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Bose-Einstein Distribution

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The Bose-Einstein distribution arises in the study of integer-spin particles in physics. For real s and mu<=0 and k>0, a common unnormalized form is

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

The corresponding integral, also allowing complex s, 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)

for real mu<0 and Re[s]>-1, or for mu=0 and Re[s]>0, where Re[s] is the real part of s, Gamma(z) is the gamma function, Phi(z,s,a) is the Lerch transcendent, and Li_n(z) is a polylogarithm (NIST DLMF, eqns. 25.12.15-16).

For real mu>0, the denominator vanishes at k=mu and this ordinary improper integral does not converge.

See also

Bose-Einstein Condensation, Bose Gas, Einstein Functions, Fermi-Dirac Distribution, Sigmoid Function

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References

NIST Digital Library of Mathematical Functions. "Fermi-Dirac and Bose-Einstein Integrals." Eqns. 25.12.15-16. https://dlmf.nist.gov/25.12.iii.

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Bose-Einstein Distribution

Cite this as:

Weisstein, Eric W. "Bose-Einstein Distribution." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Bose-EinsteinDistribution.html

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