Locally periodic one-dimensional harmonic-mean coefficient ($\hat{a}(x)=({\int }_0^{1}a(x,X{)}^{-1}dX{)}^{-1}$)

For $(a(x,x/ϵ)y^{\prime }{)}^{\prime }=f(x,x/ϵ)$ with positive one-periodic fast coefficient, a periodic two-scale expansion has leading term independent of the fast variable. The first corrector satisfies $y_{1X}=C(x)/a-y_0^{\prime }$. Its zero-mean derivative requires $C=\hat{a}{y}_0^{\prime }$. Averaging the next equation gives $C^{\prime }={\int }_0^{1}fdX$. 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.