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

TL;DR

This work resolves a key inconsistency in time-modulated diffusion by pushing homogenization to second order. It shows that non-reciprocal propagation arises even when only one material parameter (conductivity or capacity) is modulated, a result hidden at leading order in prior models. Using a two-scale expansion in a moving frame, the authors derive explicit second-order corrections that generate convective and dispersive terms, supported by Floquet-Bloch analyses of bilayer laminates and numerical simulations. The findings extend to density-modulated diffusion (Model 2), where an advective correction preserves non-reciprocity at second order. Overall, the study provides a general, rigorous framework for predicting non-reciprocal diffusion in arbitrary time-modulated laminates and highlights potential experimental avenues in thermal and charge diffusion systems.

Abstract

The homogenization procedure developed here is conducted on a laminate with periodic space-time modulation on the fine scale: at 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 which 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 which cancels out non-reciprocity at the leading order but not at the second order.

Figures (9)

• Figure 1: Bilaminate time-modulated in a wave-like fashion with modulation speed $v_m$ at $\tau=0$ (top) and $\tau=\tau_0$ (bottom).
• Figure 2: Dispersion diagrams for Model 1 when both parameters are modulated. Exact one in plain lines, leading-order and first-order homogenized model in dashed lines. Non-reciprocity arises from $\Re(\Omega)\neq 0$. The propagation velocity $\Re(\Omega)/\kappa$ being negative, there is propagation against the direction of the space-time modulation.
• Figure 3: Dispersion diagrams when only $\bar{\sigma}$ is modulated (in this case Model 1 and Model 2 are the same). Exact one in plain lines, leading-order and first-order homogenized model in dashed lines. Non-reciprocity arises from $\Re(\Omega)\neq 0$ but is missed by leading-order models. The propagation velocity $\Re(\Omega)/\kappa$ being positive, there is propagation in the direction of the space-time modulation.
• Figure 4: Dispersion diagrams for Model 2 ($\rho$ is modulated requiring therefore a corrective advective term in the modulated diffusion equation). Exact one in plain lines, leading-order homogenized model in dashed lines. Non-reciprocity arises from $\Re(\Omega)$ i.e. the propagation velocity in the direction of the space-time modulation but is missed by leading-order models.
• Figure 5: Snapshots of the leading-order field $\Theta_0$ at different times.