Beyond Viscosity Matching: Microscopic Relaxation in Anisotropic Flow

Abstract

When a gas of scattering particles is disturbed, each angular harmonic of its velocity distribution decays at its own rate. Hydrodynamics, and with it the shear viscosity, depends on only one of these rates, the one for the quadrupole harmonic. A system that scatters only a few times before it stops interacting remembers all of them. This is the situation of small collision systems such as $p+p$, $p+$Pb and O$+$O, where kinetic models tuned to the same viscosity nevertheless predict different anisotropic flow. We show that this difference is not a model uncertainty but a measurement of the remaining rates: to first order in the number of collisions, ratios of flow responses equal ratios of relaxation rates, independently of the initial geometry. For the QCD collision kernel, where the rates also depend on momentum, we obtain the full spectrum of relaxation modes in closed form and show that dilute flow, viscosity, and late-time decay probe three different averages of it. A momentum-resolved kinetic solver confirms that the angular hierarchy is lost first at soft momenta and that the flow response settles at a value no single-rate model produces. The same structure governs two-dimensional electron and atomic Fermi gases.