Energy Fluctuations in Canonical Ensemble

In summary, the conversation is about deriving <E2>-<E>2 from the equation <E>=U=sum(Eiexp(-beta Ei))/sum(exp(-beta Ei), while holding volume constant. The person is trying to take the derivative of U with respect to both temperature and volume, but is struggling to combine and cancel terms. They are also questioning the need to take the derivative with respect to volume.
  • #1
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Homework Statement



In deriving <E2>-<E>2

starting from <E>=U=sum(Eiexp(-beta Ei))/sum(exp(-beta Ei). the taking derivative of U with respect to beta, the book always notes E (thus Volume) is held constant. what i am trying to do is taking the derivative of U with respect to beta or T (temp) and V (volume). but i get stuck

Homework Equations



<E>=U= sum(Ei*exp(-beta*Ei)/sum(exp(-beta*Ei)
dU=dU/dT+dU/dV

The Attempt at a Solution



I applied the above equation dU to U as listed above. i performed the quotient and product rule obtaining terms partial derivative with respect to V and T. I tried to look for a way to combine terms and cancel terms. but I cannot. anyone help thank you.
 
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  • #2
Why are you taking the derivative of U with respect to V? It only asked you to take the derivative with respect to beta.
 
  • #3
in the notes in the book its hold Ei constant, i want to perform the derivative more generally allowing Ei to vary
 
  • #4
But I am pretty sure, they take a partial derivative with respect to beta. So you don't worry about N or V.
 
  • #5


It seems like you are on the right track in terms of using the quotient and product rule to take the derivative of U with respect to both temperature and volume. However, it is important to note that in the canonical ensemble, the volume is typically held constant. Therefore, when taking the derivative with respect to T and V, the term for dU/dV should be zero since the volume is not changing.

To find <E2>-<E>2, you can use the formula <E2>=sum(Ei2*exp(-beta*Ei)/sum(exp(-beta*Ei). Then, using the formula for variance, you can subtract <E>2 from <E2> to get the desired result.

It is also important to note that in the canonical ensemble, energy fluctuations are dependent on temperature and the specific heat capacity. Therefore, when taking the derivative with respect to T, you may need to consider the specific heat capacity term.

Overall, it is important to carefully consider the conditions and assumptions of the canonical ensemble when deriving equations and taking derivatives. I would suggest reviewing the relevant equations and their derivations to ensure accuracy in your calculations.
 

1. What is the canonical ensemble?

The canonical ensemble is a statistical mechanics concept used to describe the behavior of a system in thermal equilibrium with a heat reservoir. It considers a fixed number of particles in a fixed volume with a fixed temperature, and allows for the exchange of energy between the system and the heat reservoir.

2. What are energy fluctuations in the canonical ensemble?

Energy fluctuations refer to the variations in the energy of a system within the canonical ensemble. These fluctuations occur due to the exchange of energy between the system and the heat reservoir, and are described by the Boltzmann distribution.

3. How are energy fluctuations calculated in the canonical ensemble?

The energy fluctuations in the canonical ensemble can be calculated using the equipartition theorem, which states that the average energy of a system is equal to the number of degrees of freedom multiplied by the temperature and the Boltzmann constant. The fluctuations can also be calculated using statistical methods, such as the variance of energy.

4. Why are energy fluctuations important to study in the canonical ensemble?

Energy fluctuations are important to study in the canonical ensemble because they provide insights into the thermodynamic properties of a system. They can also affect the stability and behavior of a system, and understanding them is crucial in predicting and controlling the behavior of complex systems.

5. How do energy fluctuations affect the properties of a system in the canonical ensemble?

Energy fluctuations can affect the properties of a system in the canonical ensemble by changing its temperature, pressure, and other thermodynamic variables. They can also lead to phase transitions, where the system undergoes a sudden change in its properties due to a critical energy fluctuation. Additionally, energy fluctuations can affect the speed of chemical reactions and the stability of a system.

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