On attractor and fixed points in Bjorken flows

Abstract

We consider a plasma of massless particles undergoing Bjorken expansion, mimicking the matter created in ultra-relativistic heavy ion collisions. We study the transition to hydrodynamics using kinetic theory in the relaxation time approximation. By allowing the relaxation time to depend on time, we can monitor the speed of the transition from the collisionless regime to hydrodynamics. By using a special set of moments of the momentum distribution, we reduce the kinetic equation to a coupled mode problem which encompasses all versions of second order viscous hydrodynamics for Bjorken flows. This coupled mode problem is analysed first using techniques of linear algebra. Then we transform this two mode problem into a single non linear differential equation and proceed to a fixed point analysis. We identify an attractor solution as the particular solution of this non linear equation that joins two fixed points: one corresponding to the collisionless, early time regime, the other corresponding to late time hydrodynamics. We exploit the analytic solution of this equation in order to test several approximations and to identify generic features of the transition to hydrodynamics. We argue that extending the accuracy of hydrodynamics to early time, i.e. to the region of large gradients, amounts essentially to improve the accuracy of the location of the collisionless fixed point. This is demonstrated by showing that a simple renormalisation of a second order transport coefficient puts the free streaming fixed point at the right location, and allows us to reproduce accurately the full solution of the kinetic equation within second order viscous hydrodynamics, even in regimes far from local equilibrium.

53 pages and 12 figures

References in corpus (10)

• Relativistic viscous hydrodynamics, conformal invariance, and holography
• Second order hydrodynamic coefficients from kinetic theory
• Viscous hydrodynamics relaxation time from AdS/CFT
• Do nuclear collisions create a locally equilibrated quark-gluon plasma?
• Early Time Dynamics in Heavy Ion Collisions from AdS/CFT Correspondence
• Free-streaming approximation in early dynamics of relativistic heavy-ion collisions
• How much entropy is produced in strongly coupled Quark-Gluon Plasma (sQGP) by dissipative effects?
• Dynamical systems and nonlinear transient rheology of the far-from-equilibrium Bjorken flow
• Hydrodynamics from free-streaming to thermalization and back again
• Transasymptotics and hydrodynamization of the Fokker-Planck equation for gluons

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• Hydrodynamic attractors in heavy ion collisions: a review
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• Establishing the Range of Applicability of Hydrodynamics in High-Energy Collisions
• Perturbative approaches in relativistic kinetic theory and the emergence of first-order hydrodynamics
• Scaling and adiabaticity in a rapidly expanding gluon plasma
• Bjorken flow attractors with transverse dynamics
• Opacity dependence of transverse flow, pre-equilibrium and applicability of hydrodynamics in heavy-ion collisions
• Far-from-equilibrium attractors for massive kinetic theory in the relaxation time approximation
• Fast kinetic simulator for relativistic matter
• Hydrodynamic attractor in the non-conformal Bjorken flow
• Novel features of attractors and transseries in non-conformal Bjorken flows
• Metric anisotropies and nonequilibrium attractor for expanding plasma