Minimal of free energy equation

  • #1

Homework Statement

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

Homework Equations

Nothing really

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.
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  • #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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