Early time behavior of spatial and momentum anisotropies in a kinetic approach to nuclear collisions

Abstract

We derive a general formula for the early time dependence of a phase space distribution evolving according to the kinetic Boltzmann equation. Assuming that the early evolution of the system created in high-energy nuclear collisions can be described by kinetic theory, we calculate the scaling behaviors for the onset of various characteristics of the transverse dynamics. In particular, we show that the scaling behavior of the anisotropic flow coefficients vn at early times does not depend on the details of the collision kernel or the system composition, while at the same time it differs from the prediction of fluid dynamics.

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

“…As argued in Ref. [28], a non-vanishing v 3 can lead to a (small) linear rise of v 3 (t) at early times, which is indeed what we observe.…”

“…In Ref. [28] we showed that the contributions at O(σ) show up at odd orders starting from t 3 , those at O(σ 2 ) start at order t 4 and show up at even order in t, while O(σ 3 )-terms again show up in odd powers of t from order t 5 on.…”

“…Again, the time derivatives of the collision kernel can be re-expressed using the Boltzmann equation, so that only f (0) and its spatial derivatives are involved. Note that these time derivatives of C[f ] actually involve increasing powers of the cross section characterizing the rescatterings [28], as will be illustrated in the following section.…”

“…Irrespective of any approximation, it was pointed out in Ref. [28] that one can start from a Taylor expansion of the phase-space distribution at early times…”

“…one can replace every time derivative in Eq. ( 3), so that given the form of the collision kernel, the whole evolution of f (t, x, p) is governed by the initial distribution and its spatial derivatives [28]:…”