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

Abstract

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.

Exact solutions and attractors of higher-order viscous fluid dynamics for Bjorken flow

“…For the analytical solutions (33), (39), (45) this prescription reproduces the same results as obtained from the late-time behavior (35) of the Whittaker functions but its advantage is that it can also be used numerically where exact solutions forπ are not available (such as for the numerical solutions of the generic equation (11)). For the case shown in Fig.…”

“…The structural similarity of Eqs. (45) and (39) shows that for the solution (45) the fixed points are instead located at w 0 = 0, which corresponds to a negative (i.e. unphysical) longitudinal proper timeτ 0 = −a/2.…”

Hydrodynamic Attractors, Initial State Energy, and Particle Production in Relativistic Nuclear Collisions

“…By matching the multiplicity in Eq. (18) to experimental data, we find that the dimensionless combination of pre-factors Vice versa, to determine the initial state energy per unit rapidity shown in Fig. 3…”

Hydrodynamics from free-streaming to thermalization and back again

“…Qualitatively, the hydrodynamic attractors associated with these partially or fully resummed hydrodynamic theories are all very similar [11,45] but differ in detail depending on the underlying microscopic dynamics and the approximations made when coarse-graining it to obtain a macroscopic hydrodynamic description. For systems whose microscopic dynamics can be described by classical kinetic theory using the relativistic Boltzmann equation it was found that the evolution of non-hydrodynamic moments of the distribution function, and of that function itself, is also controlled by attractors [47], and that both anisotropic [13,15,48] and third-order Chapman-Enskog hydrodynamics [35] describe this kinetic attractor with precision for both Bjorken [43] and Gubser [49] flows. The latter observation is of particular importance since, in contrast to the purely 1-dimensional longitudinal expansion of Bjorken flow, Gubser flow is intrinsically 3-dimensional, with such rapid transverse expansion that the system actually moves away from local equilibrium and becomes asymptotically free-streaming [50,51].…”