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.
Highlights
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
The effective temperature T can be obtained via the Landau matching condition which demands that the energy density calculated from the distribution function f is equal to the energy density determined from an equilibrium distribution, feq
In anisotropic hydrodynamics (aHydro), one uses the first and second moments of the Boltzmann equation to solve for the evolution of α and Λ with the one-particle distribution function assumed to be of the form [67, 68]
Summary
I review how to obtain the exact solutions to the 0+1d RTA Boltzmann equation. The equilibrium distribution function feq may be taken to be a Bose-Einstein, Fermi-Dirac, or Boltzmann distribution. I will assume that f is given by a Boltzmann distribution p·u feq = exp − T. The effective temperature T can be obtained via the Landau matching condition which demands that the energy density calculated from the distribution function f is equal to the energy density determined from an equilibrium distribution, feq. In Milne coordinates, Bjorken flow is static, i.e. uτ = 1 and ux,y,ς = 0. The use of this simple form of the kinetic equation given by eqs. Perhaps most importantly, in this simple case it is possible to solve the kinetic equation exactly using straightforward numerical algorithms
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