Active matter continuously converts energy from its environment into directed motion, giving rise to dynamics that intrinsically violate detailed balance. A paradigmatic example for such far-from-equilibrium dynamics is the active Brownian particle model. This raises a fundamental question: how does the breakdown of detailed balance affect theories of coarse-graining such as the projection-operator technique, which provides a powerful framework for describing dynamical arrest in equilibrium systems [1]?
To address this question, I consider a passive tracer immersed in a bath of chiral active particles [2]. I demonstrate that conventional coarse-graining approaches based on orthogonal projection operators fundamentally fail to capture key nonequilibrium features of the tracer dynamics. To overcome this limitation, I develop a projection-operator formalism based on non-orthogonal projections. This framework enables the systematic derivation of coarse-grained equations of motion while preserving the nonequilibrium structure of the microscopic dynamics, including the violation of the fluctuation-dissipation relation. As a result, the coarse-grained description predicts the emergence of odd transport as well as a confinement-induced shift of the tracer's emergent chiral frequency.
I conclude by discussing the broader implications of these findings for theories of dynamical arrest, in particular, for extending mode-coupling theory to active matter.
REFERENCES
[1] U. Bengtzelius, W. Götze and A. Sjölander, "Dynamics of supercooled liquids and the glass transition", J. Phys. C: Solid State Phys. 17, 5915 (1984)
[2] C. Hargus, F, Ghimenti, J. Tailleur and F. van Wijland, "Odd dynamics of passive objects in a chiral active bath", Phys. Rev. Lett. 135, 167102 (2025)