Diffusive circulation via spatiotemporal modulation

• Diffusive circulation is achieved via spatiotemporal modulation without material motion. • Traveling-wave modulation induces an effective convective bias in diffusion. • A rigorous analytical framework exactly solves the modulated diffusion equation. • Multi-port flux routing and circulation are demonstrated in diffusive systems. Controlling diffusive fluxes underpins efficient energy management, thermal regulation, and emerging information processing technologies. However, diffusion in conventional materials is inherently reciprocal and always drives flux along potential gradients, preventing directional routing and circulation. Realizing a circulator in diffusion, which traditionally relies on directional in-plane motion of the medium, is fundamentally challenging because physically moving material conflicts with the requirement of fixed boundary ports. Here we demonstrate diffusive circulation based on spatiotemporal modulation. Conductivity and capacity are modulated as traveling waves along a closed loop, generating an effective convective bias without material movement. We develop a rigorous analytical framework that exactly solves the modulated diffusion equation and introduce effective quantities that characterize intrinsic circulation. A rectification ratio is defined to quantify flux rectification under different port boundary conditions. This work not only clarifies the fundamental mechanisms underlying diffusive nonreciprocity but also provides a general strategy to control and route flux, establishing a versatile framework for diffusive circulation in mass, charge, and thermal systems.