Far-off-equilibrium early-stage dynamics in high-energy nuclear collisions

. We explore the far-o ff -equilibrium aspects of the (1 + 1)-dimensional early-stage evolution of a weakly-coupled quark-gluon plasma using kinetic theory and hydrodynamics. For a large set of far-o ff -equilibrium initial conditions the system exhibits a peculiar phenomenon where its total equilibrium entropy decreases with time. Using a non-equilibrium definition of entropy based on Boltzmann’s H-function, we demonstrate how this apparently anomalous behavior is consistent with the second law of thermodynamics. We also use the H-function to formulate ‘maximum-entropy’ hydrodynamics, a far-o ff - equilibrium macroscopic theory that can describe both free-streaming and near-equilibrium regimes of quark-gluon plasma in a single framework.


Introduction
Precise determination of transport coefficients like the specific viscosities η/s and ζ/s of the quark-gluon plasma formed in high-energy nucleus-nucleus collisions hinges upon accurately modeling the stress tensor (T µν ) evolution during the system's early stage.This stage is characterised by far-off-equilibrium dynamics which may be modeled by weakly coupled kinetic theory until O(1) fm/c [1,2].This approach is, however, numerically daunting as solving kinetic theory amounts to tackling a 7-dimensional problem in phase-space.Moreover, if one is only interested in the evolution of macroscopic quantities like T µν , solving for the full kinetic distribution is likely unnecessary.It is thus desirable to have a macroscopic framework which can model the far-off-equilibrium evolution of T µν both physically accurately and numerically efficiently.In this work, we first explore the sensitivity of the T µν evolution in kinetic theory to initial state momentum anisotropies of the plasma.By considering extreme off-equilibrium initial conditions for a quark-gluon gas undergoing Bjorken expansion [3], we point out non-intuitive out-of-equilibrium effects arising in kinetic theory.In the second part we formulate a new macroscopic theory (ME-hydrodynamics) which can be used to describe in a single framework both the far-off-equilibrium pre-hydrodynamic and the near-equilibrium dissipative hydrodynamic regimes of the plasma.

Kinetic theory of a massive quark-gluon gas
For a weakly interacting gas of quarks, anti-quarks, and gluons undergoing boost-invariant Bjorken expansion along the beam axis, we solve the Boltzmann equation in a relaxation-time approximation, Here τ R is the microscopic relaxation time, and the superscript i ∈ {q, q, g} on the kinetic distributions distinguishes between particle species.f i eq are given by Fermi-Dirac (for quarks and anti-quarks) or Bose-Einstein (for gluons) distributions which involve the Landau matched effective temperature and quark chemical potential (T, µ).Symmetries of Bjorken flow imply vanishing net-quark diffusion, i.e. n(τ) ∝ 1/τ and T µν = i p i p µ i p ν i f i = diag(e, P T , P T , P L ), where P T and P L are effective transverse and longitudinal pressures.A physical quantity that is of particular interest is the non-equilibrium entropy density (in the fluid rest frame), obtained from Boltzmann's H-function: where a q, q = −1 and a g = 1.In equilibrium s → s eq = (e + P − µn)/T .In Fig. 1 we show solutions of kinetic theory for two sets of extreme far-off-equilibrium initial conditions (see figure caption) which were set up using a Romatschke-Strickland (RS) distribution [5,6].Although all curves start with the same effective (T, µ B ), the phase trajectories are quite sensitive to the choice of initial momentum space anisotropy.In Bjorken flow, Navier-Stokes hydrodynamics predicts that the ratio s eq /n must increase over time due to viscous heating.While this is indeed the case for panel (a) (see dotted lines for s eq /n evolution in (b)), this expectation is not borne out for the trajectories in panel (c).Here, s eq /n decreases for a certain duration of time.However, this does not imply a violation of the second-law of thermodynamics as the total entropy per baryon which includes non-equilibrium effects never decreases.The feature of decreasing equilibrium entropy per baryon density results in a peculiar phenomena which we call 'non-equilibrium cooling' (see Fig. 2).Here, the effective temperature falls even faster than what is expected for an ideal (inviscid) fluid.

Maximum-entropy truncation of the Boltzmann equation
The Boltzmann equation can be expressed as an infinite hierarchy of equations for momentum moments of f (x, p) [7] where low-order moments corresponding to components of T µν (upper panels), and the unstable fixed point P T /P ≪ 1 (lower panels).
The magenta curves depict a very weakly interacting gas whereas the green curve (τ R → 0) is for a strongly interacting system (perfect fluid).Non-equilibrium cooling: the solid (dashed) blue curve, initialised with non-equilibrium (equilibrium) initial conditions, cools more rapidly (slowly) than a dissipationless system.are coupled to higher-order 'non-hydrodynamic' moments.To obtain a macroscopic description solely in terms of T µν , the infinite hierarchy has to be truncated by expressing the non-hydrodynamic moments in terms of an approximate kinetic distribution using only information contained in T µν .Based on Jaynes's insights on the connections between statistical mechanics and information theory [8], Everett et al. [9] recently proposed a novel way of reconstructing a kinetic distribution from the energy-momentum tensor using the maximum entropy principle.The idea is to find the distribution function f (x, p) that maximizes the non-equilibrium entropy density ( 2), subject to the information (constraint) that f (x, p) reproduces the given 10 components of T µν .For a single component gas the maximum entropy distribution is [9] where Λ µν are Lagrange multipliers corresponding to T µν .Landau matching conditions further simplify the argument of the exponential [10].Unlike the commonly used distributions for Grad or Chapman-Enskog (CE) truncation, f ME is positive definite for all momenta and allows for non-equilibrium matching to conserved currents for a wide range of non-equilibrium stresses.It also ensures that the resulting macroscopic framework, which we call ME-hydro, has a non-negative entropy production rate [11], and that in the limit of small viscous stresses ME-hydro reduces to second-order Chapman-Enskog fluid dynamics [9].

ME-hydro vs. RTA kinetic theory in Bjorken and Gubser flows
The exact evolution equations for the 3 independent components of T µν = diag(e, P T , P T , P L ) in Bjorken flow are given by The terms (ζ T , ζ L ) introduce couplings to 'non-hydrodynamic' moments of f (τ, p T , p z ); for example, ζ L = p E −2 p p 4 z f .To truncate we replace f → f ME where f ME is constructed using the instantaneous values of (e, P T , P L ) [12].In Gubser flow [13] the exact evolution equations  for the two independent (dimensionless) variables (ê, PT ) as functions of de-Sitter 'time' ρ are similarly truncated using f ME [12].Figures 3-4 show that ME-hydro is in excellent agreement with the underlying kinetic theory for both of these profiles even when the system  is far-off-equilibrium. Figure 4a shows that Chapman-Enskog hydrodynamics [14] fail to capture the late-time transverse free-streaming regime of Gubser flow.The only framework that performs slightly better than ME-hydro is anisotropic hydrodynamics [15,16] (shown in panel (b)) which uses the RS ansatz as a truncation distribution.Further (numerical) analysis will determine which of these two frameworks better captures the far-off-equilibrium aspects of kinetic theory in generic flows without the restrictive symmetry constraints of Bjorken and Gubser flows.

Figure 1 .
Figure1.Phase trajectories and corresponding entropy evolution of a quark-gluon gas initialized near the stable fixed point of early-time Bjorken dynamics, i.e., P L /P ≪ 1[4] (upper panels), and the unstable fixed point P T /P ≪ 1 (lower panels).The magenta curves depict a very weakly interacting gas whereas the green curve (τ R → 0) is for a strongly interacting system (perfect fluid).
Figure 2.Non-equilibrium cooling: the solid (dashed) blue curve, initialised with non-equilibrium (equilibrium) initial conditions, cools more rapidly (slowly) than a dissipationless system.
BE dashed: ME-hydro

Figure 4 .
Figure 4. Evolution of the pressure anisotropy in Gubser flow.At early and late times the system approaches longitudinal and transverse freestreaming regimes, respectively.