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Fermi-Dirac Distribution

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The Fermi-Dirac distribution is a continuous distribution arising in the study of half-integer spin particles. For k>=0 and s>-1, a common form has probability density proportional to

p(k) proportional (k^s)/(e^(k-mu)+1),
(1)

where mu is a real number.

The corresponding normalizing integral is an unnormalized Fermi-Dirac integral and is given by

int_0^infty(k^sdk)/(e^(k-mu)+1)=e^muGamma(s+1)Phi(-e^mu,s+1,1)
(2)
=-Gamma(s+1)Li_(1+s)(-e^mu),
(3)

where Phi(z,s,a) is the Lerch transcendent and Li_n(z) is a polylogarithm.

See also

Bose-Einstein Distribution, Einstein Functions, Fermi-Dirac Integral, Sigmoid Function

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References

National Institute of Standards and Technology. "Fermi-Dirac and Bose-Einstein Integrals." §25.12(iii) in Digital Library of Mathematical Functions. https://dlmf.nist.gov/25.12#iii.

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Fermi-Dirac Distribution

Cite this as:

Weisstein, Eric W. "Fermi-Dirac Distribution." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Fermi-DiracDistribution.html

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