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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 \epsilon_1 and \epsilon_2 with \epsilon_1<\epsilon_2.
a) With no explicit computation, draw a qualitative representation of the mean energy \bar{E} of the system as a function of the temperature T. What happens to \bar{E} in the limit of very small and very large temparatures? Approximately at which value of T does \bar{E} changes from its low temperature limit to its high temperature limit?
He drew a curve that starts at T=0 with \bar{E}(0)=N\epsilon_1, rises up, appears to have an inflexion point at (\epsilon_2-\epsilon_1)/k, and approaches N(\epsilon_1+\epsilon_2)/2 as T\rightarrow +\infty.
I just really don't know how he knows all that. The only relation btw T and E I know is really not helpful:
\frac{1}{kT}=\frac{\partial \ln(\Omega)}{\partial E}
There's one in term of the partition function too but we're not allowed to calculate it.
A system contains N weakly interacting particles, each of which can be in either one of two states of respective energies \epsilon_1 and \epsilon_2 with \epsilon_1<\epsilon_2.
a) With no explicit computation, draw a qualitative representation of the mean energy \bar{E} of the system as a function of the temperature T. What happens to \bar{E} in the limit of very small and very large temparatures? Approximately at which value of T does \bar{E} changes from its low temperature limit to its high temperature limit?
He drew a curve that starts at T=0 with \bar{E}(0)=N\epsilon_1, rises up, appears to have an inflexion point at (\epsilon_2-\epsilon_1)/k, and approaches N(\epsilon_1+\epsilon_2)/2 as T\rightarrow +\infty.
I just really don't know how he knows all that. The only relation btw T and E I know is really not helpful:
\frac{1}{kT}=\frac{\partial \ln(\Omega)}{\partial E}
There's one in term of the partition function too but we're not allowed to calculate it.
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