Another question from Ashcroft and Mermin: Fermi-Dirac Distribution

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Discussion Overview

The discussion revolves around the relationship between the Helmholtz free energy and the energy expression E(a,N) as presented in the Ashcroft and Mermin textbook, specifically following equation 2.38. Participants are seeking clarification on how these two terms are related within the context of statistical thermodynamics.

Discussion Character

  • Technical explanation
  • Conceptual clarification

Main Points Raised

  • One participant asks about the relationship between the Helmholtz free energy F and the energy expression E(a,N), noting that F is defined as U - TS.
  • Another participant inquires about the meaning of the variable 'a' in the context of the discussion.
  • A later reply clarifies that 'a' refers to the stationary state of an N-electron system.
  • One participant summarizes that the Helmholtz free energy can be expressed in terms of the canonical ensemble partition function Q(N, V, T), stating that F(N, V, T) = -kT lnQ(N, V, T).

Areas of Agreement / Disagreement

The discussion does not appear to reach a consensus on the relationship between F and E, as participants are still seeking clarification and additional references.

Contextual Notes

There are unresolved aspects regarding the definitions and implications of the terms used, particularly the variable 'a' and its context within the energy expression.

hagopbul
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TL;DR
this time about fermi -dirac distribution
Good Day :

i reached the page 40 of Ashcroft Mermin book and after the equation 2.38 there is this expression of E(a,N) which is equal to Helmoltez Free energy F = U - TS , how this two terms F , E are related ? anyone can provide adequate explanation , and few useful references

Best Regards
HB
 
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I do not have the book. What is a ?
 
ath stationary state of N-electron system
 
hagopbul said:
Summary:: this time about fermi -dirac distribution

i reached the page 40 of Ashcroft Mermin book and after the equation 2.38 there is this expression of E(a,N) which is equal to Helmoltez Free energy F = U - TS , how this two terms F , E are related ? anyone can provide adequate explanation , and few useful references
The sum in the denominator of eq. 2.38 is the “canonical ensemble partition function” Q(N, V, T) of the considered N-electron system at volume V and temperature T. From statistical thermodynamics one has for the Helmholtz free energy F(N, V, T) of a closed system: F(N, V, T) = -kT lnQ(N, V, T)

https://en.wikipedia.org/wiki/Helmholtz_free_energy
 
Last edited:

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