Stat Mech, question about average of function of ith state.

[Zacarias Nason]

I just started learning some Stat. Mech by Leonard Susskind's lectures and am in a part briefly overviewing basic probability in general; One of the things brought up was F(i), some quantity associated with the ith state of a system, and the important average of F(i) averaged over the probability distribution, or:

$$< F(i)> \ = \sum_{i} F(i)P(i)$$

where F(i) is again some quantity associated with the ith state and P(i) is the probability of that particular outcome; Susskind emphasizes that the average of F(i) does not have to be any of the values it can take on, e.g. if your system is a coin toss where heads and tails are assigned values of F(H) = 1, F(T) = -1 respectively, <F(i)> when you have a very large number of tosses should approach zero. If this can't represent one of the values of F(i), what is the value or purpose of <F(i)>? What does it tell you? Why is it important?

 

[DrClaude]

If this can't represent one of the values of F(i), what is the value or purpose of <F(i)>? What does it tell you? Why is it important?

It can tell you a lot. For instance, in the example of the coin, having an average of zero tells you that the coin is fair. Having an average of 1/2 tells you that the probability for head is higher than the probability for tail, that the coin is not fair. For a certain physical system, this could tell you that the energy of the "head" state is lower than the energy of the "tail" state.

What you have to keep in mind is that statistical physics often deals with very large systems (~10²³ particles), where all that you will be able to measure are average values on that ensemble, not the state of individual particles making up that system. Averages are therefore closely linked to measurements.