Statistical Mechanics Derivation

In summary, the equation \langle (\Delta f)^{2} \rangle = \overline{f^{2}} - (\overline{f})^{2} from Landau and Lifgarbagez states that the mean square deviation of a generic quantity f can be calculated by subtracting the square of the mean value of f from the mean value of its square. This equation is not derived, but rather stated, and it is important to note that \overline{f} represents a constant value. To calculate the mean value, you can use the equation \Delta f = f - \overline{f} and the fact that \langle \alpha f+\beta g\rangle = \alpha\langle f\rangle
  • #1
darkchild
155
0

Homework Statement



From Landau and Lifgarbagez:

[tex]\langle (\Delta f)^{2} \rangle = \overline{f^{2}} - (\overline{f})^{2}[/tex]

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



[tex]\Delta f = f - \overline{f}[/tex]

The Attempt at a Solution



[tex](\Delta f)^{2} = (f-\overline{f})^{2} = f^{2} - 2f \overline{f} + \overline{f}^{2}[/tex]

Basically, I don't know how to do the averaging (not without explicit values of f, anyhow).
 
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  • #2
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 [tex]\overline{f}[/tex] is a constant, and use the fact that

[tex]\langle \alpha f+\beta g\rangle = \alpha\langle f\rangle+\beta\langle g\rangle[/tex]

where [itex]\alpha[/itex] and [itex]\beta[/itex] are constants.
 

1. What is the purpose of a statistical mechanics derivation?

A statistical mechanics derivation is used to describe the behavior of a large number of particles or systems using statistical methods. It allows us to predict the macroscopic properties of a system based on the microscopic characteristics of its individual components.

2. How does a statistical mechanics derivation differ from classical mechanics?

In classical mechanics, the behavior of a system is determined by solving equations of motion for each individual particle. In contrast, a statistical mechanics derivation considers the statistical distribution of particles to predict the behavior of the system as a whole.

3. What are the key assumptions made in a statistical mechanics derivation?

The key assumptions made in a statistical mechanics derivation include the system being in thermal equilibrium, the particles being indistinguishable, and the particles obeying classical or quantum mechanics. Additionally, the system must be large enough for statistical methods to be applicable.

4. What is the mathematical framework used in a statistical mechanics derivation?

The mathematical framework used in a statistical mechanics derivation is probability theory. This includes concepts such as statistical ensembles, partition functions, and probability distributions.

5. How is entropy involved in a statistical mechanics derivation?

Entropy is a key concept in statistical mechanics and is used to measure the disorder or randomness of a system. In a statistical mechanics derivation, entropy is used to calculate the probability of a system being in a particular state, and to predict the equilibrium state of a system.

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