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In statistical mechanics, we define

[tex] \frac{1}{\tau}=\left( \frac{\partial \sigma}{\partial U} \right)_N[/tex]

This formula gives the temperature as a function of the energy of the system and N. So knoweldge of U and N determines the temperature of the system. Conversely, at least when the above equation can be inverted, knowledge of the temperature and N determines the energy exactly.

This is contrasted with the ensemble average: [tex] U = \tau^2 \frac{\partial \log Z}{\partial \tau} [/tex]. This seems to imply that knowledge of the temperature and N only determines the ensemble average energy, whereas before, the energy was determined exactly.

For concreteness, consider a system of N spin 1/2 particles. In the large N limit,

[tex] \frac{1}{\tau}= \frac{-U}{Nm^2B^2}[/tex]

whereas

[tex] <U> = -2mBN \sinh (mB/\tau)[/tex].

How is it that these two formulas are not in conflict with each other?

Edit: I just realized that the second formula is actually an exact expression and in the appropriate limit reduces to the first. However, I am still confused about this. Why is averaging necessary in one case, while not in the other?

[tex] \frac{1}{\tau}=\left( \frac{\partial \sigma}{\partial U} \right)_N[/tex]

This formula gives the temperature as a function of the energy of the system and N. So knoweldge of U and N determines the temperature of the system. Conversely, at least when the above equation can be inverted, knowledge of the temperature and N determines the energy exactly.

This is contrasted with the ensemble average: [tex] U = \tau^2 \frac{\partial \log Z}{\partial \tau} [/tex]. This seems to imply that knowledge of the temperature and N only determines the ensemble average energy, whereas before, the energy was determined exactly.

For concreteness, consider a system of N spin 1/2 particles. In the large N limit,

[tex] \frac{1}{\tau}= \frac{-U}{Nm^2B^2}[/tex]

whereas

[tex] <U> = -2mBN \sinh (mB/\tau)[/tex].

How is it that these two formulas are not in conflict with each other?

Edit: I just realized that the second formula is actually an exact expression and in the appropriate limit reduces to the first. However, I am still confused about this. Why is averaging necessary in one case, while not in the other?

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