Derivation of Average Square Energy Fluctuation in a Canonical System

In summary, the canonical (Boltzmann) distribution law describes the probability of a state ##v## in a canonical system as ##P_v = Q^{-1}e^{-\beta E_v}##, where ##Q^{-1}## is the normalization constant and ##Q = \sum_{v}e^{-\beta E_v}##. Chandler then derives ##\langle(\delta E_v)^2\rangle## in the attachment. To understand the derivation, we can work backwards and calculate the derivatives of ##(\partial ^2 \ln Q / \partial \beta^2)_{N,V}##.
  • #1
phun_physics
2
1
TL;DR Summary
I am currently reading through Chandler's Introduction to Modern Statistical Mechanics and would like insight into this derivation
The canonical ( Boltzmann) distribution law for a canonical system is described the probability of state ##v## by ##P_v = Q^{-1} e^{-\beta E_v} ## where ##Q^{-1}## is the normalization constant of ##\sum_v P_v = 1## and therefore ##Q = \sum_{v}e^{-\beta E_v}##. Chandler then derives ## \langle( \delta E_v)^2 \rangle## in the attachment.

I am confused on how he went from these steps: ##Q^{-1}(\partial^2 Q / \partial \beta^2)_{N, V} - Q^{-2}(\partial Q / \partial \beta)^{2}_{N,V}## = ##(\partial ^2 lnQ / \partial \beta^2)_{N,V}##

Any help would be extremely appreciated! Thanks!
 

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  • #2
phun_physics said:
I am confused on how he went from these steps: ##Q^{-1}(\partial^2 Q / \partial \beta^2)_{N, V} - Q^{-2}(\partial Q / \partial \beta)^{2}_{N,V}## = ##(\partial ^2 lnQ / \partial \beta^2)_{N,V}##
Do it backwards. Start from ##(\partial ^2 \ln Q / \partial \beta^2)_{N,V}## and calculate the derivatives.
 

1. What is the "average square energy fluctuation" in a canonical system?

The average square energy fluctuation in a canonical system is a measure of the variation or spread of energy values within the system. It is calculated by taking the average of the squared differences between each energy value and the mean energy value of the system.

2. Why is the derivation of average square energy fluctuation important in scientific research?

The derivation of average square energy fluctuation is important because it allows scientists to understand the behavior and stability of a system. It provides insights into the distribution of energy values and can help in predicting the behavior of the system under different conditions.

3. How is the average square energy fluctuation related to the concept of entropy?

The average square energy fluctuation and entropy are closely related concepts. Entropy is a measure of the disorder or randomness of a system, and the average square energy fluctuation is a measure of the variation or spread of energy values within the system. As the average square energy fluctuation increases, the entropy also increases, indicating a more disordered and less stable system.

4. What factors can affect the average square energy fluctuation in a canonical system?

The average square energy fluctuation in a canonical system can be affected by various factors such as temperature, pressure, and the number of particles in the system. Changes in these factors can lead to changes in the average energy values and, consequently, the average square energy fluctuation.

5. How is the derivation of average square energy fluctuation performed?

The derivation of average square energy fluctuation involves using statistical mechanics principles and mathematical equations to calculate the average energy values and then taking the average of the squared differences between each energy value and the mean energy value. This process can be complex and may require advanced mathematical skills and knowledge of statistical mechanics.

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