Active-hydraulic flows solve the 6-vertex model (and vice versa)

TL;DR

This paper establishes a precise correspondence between active-hydraulic flows in microchannel networks and the six-vertex model, enabling prediction and control of flow configurations and extending the framework to arbitrary networks.

Contribution

It introduces a novel mapping between active fluid flows and the six-vertex model, providing a predictive framework for flow geometries in complex networks.

Findings

Active flows correspond to six-vertex model configurations.

Lagrangian trajectories form packed loops predictable by the model.

Framework extends to arbitrary network geometries.

Abstract

By confining colloidal active fluids in microchannel networks, we demonstrate that their degenerate flows corresponds to the configurations of the six-vertex model. We use this quantitative correspondence to control and explain the active flows that emerge in square grid networks. In particular, we show that the Lagrangian trajectories of active particles realize the Baxter-Kelland-Wu mapping and form completely packed loops, whose geometry can be exactly predicted and explained. We then go beyond the square-grid geometry and introduce a general framework for predicting the geometry of active-hydraulic flows in arbitrary networks.