= Landau quasiparticle energy functional
{c}
{title2=$\delta(E-\mu N)$}
Changing one <Landau quasiparticle> occupation changes the energies of others. To second order around equilibrium,
$$
\delta(E-\mu N)=\sum_{p\sigma}\xi_p\delta n_{p\sigma}+\frac{1}{2\mathcal V}\sum_{p\sigma,p'\sigma'}f_{\sigma\sigma'}(p,p')\delta n_{p\sigma}\delta n_{p'\sigma'}.
$$
Here $\mathcal V$ is <volume>, occupations are per <momentum> and <spin> state, and $f$ has units of <energy> times <volume>.<On the Theory of the Fermi Liquid by Lev Davidovich Landau>[p.70, equation (1); integrate the energy shift to obtain the quadratic functional]{ref}
For a toy two-state change at fixed <volume>, take $\delta n_a=\delta n_b=\eta$, $f_{aa}=f_{bb}=0$, $f_{ab}=f_{ba}=g\mathcal V$, and unperturbed energies $\xi_a=\xi_b=e$. The expression gives $2e\eta+g\eta^2$. The factor $1/2$ counts the interaction of the same pair once; it does not halve each excitation's energy. This finite-state illustration only explains the bookkeeping of the <Landau quasiparticle energy functional>.
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