Early time behavior of spatial and momentum anisotropies in kinetic theory across different Knudsen numbers

Abstract

We investigate the early time development of the anisotropic transverse flow and spatial eccentricities of a fireball with various particle-based transport approaches using a fixed initial condition. In numerical simulations ranging from the quasi-collisionless case to the hydrodynamic regime, we find that the onset of vn and of related measures of anisotropic flow can be described with a simple powerlaw ansatz, with an exponent that depends on the amount of rescatterings in the system. In the few-rescatterings regime we perform semi-analytical calculations, based on a systematic expansion in powers of time and the cross section, which can reproduce the numerical findings.

On differences between even and odd anisotropic-flow harmonics in non-equilibrated systems

“…Indeed, we have shown in Ref. [20] -yet only for early timesthat pushing the analytical calculation to higher order in σ improves the agreement with the 2 → 0 results. In contrast, for t/R ≥ 2, when fewer collisions take place, the results of both approaches are again very parallel.…”

“…Remarkably, the analytical approach at order O(σ ) yields v n = 0 for all odd coefficients, but finite values for even ones, which hints at a fundamental difference between odd and even harmonics. Note that we have found elsewhere that odd v n harmonics can be non-zero at order O(σ 2 ) [20].…”

“…The test particle positions are sampled from the distribution function ( 9), while for their momenta we use a Boltzmann distribution with a position-independent temperature -in contrast to Ref. [20]. Since the simulations are performed with a finite test particle number N p ranging between 2 × 10 5 and 2 × 10 6 , neither perfect isotropy in momentum space nor uniformity of the momentum distribution across the whole geometry can be achieved.…”

“…3.5, this is due to a cancellation between different regions in the special casewhich we consider throughout the paper -where the local momentum distribution is the same at every point of the transverse plane in the initial condition. To be more precise, one finds that v 3 , and more generally every odd flow harmonic, is zero at first order in σ , but at higher orders it can be non-zero [20].…”

“…Thus, alternative descriptions based on microscopic kinetic transport theory, which is known to reproduce fluid-dynamical results when particles undergo many rescatterings [12], are being explored again, in particular with a view to small systems. A number of recent attempts start from semi-realistic initial geometries, which allow to isolate the flow harmonics of interest and study their origin [13][14][15][16][17][18][19][20].…”