# Is the normal mode frequency of harmonic oscillator related to temperature?

It is well known that for an isolated system, the normal mode frequency of a N-body harmonic oscillator satisfies Det(T-$$\omega^{2}$$V)=0. How about a non-isolated, fixed temperature system?
In solid state physics I have learned that in crystal the frequency does not change, but the amplitude of each mode changes. But how about the free energy? Shouldn't the free energy be always at the minimum?
I am confused...
Thanks so much.

Your question is not very clear.
What are T and V in your dispersion relation?

If the system has a given fixed temperature, what would change if it is isolated or not? Why would the normal mode be affected?

Concerning the free energy. A system at equilibrium, like your collection of oscillators at a given temperature, has already reached the minimum free energy.
How could the free energy be further decreased?
Which parameter/variable could you modify to decrease its free energy?
I see none.

If you collection of oscillators contained two sub-collections each with its own temperature, then the free energy could still decrease when this out-of-equilibrium system would reach equilibrium. This would happen by the system reaching one uniform temperature. This would imply an exchange of energy between the different modes until free energy is minimum (for a fixed total energy).

Remember that the distibution of energy on the different modes is precisely the result of reaching equilibrium with a minimum free energy. However, the modes themselves are not influenced.

Hmm, I see. Your word "distribution of energy on the different modes is precisely the result of reaching equilibrium with a minimum free energy" made me clear. Thanks a lot.