Active phase-space topology unifies depletion and alignment in bacterial flows
Transport at small scales is classically understood within an equilibrium framework, where dispersion theory successfully describes shear-enhanced diffusion for passive particles in the continuum limit. However, as most bacteria can move on their own, their motility in flows, inherently out of thermal equilibrium, fundamentally challenges this framework. A minimal, predictive unified theory of bacterial transport in low-Reynolds-number flows remains lacking. Here, from first principles, we develop an analytical hydrodynamic model that enforces consistent no-flux boundary conditions and uses the method of images to characterize the flow-wall coupling. The model quantitatively reproduces measured bacterial distributions and reveals a hydrodynamic locking mechanism accompanied by mean-drift invariance -- an active counterpart to Taylor dispersion. We clarify that shear-induced depletion and alignment are dual manifestations of a single active phase-space topology, ruling out explanations based solely on the local shear magnitude. The theory is validated against microfluidic experiments spanning multiple bacterial species and shear geometries, from one-dimensional to fully three-dimensional flows. Our findings establish a unified phase-space framework for bacterial hydrodynamics, advancing the fundamental understanding of active matter.