TOPICS
Search

Fermi-Dirac Distribution

DOWNLOAD Mathematica NotebookDownload Wolfram Notebook

A distribution which arises in the study of half-integer spin particles in physics,

P(k)=(k^s)/(e^(k-mu)+1).
(1)

Its integral 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, Sigmoid Function

Explore with Wolfram|Alpha

Cite this as:

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

Subject classifications