This paper studies how spatial variations across the plasma affect the rapid onset of hydrodynamic behavior in an expanding quark–gluon plasma. The authors relax the usual simplifying assumption that the system is the same at every point across the collision plane. They add transverse waves with a finite wave number k and track how these waves change the route the system takes from a far-from-equilibrium state toward the simpler dynamics described by hydrodynamics.

To do this they use a kinetic theory model in the relaxation-time approximation (RTA). RTA is a simplified collision rule that pushes the particle distribution toward local equilibrium over a characteristic time τ_R. The system is taken to be boost-invariant along the beam direction, which means it looks the same under a certain longitudinal rescaling useful for heavy-ion collisions. The authors expand the particle distribution in angular functions (spherical harmonics) and collect the coefficients into moments. The first set of moments corresponds directly to energy and momentum and so to the usual hydrodynamic variables.

Writing the kinetic equation in this spherical-harmonic basis turns the problem into a matrix evolution equation. Transverse gradients introduce couplings between different angular moments. The dynamics is then controlled by a three-way competition among the longitudinal expansion rate (1/τ), the collision rate (1/τ_R), and the gradient scale set by k. At early times the fast longitudinal expansion weakens the coupling between moments, so each angular sector follows the same “homogeneous” attractor known from previous studies.

At later times, and for sufficiently small transverse wave number k, these gradient-induced couplings become important. The system is driven toward a global attractor manifold that is built from the hydrodynamic sound and shear modes. The approach to that manifold happens on a timescale τ_D that depends on the azimuthal harmonic m and on the combination kτ_R. However, when kτ_R is large, the authors find that a spectral gap in the linearized evolution closes. In that regime perturbations still decay (they equilibrate), but they do so without following a simple, isolated hydrodynamic attractor. In other words, no short reduced description with only a few hydrodynamic modes applies.