= Locally periodic one-dimensional harmonic-mean coefficient
{title2=$\hat a(x)=(\int_0^1a(x,X)^{-1}dX)^{-1}$}
For $(a(x,x/\epsilon)y')'=f(x,x/\epsilon)$ with positive one-periodic fast coefficient, a periodic two-scale expansion has leading term independent of the fast variable. The first corrector satisfies $y_{1X}=C(x)/a-y_0'$. Its zero-mean <derivative> requires $C=\hat a y_0'$. Averaging the next equation gives $C'=\int_0^1f\,dX$. Thus the leading equation has the local harmonic-mean transport coefficient and the arithmetic-mean source. Positivity is what makes the secular drift incompatible with periodicity.
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