 Fermi-window convolution of the grand potential

For noninteracting fermions with density of states $g$, define ${\Phi }_0(x)=-{\int }_0^x(x-y)g(y)dy$. The grand potential is ${\int }_0^{\infty }g(ϵ)F(ϵ)dϵ$, where $F=k_{B}T\ln (1-n_F)$ has derivative $F^{\prime }=n_F$. Integrating by parts twice gives the title expression, since ${\Phi }_0(0)={\Phi }_0^{\prime }(0)=0$ and upper boundary terms vanish for ordinary polynomially growing densities of states. The Fermi window therefore thermally averages the zero-temperature grand potential as a function of trial Fermi energy.