How Is the Variance of a Quantity Derived in Statistical Mechanics?

[darkchild]

Homework Statement

From Landau and Lifgarbagez:

$$\langle (\Delta f)^{2} \rangle = \overline{f^{2}} - (\overline{f})^{2}$$

This isn't derived, just stated, and I'd like to understand how it comes about. f is a generic quantity "relating to a macroscopic body or to a part of it."

Homework Equations

$$\Delta f = f - \overline{f}$$

The Attempt at a Solution

$$(\Delta f)^{2} = (f-\overline{f})^{2} = f^{2} - 2f \overline{f} + \overline{f}^{2}$$

Basically, I don't know how to do the averaging (not without explicit values of f, anyhow).

 

[vela]

Basically, I don't know how to do the averaging (not without explicit values of f, anyhow).

You put angle brackets around it or put a bar over it. ;)

Remember that $$\overline{f}$$ is a constant, and use the fact that

$$\langle \alpha f+\beta g\rangle = \alpha\langle f\rangle+\beta\langle g\rangle$$

where $\alpha$ and $\beta$ are constants.