Helmholtz free energy in the canonical ensemble

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Discussion Overview

The discussion centers around the derivation of the Helmholtz free energy in the context of the canonical ensemble within statistical mechanics. Participants explore the relationship between the Helmholtz free energy, internal energy, and entropy, as well as the formula involving the partition function.

Discussion Character

  • Exploratory, Technical explanation, Conceptual clarification

Main Points Raised

  • One participant questions how to derive the Helmholtz free energy given the canonical ensemble distribution, expressing uncertainty about using the formula A = U - TS due to not knowing how to express entropy S.
  • Another participant proposes the formula A = -kT ln Z as a solution for calculating the Helmholtz free energy.
  • A later reply indicates that the derivation of the proposed formula can be found in statistical physics textbooks or on Wikipedia, suggesting that it is a well-documented result.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the derivation process, with one participant expressing uncertainty about the use of the standard formula and another providing a formula without detailing its derivation.

Contextual Notes

The discussion highlights a potential limitation in the understanding of how to express entropy in the context of the canonical ensemble, which may affect the application of the formula A = U - TS.

amedeo_fisi
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Hello everybody :D
My question is: given the distribution of the canonical ensemble, how do we get the helmoltz free energy?
I think we can't use A = U-TS because we don't know how to write S. So what's the solution? Thanks
 
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The simplest way is to use
$$
A = - k T \ln Z
$$
 
Thanks DrClaude, but I already know the result! How do you get that formula?
 
You should be able to find the derivation in any statistical physics textbook. You can also find a derivation on Wikipedia.
 

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