In statistical mechanics, we define(adsbygoogle = window.adsbygoogle || []).push({});

[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?

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Conceptual Difficulty with Stat Mech

**Physics Forums | Science Articles, Homework Help, Discussion**