# Homework Help: Minimal of free energy equation

1. Feb 1, 2015

### finchesarecool

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. Feb 1, 2015

### Goddar

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...