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Plot Heat Capacity vs Temperature for a 2 state system microcananonical ensemble

  1. Dec 26, 2016 #1
    1. The problem statement, all variables and given/known data

    I have ##C= NK_B (\frac{\epsilon}{K_B T})^{2}e^{\frac{\epsilon}{K_B T}}\frac{1}{(e^{\frac{\epsilon}{K_BT}}+1)^2} ##

    and need to sketch ##C## vs. ##T##

    2. Relevant equations

    See above

    3. The attempt at a solution

    I have ##C= NK_B (\frac{\epsilon}{K_B T})^{2}e^{\frac{\epsilon}{K_B T}}\frac{1}{(e^{\frac{\epsilon}{K_BT}}+1)^2} ##

    Considering asymptotic limits I have:

    ##C \to e^{-\frac{\epsilon}{K_{B}T}} ## as ##T \to 0##
    ##C \to \frac{1}{T^{2}} ## as ##T \to \infty##

    The solution is attached.

    So from these limits I get the shape at these ends, and deduce there is a maximum to allow me to sketch the rest of it.

    I am unsure how to deduce this maximum?

    Differentiating gives quite a mess and it seems that it should be obvious to conclude the maximum is at ## \epsilon / K_{B} ##, or at least a better method to find this point? (My knowledge of graph sketching is quite poor).

    Many thanks in advance.
     

    Attached Files:

    Last edited: Dec 26, 2016
  2. jcsd
  3. Dec 27, 2016 #2

    bump
     
  4. Dec 27, 2016 #3

    vela

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    The only suggestion I'd have is to let ##u=\frac{\epsilon}{k_\text{B}T}## and find the extremum of
    $$\frac{u^2 e^u}{(1+e^u)^2}.$$ It shouldn't be that messy.
     
  5. Dec 30, 2016 #4
    I have:

    ##2e^u+2+u-e^u u =0 ## , unsure of where to go now...
     
  6. Dec 30, 2016 #5

    vela

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    You'd have to solve that numerically. To get a qualitative idea of where the root lies, you can rewrite that equation as
    $$e^u = \frac{u+2}{u-2}.$$ Plot graphs of the two sides of the equations and see where they intersect.
     
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