Fermi and Bose functions

The Fermi and Bose functions or complete Fermi and Bose integrals are classes of functions that make quantum ensembles easier to manage mathematically. The Fermi functions fkf_{k} and Bose functions gkg_{k} are defined as

fk(z)≡∑l=1∞(−1)l+1zllk,gk(z)≡∑l=1∞zllkf_{k}(z)\equiv \sum_{l=1}^{\infty}(-1)^{l+1} \frac{z^{l}}{l^{k}},\qquad g_{k}(z)\equiv \sum_{l=1}^{\infty} \frac{z^{l}}{l^{k}}

where kk is a real number. Both of these are specific forms of the polylogarithm:

fk(z)=−Lik(−z),gk(z)=Lik(z)f_{k}(z)=-\text{Li}_{k}(-z),\qquad g_{k}(z)=\text{Li}_{k}(z)

and so

gk(z)=−fk(−z)g_{k}(z)=-f_{k}(-z)

They obey the recurrence relations

zddzfk(z)=fk−1(z),zddzgk(z)=gk−1(z)z \frac{d}{dz}f_{k}(z)=f_{k-1}(z),\qquad z \frac{d}{dz}g_{k}(z)=g_{k-1}(z)

They have integral representations through the Gamma function:

fk(z)gk(z) }=1Γ(k)∫0∞xk−1z−1ex±1dx=22k−1πΓ(k−12)Γ(2k−1)∫0∞x2k−1z−1ex2±1dx\left.\begin{align} f_{k}(z) \\ g_{k}(z) \end{align}\ \right\}= \frac{1}{\Gamma(k)}\int_{0}^{\infty} \frac{x^{k-1}}{z^{-1}e^{x}\pm 1}dx=\frac{2^{2k-1}}{\sqrt{ \pi }} \frac{\Gamma\left( k- \frac{1}{2} \right)}{\Gamma(2k-1)}\int_{0}^{\infty} \frac{x^{2k-1}}{z^{-1}e^{x^{2}}\pm 1}dx

where ++ is for Fermi functions and −- for Bose functions. In a physical context, zz is the fugacity and the integration variable is typically momentum. The Fermi-Dirac and Bose-Einstein distribution appear in the integral.

Although it does not appear in the definition, due to the physical constraints imposed by bosons, gk(z)g_{k}(z) is defined only in 0≤z≤10\leq z\leq 1. It is therefore bounded both below, where it is gk(0)=0g_{k}(0)=0, and above, where it is gk(1)=ζ(k)g_{k}(1)=\zeta(k), where ζ(k)\zeta(k) is the Riemann Zeta function.

Origin

These functions are convenience functions defined from integrals that occur in quantum statistical mechanics when dealing with fermions and bosons. They appear, for instance, in the equation of state of a quantum ideal gas in both the quantum microcanonical ensemble and the quantum grand canonical ensemble derivations. For example, in the latter, they come up when taking the derivative of the grand canonical partition function Z\mathcal{Z} in the thermodynamic limit:

ln⁡Z=4πVh3∫0∞p2ln⁡(1+ze−βεp)dp\ln \mathcal{Z}=\frac{4\pi V}{h^{3}}\int_{0}^{\infty}p^{2}\ln(1+ze^{-\beta \varepsilon_{p}})dp

This integral can't be solved analytically, but we can expand ln⁡(1+x)\ln(1+x) in a Taylor series about 00 to find

1Vln⁡Z=4πh3∫0∞p2∑j=1∞(−1)j+1zje−jβεpj=4πh3∑j=1∞(−1)j+1zjj∫0∞p2e−jβp2/2m=4πh3∑j=1∞(−1)j+1zjjπ4(jβ/2m)3/2=(2mπβh2)3/2∑j=1∞(−1)j+1zjj5/2⏟f5/2(z)\begin{align} \frac{1}{V}\ln \mathcal{Z}&=\frac{4\pi}{h^{3}}\int_{0}^{\infty}p^{2}\sum_{j=1}^{\infty} \frac{(-1)^{j+1}z^{j}e^{-j\beta\varepsilon_{p}}}{j} \\ &=\frac{4\pi}{h^{3}}\sum_{j=1}^{\infty} \frac{(-1)^{j+1}z^{j}}{j}\int_{0}^{\infty}p^{2}e^{-j\beta p^{2}/2m} \\ &=\frac{4\pi}{h^{3}}\sum_{j=1}^{\infty} \frac{(-1)^{j+1}z^{j}}{j} \frac{\sqrt{ \pi }}{4(j\beta/2m)^{3/2}} \\ &=\left( \frac{2m\pi}{\beta h^{2}} \right)^{3/2}\underbrace{ \sum_{j=1}^{\infty} \frac{(-1)^{j+1}z^{j}}{j^{5/2}} }_{ f_{5/2}(z) } \end{align}

We can see the Fermi function f5/2(z)f_{5/2}(z). From this we can retroactively state

f5/2(z)=4π(β2m)3/2∫0∞p2ln⁡(1+ze−βεp)dpf_{5/2}(z)=\frac{4}{\sqrt{ \pi }} \left( \frac{\beta}{2m} \right)^{3/2}\int_{0}^{\infty}p^{2}\ln(1+ze^{-\beta \varepsilon_{p}})dp

Sommerfeld expansion

Fermi functions have an alternative power series representation known as a Sommerfeld expansion that comes in handy in physical applications. It is accurate for high zz. For f3/2(z)f_{3/2}(z) and f5/2(z)f_{5/2}(z) it yields

f3/2(z)≃43π(ln⁡z)3/2[1+π281(ln⁡z)2+…]f_{3/2}(z)\simeq \frac{4}{3\sqrt{ \pi }}(\ln z)^{3/2}\left[ 1+ \frac{\pi ^{2}}{8} \frac{1}{(\ln z)^{2}}+\ldots \right] f5/2(z)=815π(ln⁡z)5/2[1+5π281(ln⁡z)2+…]f_{5/2}(z)=\frac{8}{15\sqrt{ \pi }}(\ln z)^{5/2}\left[ 1+ \frac{5\pi^{2}}{8} \frac{1}{(\ln z)^{2}}+\ldots \right]

The full proof is in the Sommerfeld expansion page.

Low zz approximation

For low zz, there is no need for a fancy method like the Sommerfeld expansion: the first few terms of the definition are accurate:

fk(z)=z−z22k+z33k−…f_{k}(z)=z- \frac{z^{2}}{2^{k}}+ \frac{z^{3}}{3^{k}}-\ldots