Another question from Ashcroft and Mermin: Fermi-Dirac Distribution

In summary, the conversation discusses the relationship between the Helmholtz free energy and the canonical ensemble partition function in a N-electron system, as described in the Ashcroft Mermin book. The Helmholtz free energy is equal to U - TS and is calculated using the canonical ensemble partition function. Additional references and explanations are requested for further understanding.
  • #1
hagopbul
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TL;DR Summary
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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  • #2
I do not have the book. What is a ?
 
  • #3
ath stationary state of N-electron system
 
  • #4
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
 
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FAQ: Another question from Ashcroft and Mermin: Fermi-Dirac Distribution

What is the Fermi-Dirac distribution?

The Fermi-Dirac distribution is a probability distribution that describes the distribution of fermions (particles with half-integer spin) in a system at thermal equilibrium. It is named after Enrico Fermi and Paul Dirac, who first proposed this distribution in the 1920s.

How is the Fermi-Dirac distribution different from the Maxwell-Boltzmann distribution?

The Fermi-Dirac distribution takes into account the exclusion principle, which states that no two fermions can occupy the same quantum state at the same time. This leads to a lower probability of finding fermions at higher energy states, unlike the Maxwell-Boltzmann distribution which assumes that all energy states are equally probable.

What is the significance of the Fermi energy in the Fermi-Dirac distribution?

The Fermi energy is the energy level at which the probability of finding a fermion is 0.5. It is a measure of the highest energy level that can be occupied by fermions at absolute zero temperature. The Fermi energy also determines the Fermi level, which is an important concept in solid-state physics.

How does temperature affect the Fermi-Dirac distribution?

As temperature increases, the Fermi-Dirac distribution shifts towards higher energy states, leading to a decrease in the number of fermions at the lowest energy levels. This is because at higher temperatures, more energy states become available for fermions to occupy, reducing the probability of finding them at the lowest energy levels.

What are some applications of the Fermi-Dirac distribution?

The Fermi-Dirac distribution is used in various fields of physics, such as solid-state physics, statistical mechanics, and quantum mechanics. It is also important in understanding the behavior of electrons in metals and semiconductors, and in predicting the properties of degenerate matter in astrophysics.

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