Two immiscible fluids flowing through porous rock — oil and water through sandstone, for instance — can reach a steady state where both fluids flow simultaneously through different pore channels. The flow rate depends on the pressure gradient driving it. At low pressure gradients, the relationship is linear: double the pressure, double the flow. This is Darcy's law, the fundamental equation of porous media flow, and it is supposed to hold universally in the low-Reynolds-number regime where these flows operate.
Sinha, Carmona, Andrade, and Hansen (arXiv:2603.08586, March 2026) show that the two-phase flow undergoes a phase transition at a critical pressure gradient. Below the transition, the flow is linear and well-behaved. Above it, the flow becomes nonlinear, exhibits hysteresis, and fluctuates across a wide range of timescales. The authors mapped the two-phase flow patterns onto a spin model using the maximum entropy principle and found that the transition corresponds exactly to a paramagnetic-to-spin-glass phase transition. The flow doesn't just behave like a glass. It is a glass, in the precise statistical-mechanical sense: the configuration space of flow pathways through the pore network has the same frustrated, rugged-landscape structure as a spin glass.
The mapping works because each pore throat can be occupied by either fluid — a binary variable, like a spin. The disorder in the pore network (random throat sizes, random connectivity) plays the role of quenched disorder in the spin model. The competition between the two fluids for access to pore channels creates frustration: the optimal pathway for one fluid blocks pathways for the other. Below the critical pressure gradient, one dominant flow configuration satisfies most of the constraints. Above it, many nearly equivalent configurations compete, the system can't settle on one, and the flow wanders between metastable states — glassy dynamics in a flowing fluid.
The hysteresis is the diagnostic signature. Increase the pressure gradient through the transition, then decrease it: the flow rate doesn't retrace the same curve. The system remembers which configuration it was trapped in. A flowing fluid, at steady state, exhibiting memory. Not because of any exotic mechanism, but because the competition between two ordinary fluids in a disordered medium generates the same frustrated energy landscape that produces memory in glasses.
Sinha, Carmona, Andrade, and Hansen, "Glassy phase transition in immiscible steady-state two-phase flow in porous media," arXiv:2603.08586 (March 2026).