Propagation and non-reciprocity in time-modulated diffusion through the lens of high-order homogenization

The homogenization procedure developed here is conducted on a laminate with periodic space-time modulation on the fine scale: at the leading order, this modulation creates convection in the low-wavelength regime if both parameters are modulated. However, if only one parameter is modulated, which is more realistic, this convective term disappears and one recovers a standard diffusion equation with effective homogeneous parameters; this does not describe the non-reciprocity and the propagation of the field observed from exact dispersion diagrams. This inconsistency is corrected here by considering second-order homogenization that results in a non-reciprocal propagation term that is proved to be non-zero for any laminate and verified via numerical simulation. The same methodology is also applied to the case when the density is modulated in the heat equation, leading, therefore, to a corrective advective term that cancels out non-reciprocity at the leading order but not at the second order.