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The non-equilibrium attractor for kinetic theory in relaxation time approximation

I demonstrate that the concept of a non-equilibrium attractor can be extended beyond the lowest-order moments typically considered in hydrodynamic treatments. Using a previously obtained exact solution to the relaxation-time approximation Boltzmann equation for a transversally homogeneous and boost-invariant system subject to Bjorken flow, I derive an equation obeyed by all moments of the one-particle distribution function. Using numerical solutions, I show that, similar to the pressure anisotropy, all moments of the distribution function exhibit attractor-like behavior wherein all initial conditions converge to a universal solution after a short time with the exception of moments which are sensitive to modes with zero longitudinal momentum and high transverse momentum. In addition, I compute the exact solution for the distribution function itself on very fine lattices in momentum space and demonstrate that (a) an attractor for the full distribution function exists and (b) solutions with generic initial conditions relax to this solution, first at low momentum and later at high momentum.