Conceptual Difficulty with Stat Mech

[Euclid]

In statistical mechanics, we define

$$\frac{1}{\tau}=\left( \frac{\partial \sigma}{\partial U} \right)_N$$

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: $$U = \tau^2 \frac{\partial \log Z}{\partial \tau}$$. 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,

$$\frac{1}{\tau}= \frac{-U}{Nm^2B^2}$$

whereas

$$<U> = -2mBN \sinh (mB/\tau)$$.

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?

 

[quetzalcoatl9]

an instantaneous energy and the average energy are not the same thing.

your first energy, the one involving the derivative of the ln of the partition function, is infact an average (although you didn't explicitly write it this way for some reason).

This seems to imply that knowledge of the temperature and N only determines the ensemble average energy, whereas before, the energy was determined exactly.

Don't forget the partition function! Without that you wouldn't know how the energy is actually partitioned amongst the possible states, and therefore wouldn't know the average energy!

 

[Euclid]

I am aware of the distinction between the energy of a state and the ensemble average energy of a system. In fact, that's basically the cause of my confusion. In the definition of temperature, there no statistical averaging going on, yet you get the same relationship between T, U and N as you do when you take averages, as in the second formula.

I think about this as follows. Given a system S in thermal equilibrium with a reservoir R at temperature T, it follows from the 1st L.T. that S is at temperture T. Therefore, from the relation $$1/T = \partial_U \sigma(U,N)$$ (assuming I can invert this relation to find U in terms of T) we know the energy of the system S. Now on the other hand, we are taught that there is no definite relationship between these quantities. The system S could have any energy with relative probability given by the Boltzmann factor. All we can know is the average energy. It seems to me that there is a genuine conflict here. Am I wrong?

 

[quetzalcoatl9]

but you are in the canonical ensemble (constant N,V,T) so the energy will vary..i'm afraid i don't understand your question, can you phrase it differently?

 

[Euclid]

I think I figured out my problem (and now that I have it is quite clear why you didn't understand my question). The problem was that I was thinking of two systems in thermal equilibrium as necessarily having equal temperatures, while in principle, equal temperature is only the most likely outcome.