= Free-transport phase mixing
{title2=$\widehat f_t(k,\xi)=\widehat f_0(k,\xi+kt)$}
= Kinetic phase mixing
{synonym}
On the unit-period spatial circle, free streaming has solution $f_t(x,v)=f_0(x-tv,v)$. With Fourier phase $e^{-2\pi i(kx+\xi v)}$, its mixed <Fourier transform> is $\widehat f_t(k,\xi)=\widehat f_0(k,\xi+kt)$. A nonzero spatial mode of the velocity-integrated density therefore samples increasingly large velocity frequencies. The conserved spatial zero mode remains, while extra velocity regularity supplies quantitative decay of the other modes.
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