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.