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Minimal of free energy equation

  1. Feb 1, 2015 #1
    1. The problem statement, all variables and given/known data
    Show that the free energy, F=SUM(E_i*p_i)+T*SUM(p_i*ln(p_i)) is minimal when p_i=e^(-E_i/T)/SUM(e^(-E_i/T))

    Where E is energy, T is temperature

    2. Relevant equations
    Nothing really

    3. The attempt at a solution
    So if I just try and find the derivative of the force I arrive at something along the lines of p_i=e^((-E_i/T)-1).
    I'm assuming I'm supposed to use some constraints to show that the entropy is fixed but I'm not sure how to go about doing that.
     
  2. jcsd
  3. Feb 1, 2015 #2
    Hi. Note that F is not a force but the Helmholtz free energy defined as F = U–TS;
    Also note that you do have a minimum when pi is proportional to exp(–1–Ei/T) as you vary F and set δF = 0, but pi are probabilities and therefore they need to be normalized to 1...
     
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