Derive the many-body circulation sign-change value

Derive the many-body sign-change value of the intrinsic chirality ratio at which the mean circulation changes sign in the confined active-particle system with nematic bulk alignment and mixed polar–nematic wall coupling, using a coarse-grained theory.

Background

The paper finds that, for predominantly nematic wall coupling, the interacting suspension undergoes a circulation reversal near ω0/cθ≃0.15\omega_0/c_\theta\simeq 0.15. This reversal is nearly independent of the wall-symmetry mixing parameter and disappears when particle–particle alignment is removed, indicating that it is a collective phenomenon rather than a consequence of the isolated-particle wall-orientation condition.

Although the simulations associate the reversal with a pronounced decrease in nematic order, the authors explicitly state that the observations do not provide an analytic prediction for the numerical value of the sign change. A coarse-grained theory capable of deriving this value remains to be developed.

References

These observations show that bulk alignment is required for the sign change near \omega_0/c_\theta\simeq0.15, but they do not provide an analytic prediction for this value. They also distinguish two testable responses. Changing a bulk parameter should alter both the many-body sign change and the associated decrease in nematic order. Changing the polar component of the wall torque instead shifts the isolated-particle tangent condition according to Eq.~eq:boundary_line. The similar decrease in order for the three small-$\lambda$ curves and the result obtained without bulk alignment support this distinction. Deriving the many-body sign-change value from a coarse-grained theory remains an open problem.

— Collective Order Decouples Boundary Selection from Macroscopic Chirality in Confined Active Matter  (2608.23942 - Sepúlveda, 25 Aug 2026) in Section 4.2, “Collective origin”