Effective kinetic description of event-by-event pre-equilibrium dynamics in high-energy heavy-ion collisions

We develop a macroscopic description of the space-time evolution of the energy-momentum tensor during the pre-equilibrium stage of a high-energy heavy-ion collision. Based on a weak coupling effective kinetic description of the microscopic equilibration process (à la “bottom-up”), we calculate the nonequilibrium evolution of the local background energy-momentum tensor as well as the nonequilibrium linear response to transverse energy and momentum perturbations for realistic boost-invariant initial conditions for heavy-ion collisions. We demonstrate how this framework can be used on an event-by-event basis to propagate the energy-momentum tensor from far-from-equilibrium initial-state models to the time ${\tau }_{\text{hydro}}$ when the system is well described by relativistic viscous hydrodynamics. The subsequent hydrodynamic evolution becomes essentially independent of the hydrodynamic initialization time ${\tau }_{\text{hydro}}$ as long as ${\tau }_{\text{hydro}}$ is chosen in an appropriate range where both kinetic and hydrodynamic descriptions overlap. We find that for $\sqrt{s_{NN}}=2.76\text{TeV}$ central Pb-Pb collisions, the typical timescale when viscous hydrodynamics with shear viscosity over entropy ratio $\eta /s=0.16$ becomes applicable is ${\tau }_{\text{hydro}}\sim 1\mathrm{fm}/\mathrm{c}$ after the collision.

Figure 1

The transverse energy density distribution for boost-invariant IP-Glasma initial conditions at the start of the kinetic theory pre-equilibrium evolution at ${\tau }_{\text{EKT}}=0.2\text{fm}$ and after the linearized kinetic evolution given by Eq. (4) at the hydrodynamic initialization time ${\tau }_{\text{hydro}}=1.2\text{fm}$. The white circle indicates the size of the causal neighbourhood in the transverse plane, Eq. (2).

Figure 2

Different phases of weak coupling equilibration (à la “bottom-up”) for a Bjorken expanding plasma. Different panels show the gluon-momentum distribution function $\overline{f}(\tau ,\mathbf{p})$ in the $(p^x,p^z)$ plane as a function of scaled time $\tau T_{\text{id.}}/(4\pi \eta /s)$: (a) The initial distribution is broadened by elastic $2\leftrightarrow 2$ scattering; (b) inelastic $1\leftrightarrow 2$ splitting and minijet quenching build up a low-momentum thermal bath; (c) the distribution approaches local thermal equilibrium but has significant corrections which are described by viscous hydrodynamics. The initial conditions for $\overline{f}$ are given by Eq. (7) and the evolution is done for the coupling constant $\lambda =10$ (with $\eta /s\approx 0.62$). The color scale represents the distribution function in the range $0.1<\overline{f}<5$.

Figure 3

Equilibration of the different components of the background energy-momentum tensor. After normalizing the vertical axis by the asymptotic values, Eq. (12), and expressing time on the horizontal axis in terms of the scaled time variable, Eq. (16), the entire evolution collapses onto a single curve for a range of coupling constants $\lambda =10-25$ corresponding to $\eta /s\approx 0.62-0.16$.

Figure 4

The universal scaling function, Eq. (17), fit to the kinetic theory evolution; see Fig. 3. The early time behavior is approximately described by free streaming, while the late time behavior matches smoothly onto the second-order hydrodynamic asymptotics, Eq. (14).

Figure 5

The entropy and gluon number density per rapidity as a function of scaled time $\frac{\tau T_{\text{id.}}}{\eta /s}$. The kinetic theory results for the entropy are compared to viscous hydrodynamics (see text).

Figure 6

(a) The universal scaling function ${\tilde{G}}_s^s$ [see Eq. (42)] for the energy response to an initial energy perturbation as a function of the phase $\left|\mathbf{k}\right|(\tau -{\tau }_0)$ at a fixed scaling time, $\frac{\tau T_{\text{id.}}}{\eta /s}$. The different curves correspond to different coupling strengths (or $\eta /s$) which collapse onto a universal curve. The response functions at different scaling times $\tau T_{\text{id.}}/(4\pi \eta /s)=0.5,1.5,2.0$ exhibit an equally good overlap (not shown). (b) The analogous plot for the energy response to an initial momentum perturbation, ${\tilde{G}}_s^v$.

Figure 7

Comparison of the response functions for $\delta T^{ij}$ from an initial energy perturbation [see ${\tilde{G}}_s^{t,\delta }$ and ${\tilde{G}}_s^{t,k}$ in Eq. (29)] with the constitutive relations of first- and second-order hydrodynamics [Eqs. (43) and (44)].

Figure 8

Profile of the energy density $T^{\tau \tau }$ along the $y$ direction for a kinetic theory evolution with and without a regulator, Eq. (56). For points with large gradients, the system response is driven toward free-streaming evolution. Note that only a small fraction of the total number of grid points in the transverse plane are affected; see Table 1. Here IP-Glasma initial conditions are used with pre-equilibrium evolution from ${\tau }_{\text{EKT}}=0.2\text{fm}$ to ${\tau }_{\text{hydro}}=1.0\text{fm}$.

Figure 9

Transverse density of $\tau e^{3/4}\sim s\tau$ for MC-Glauber event used in Sec. 5a at initial time ${\tau }_{\text{EKT}}=0.1\text{fm}$ corresponding to a central PbPb event at center-of-mass energy $\sqrt{s_{NN}}=2.76\text{TeV}$.

Figure 10

(top row) The transverse averages [as defined in Eq. (59)] of (a) $\tau {\epsilon }^{3/4}$ and (b) transverse velocity $v_{\perp }$, and (c) momentum eccentricity [as defined by Eq. (61)] in the hydrodynamic phase as a function of time $\tau$. The different lines correspond to different hydrodynamic initialization time ${\tau }_{\mathrm{hydro}}$, i.e., different duration of kinetic pre-equilibrium evolution. The initial condition of the effective kinetic theory at time ${\tau }_{\text{EKT}}=0.1\text{fm}$ is a central participant MC-Glauber event normalized to correspond to a $\sqrt{s_{NN}}=2.76$ TeV Pb-Pb collisions (see Fig. 9). (bottom row) the same as [(a)–(c)], except a conformal equation of state is used in the hydrodynamics evolution instead of a lattice QCD one (see Fig. 11).

Figure 11

Deviation of the QCD equation of state [62] from conformality, as quantified by the ratio $e/\left(3p\right)$.

Figure 12

Single event profiles along the $x$ axis ($y=0$) of $\tau e^{3/4}$ (top row) and velocity $v^x$ (bottom row) for different hydrodynamics transition times ${\tau }_{\text{hydro}}$. Different columns correspond to three different times in hydrodynamic evolution: $\tau =1.2,2.0$, and 5.0 fm. The same EKT initialization time ${\tau }_{\text{EKT}}=0.1$ fm was used. The equation of state is a realistic QCD one. The transverse velocity is not shown for very low energy densities ($\tau e^{3/4}<0.01{\text{GeV}}^2$) where numerical errors can generate spurious values of velocity.

Figure 13

(a) Comparison of the out-of-equilibrium shear stress tensor [cf. Eq. (62)] with the Navier-Stokes estimate at different hydrodynamics initialization times ${\tau }_{\text{hydro}}=0.8,1.0,1.2$ fm. (b) Scaled evolution time variable $\frac{\tau T_{\text{id.}}}{\eta /s}$ at different hydro starting times. Values of $\tau T_{\text{id.}}/(4\pi \eta /s)>1$ indicate that the system is close enough to local thermal equilibrium for hydrodynamics to become applicable (see Sec. 2d).

Figure 14

The thermal freeze-out pion (a) multiplicity $d{N}_{\pi }/dy$, (b) radial flow $\langle p_T^{\pi }\rangle$, and (c) elliptic flow $v_2^{\pi }$ as a function of hydrodynamic initialization time ${\tau }_{\text{hydro}}=0.4–1.2\text{fm}$, i.e., different duration of pre-equilibrium evolution. [Note the suppressed zero in panels (b) and (c).] The initial MC-Glauber conditions are specified at ${\tau }_{\text{EKT}}=0.1\text{fm}$ as indicated by the gray band. Different pre-equilibrium scenarios are linearized kinetic theory evolution KøMPøST, interactionless free streaming, and simple Bjorken energy rescaling with time $\propto {\tau }^{-4/3}$ with no dynamics but immediate isotropization. The open symbols show the $\langle p_T^{\pi }\rangle$ and $v_2^{\pi }$ for the free streaming and Bjorken pre-equilibrium evolution from 0.1 to 1.2 fm with energy density scaled to reproduce the same pion multiplicity as KøMPøST with ${\tau }_{\text{hydro}}=1.2\text{fm}$. Black triangles show the results for initializing the hydrodynamic simulation directly at $\tau =0.1\text{fm}$ without pre-equilibrium evolution.

Figure 15

Transverse “entropy” density $\tau e^{3/4}\sim s\tau$ for a single $\sqrt{s_{NN}}=2.76\text{TeV}$ central Pb-Pb IP-Glasma event at kinetic theory initialization time ${\tau }_{\text{EKT}}=0.2\text{fm}$.

Figure 16

Time evolution of the longitudinal and transverse stress tensor components, $〈P_L〉$ and $〈P_T〉$, averaged across the transverse plane, relative to the average background energy density of a single IP-Glasma event shown in Fig. 15. (a) $〈P_L〉$ and $〈P_T〉$ in the 2+1D Yang Mills and kinetic theory codes for a range of $\eta /s$ at early times. (b) $〈P_L〉$ and $〈P_T〉$ in the 2+1D Yang Mills, kinetic theory, and viscous hydrodynamics codes with $\eta /s=2/4\pi$ for a wide range of times. After $\sim 2\mathrm{fm}$ the 3D hydrodynamic expansion starts, and the (integrated) estimate $〈P_L〉/〈e〉\simeq \frac{1}{3}$ does not serve as a useful measure of local thermal equilibrium for a flowing fluid.

Figure 17

(Top row) Transverse average of $\tau e^{3/4}$ and of the transverse velocity $v_{\perp }$, as defined in Eq. (59), and momentum eccentricity as defined by Eq. (61), as a function of time $\tau$. The three average fields are plotted for different hydrodynamic initialization time ${\tau }_{\text{hydro}}$, i.e., different duration of kinetic pre-equilibrium evolution. The initial condition of the effective kinetic theory at time ${\tau }_{\text{EKT}}=0.2$ fm is a central IP-Glasma events normalized to correspond to a $\sqrt{s_{NN}}=2.76$ TeV Pb-Pb collisions (see Fig. 15). (Bottom row) the same as panels (a)–(c), except a conformal equation of state is used in the hydrodynamics evolution instead of a lattice QCD one.

Figure 18

Transverse average of $\tau {\epsilon }^{4/3}$ and of the transverse velocity $v_{\perp }$, as defined in Eq. (59), and momentum eccentricity, as defined by Eq. (61), as a function of time $\tau$. The three average fields are plotted for a single hydrodynamic initialization time ${\tau }_{\text{hydro}}=0.8$ fm and two kinetic theory initialization times ${\tau }_{\text{EKT}}=0.1$ fm and ${\tau }_{\text{EKT}}=0.2$ fm for IP-Glasma initial conditions. A realistic QCD equation of state is used in the hydrodynamics evolution.

Figure 19

Transverse profile of $\tau e^{3/4}$ (upper panels) and velocity $v^x$ (lower panels) for different hydrodynamics transition times ${\tau }_{\text{hydro}}$ at $y=0\text{fm}$. The different columns correspond to three different hydrodynamic evolution times, $\tau =1.2,2.0$, and 5.0 fm. The same EKT initialization time ${\tau }_{\text{EKT}}=0.2$ fm was used. The equation of state is a realistic QCD one. The transverse velocity is not shown for very low energy densities ($\tau e^{3/4}<0.01{\text{GeV}}^2$), where numerical errors can generate spurious values of velocity.

Figure 20

(a) Comparison of ${\pi }^{xx}+{\pi }^{yy}$ as defined in Eq. (62) with its Navier-Stokes counterpart at hydro initialization times ${\tau }_{\text{hydro}}=0.8,1.0,1.2\text{fm}$ with IP-Glasma initial conditions (curves offset for clarity). (b) Scaled time variable for different locations in the transverse plane and at different crossover times ${\tau }_{\text{hydro}}$. Values of $\tau T_{\text{id.}}/(4\pi \eta /s)>1$ indicate that the system is close enough to local thermal equilibrium for hydrodynamics to become applicable (see Sec. 2d).

Figure 21

The thermal freeze-out pion (a) multiplicity $d{N}_{\pi }/dy$, (b) radial flow $\langle p_T^{\pi }\rangle$, and (c) elliptic flow $v_2^{\pi }$ as a function of hydrodynamic initialization time ${\tau }_{\text{hydro}}=0.4–1.2\text{fm}$, i.e., different duration of pre-equilibrium evolution. The initial IP-Glasma conditions are specified at ${\tau }_{\text{EKT}}=0.2\text{fm}$. The different pre-equilibrium scenarios are linearized kinetic theory (i.e., KøMPøST) and free streaming. The open symbols show the $\langle p_T^{\pi }\rangle$ and $v_2^{\pi }$ for the free-streaming pre-equilibrium evolution from 0.2 to 1.2 fm with energy density scaled to reproduce the same pion multiplicity as KøMPøST with ${\tau }_{\text{hydro}}=1.2\text{fm}$. Black triangles show the results for initializing the hydrodynamic simulation directly at $\tau =0.2\text{fm}$ without pre-equilibrium evolution.

Figure 22

Linear and quadratic long wavelength response coefficients to initial energy perturbations (a) and initial momentum perturbations (b) from nonequilibrium kinetic theory evolution. Dashed lines show comparison to viscous hydrodynamic asymptotic derived in Appendix pp4-s2.

Figure 23

Comparison of hydrodynamic fields obtained in the long-wavelength (top) and free-streaming (bottom) limits, with the results of full kinetic theory pre-equilibrium (KøMPøST). Different columns show profiles of energy density $e$ (left), flow velocity $v^x$ (center), and shear stress $({\pi }^{xx}+{\pi }^{yy})$ (right).

Figure 24

Evolution of pressure anisotropy, Eq. (A1), in kinetic theory with initial distribution function Eq. (7), and fitted transport coefficients $\eta /s$ and $C_2$ (see text).

Figure 25

The universal scaling function fit, Eq. (A8), to the evolution of background energy density on (a) linear and (b) log scales.

Figure 26

Independent tensor components of kinetic theory coordinate space response functions to initial-energy (scalar) perturbations, Eq. (B1). Different lines correspond to response functions at different scaled times $\frac{\tau T_{\text{id.}}}{\eta /s}$, Eq. (16).

Figure 27

Independent tensor components of kinetic theory coordinate space response functions to initial momentum (vector) perturbations, Eq. (B3). Different lines correspond to response functions at different scaled times $\frac{\tau T_{\text{id.}}}{\eta /s}$, Eq. (16).