= Phase-diffusion coefficient from a Bloch cell problem
{title2=$D=\langle w,L_2v_0+L_1v_1\rangle$}
Suppose a <Bloch stability operator> has expansion $L(q)=L_0+iqL_1-q^2L_2+\cdots$, with a simple neutral <eigenfunction> $v_0$ and an adjoint neutral function $w$ normalized by $\langle w,v_0\rangle=1$. Expand $v(q)=v_0+iqv_1+\cdots$ and $\lambda(q)=icq-Dq^2+\cdots$. The <Fredholm alternative> gives $c=\langle w,L_1v_0\rangle$ and the cell problem $L_0v_1=cv_0-L_1v_0$, $\langle w,v_1\rangle=0$. Applying the adjoint solvability condition at second order gives the displayed phase-diffusion <coefficient>. Multiple translation <phases> or a conserved mean require a coupled matrix problem rather than this scalar formula.
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