Standard deviation of the energy of a system in a heat bath

In summary, the mean energy of a system in a heat bath is given by \bar{E} = - \frac{\partial ln(Z)}{\partial \beta}, where Z is the partition function and \beta = k_BT. The standard deviation of E is defined by (\Delta E)^2 = \frac{\partial^2 ln(Z)}{\partial \beta ^2}. This relationship can be derived from Eric Poisson's Statisical Physics II notes, specifically on page 40. These notes also provide useful information in other areas.
  • #1
Narcol2000
25
0
Given the mean energy of a system in a heat bath is

[tex]
\bar{E} = - \frac{\partial ln(Z)}{\partial \beta}
[/tex]

Where Z is the partition function and [tex]\beta = k_BT[/tex]

Why is the standard deviation of E defined by:

[tex]
(\Delta E)^2 = \frac{\partial^2 ln(Z)}{\partial \beta ^2}
[/tex]

I can't seem to find any proof of how the second derivative is related to the standard deviation.
 
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  • #2
Eric Poisson has a derivation on p40 of his Statisical Physics II notes: http://www.physics.uoguelph.ca/~poisson/research/notes.html.
 
  • #3
Thanks a lot those notes are also pretty useful in other respects as well.
 

Related to Standard deviation of the energy of a system in a heat bath

1. What is the definition of standard deviation of the energy of a system in a heat bath?

The standard deviation of the energy of a system in a heat bath is a measure of how spread out the energy values are from the average energy value of the system. It is a statistical term that indicates the variability or uncertainty in the energy of a system within a heat bath.

2. How is the standard deviation of the energy of a system in a heat bath calculated?

The standard deviation is calculated by taking the square root of the variance, which is the average of the squared differences between each energy value and the mean energy value of the system. This calculation gives a measure of the dispersion of the energy values around the mean.

3. What does a high standard deviation of the energy of a system in a heat bath indicate?

A high standard deviation indicates that the energies of the system are widely spread out from the average energy value. This could mean that the system is experiencing a lot of fluctuations and is not stable in its energy state.

4. How does the standard deviation of the energy of a system in a heat bath relate to the temperature of the system?

The standard deviation is directly proportional to the temperature of the system. As the temperature of the system increases, the energy values become more spread out, resulting in a higher standard deviation. Similarly, a decrease in temperature leads to a decrease in standard deviation.

5. Can the standard deviation of the energy of a system in a heat bath be used to predict the behavior of the system?

Yes, the standard deviation can give insight into the behavior of the system. A high standard deviation may indicate a more dynamic and unpredictable system, while a low standard deviation may suggest a more stable and predictable system.

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