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

Abstract

We consider causal higher order theories of relativistic viscous hydrodynamics in the limit of onedimensional boost-invariant expansion and study the associated dynamical attractor. We obtain evolution equations for the inverse Reynolds number as a function of Knudsen number. The solutions of these equations exhibit attractor behavior which we analyze in terms of Lyapunov exponents using several different techniques. We compare the attractors of the second-order Müller-Israel-Stewart (MIS), transient Denicol-Niemi-Molnar-Rischke (DNMR), and third-order theories with the exact solution of the Boltzmann equation in the relaxation-time approximation. It is shown that for Bjorken flow the third-order theory provides a better approximation to the exact kinetic theory attractor than both MIS and DNMR theories. For three different choices of the time dependence of the shear relaxation rate we find analytical solutions for the energy density and shear stress and use these to study the attractors analytically. By studying these analytical solutions at both small and large Knudsen numbers we characterize and uniquely determine the attractors and Lyapunov exponents. While for small Knudsen numbers the approach to the attractor is exponential, strong power-law decay of deviations from the attractor and rapid loss of initial state memory is found even for large Knudsen numbers. Implications for the applicability of hydrodynamics in far-offequilibrium situations are discussed.

Global flow structure and exact formal transseries of the Gubser flow in kinetic theory

“…There are many unknown elements of this new formulation despite a few important lessons learned in the recent years. For instance, different numerical studies indicate that the equations of motion for a class of fluids undergoing Bjorken flow [13,[22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39] can be dimensionally reduced to a single equation for the inverse Reynolds number Re −1 as a function of the Knudsen number Kn. 1 Being an explicitly time-dependent (also known as "nonautonomous") dynamical system with an attracting IR fixed point 2 at late times Kn 1, the model enjoys solutions that approach this fixed point which naturally merge before equilibriating.…”

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 attractor forπ is obtained by evolving (2,3) with initial conditions at τ 0 = 0 corresponding to zero right hand sides, i.e. withπ attr 0,aHydro = 1 4 and π attr 0,CE ≈ 0.235 [45], respectively. The first value agrees with the free-streaming limit; the second is ∼ 1% off.…”

Jet wake from linearized hydrodynamics

“…Nevertheless, explicit numerical analysis of the out-of-equilibrium dynamics of both strongly coupled and weakly coupled gauge theories (see the pioneering works [111][112][113][114][115] and see refs. [92,98,110,116,117] for recent reviews) as well as the recent studies of hydrodynamic attractors [118][119][120][121][122] suggest that hydrodynamics may be applicable even when gradients are large and the matter is away from local thermal equilibrium, although a non-hydrodynamic explanation of the attractor phenomenon in terms of a pre-hydrodynamic epoch during which the dynamics is dominated by one or a few lowlying modes in an effective Hamiltonian would delay the onset of hydrodynamization [123]. Furthermore, explicit analysis of the medium response to an energetic colored object in strongly-coupled gauge theories [79-81, 85-87, 91, 92] demonstrates that, at least in the strongly coupled fluid of N = 4 supersymmetric Yang-Mills theory, the energy-momentum tensor disturbance induced by these energetic particles behave hydrodynamically even at distances as short as 1/πT away from the jet.…”