    quasar987

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    Here's a problem the TA made but now that I look back at it, I wonder how he did it.

    A system contains N weakly interacting particles, each of which can be in either one of two states of respective energies [itex]\epsilon_1[/itex] and [itex]\epsilon_2[/itex] with [itex]\epsilon_1<\epsilon_2[/itex].

    a) With no explicit computation, draw a qualitative representation of the mean energy [itex]\bar{E}[/itex] of the system as a function of the temperature T. What happens to [itex]\bar{E}[/itex] in the limit of very small and very large temparatures? Approximately at which value of T does [itex]\bar{E}[/itex] changes from its low temperature limit to its high temperature limit?


    He drew a curve that starts at T=0 with [itex]\bar{E}(0)=N\epsilon_1[/itex], rises up, appears to have an inflexion point at [itex](\epsilon_2-\epsilon_1)/k[/itex], and approaches [itex]N(\epsilon_1+\epsilon_2)/2[/itex] as [itex]T\rightarrow +\infty[/itex].

    I just really don't know how he knows all that. The only relation btw T and E I know is really not helpful:

    [tex]\frac{1}{kT}=\frac{\partial \ln(\Omega)}{\partial E}[/tex]

    There's one in term of the partition function too but we're not allowed to calculate it.
     
    Last edited: Nov 14, 2006
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