Exploring the applicability of dissipative fluid dynamics to small systems by comparison to the Boltzmann equation

Background: Experimental data from heavy-ion experiments at RHIC-BNL and LHC-CERN are quantitatively described using relativistic fluid dynamics. Even $p+A$ and $p+p$ collisions show signs of collective behavior describable in the same manner. Nevertheless, small system sizes and large gradients strain the limits of applicability of fluid-dynamical methods.

Purpose: The range of applicability of fluid dynamics for the description of the collective behavior, and in particular of the elliptic flow, of small systems needs to be explored.

Method: Results of relativistic fluid-dynamical simulations are compared with solutions of the Boltzmann equation in a longitudinally boost-invariant picture. As the initial condition, several different transverse energy-density profiles for equilibrated matter are investigated.

Results: While there is overall a fair agreement of energy- and particle-density profiles, components of the shear-stress tensor are more sensitive to details of the implementation. The highest sensitivity is exhibited by quantities influenced by properties of the medium at freeze-out.

Conclusions: For some quantities, like the shear-stress tensor, agreement between fluid dynamics and transport theory extends into regions of Knudsen numbers and inverse Reynolds numbers where relativistic fluid dynamics is believed to fail.

Figure 1

Comparing the semianalytic “Gubser” solution of the Israel-Stewart equations [51] to the numerical solution using SHASTA with two different numerical resolutions as indicated in the plots. Panel (a) shows the ${\pi }^{zz}$ profile and panel (b) shows the ${\pi }^{xx}$ profile at different times $\tau =1.2,1.5,2.0\text{fm}$. The curves with the larger maximum absolute value of ${\pi }^{\mu \nu }$ correspond to earlier times.

Figure 2

Initial energy-density profiles.

Figure 3

Shear viscosity-to-entropy density ratio as a function of temperature for a constant isotropic cross section.

Figure 4

Knudsen-number contours for a Gaussian $w=3\text{fm}$ (top) and $w=1\text{fm}$ (bottom) profile, with $\sigma =20,5$, and $1\text{mb}$ (from left to right). The contour line shows where the test-particle number per computational cell in the BAMPS calculation assumes the value 4.

Figure 5

Energy-density (left) and $v_x$ (right) profiles for $\sigma =1\text{mb}$ and a $w=3\text{fm}$ Gaussian initial density profile for different times $t=1,2$, and 4 fm. Solid lines represent fluid-dynamical results, while dashed lines show the BAMPS solutions.

Figure 6

The ${\pi }^{xx}$ profiles scaled by energy density $e$ for $\sigma =20\mathrm{mb},5\mathrm{mb}$, and $1\mathrm{mb}$ (from left to right) and a $w=3\mathrm{fm}$ Gaussian initial density profile.

Figure 7

The ratio of longitudinal over transverse pressure $P_L/P_T$ for $\sigma =20\mathrm{mb},5\mathrm{mb}$, and $1\mathrm{mb}$ (from left to right) and a $w=3\mathrm{fm}$ Gaussian initial density profile.

Figure 8

Energy-density (left) and $v_x$ (right) profiles for $\sigma =1\text{mb}$ and a $w=1\text{fm}$ Gaussian initial density profile for different times $t=0.5,1$, and 2 fm. Solid lines represent the fluid-dynamical results, while dashed lines show the BAMPS solutions, as in Fig. 5.

Figure 9

The ${\pi }^{xx}$ profiles scaled by energy density for $\sigma =20,5$, and $1\text{mb}$ (from left to right) and a $w=1\text{fm}$ Gaussian initial density profile, as in Fig. 6.

Figure 10

The ratio of longitudinal over transversal pressure $P_L/P_T$ for $\sigma =20,5$, and $1\text{mb}$ (from left to right) and a $w=1\text{fm}$ Gaussian initial density profile, as in Fig. 7.

Figure 11

Knudsen-number contours for an nBC Glauber initial profile for $\sigma =20$, 10, and $5\text{mb}$ (from left to right). The white line corresponds to the test-particle number per computational cell $N_{\mathrm{test}}=4$ contour in the BAMPS calculation.

Figure 12

Energy-density (left) and $v_x$ (right) profiles for $\sigma =5\text{mb}$ and an nBC Glauber initial density profile for times $t=1,2,$ and 4 fm. Solid lines represent fluid-dynamical results, while dashed lines show the BAMPS solutions.

Figure 13

The ${\pi }^{xx}$ profiles scaled by energy density $e$ for $\sigma =20$, 10, and $5\text{mb}$ and an nBC Glauber initial density profile.

Figure 14

The ratio of longitudinal over transverse pressure $P_L/P_T$ for $\sigma =20$, 10, and $5\text{mb}$ and an nBC Glauber initial density profile.

Figure 15

Momentum-space eccentricity as function of time, for $\sigma =20$, 10, and $5\text{mb}$ and nBC initial conditions.

Figure 16

Transverse-momentum spectra (top panel) for a Glauber initial profile (“nBC”) and $\sigma =20$, 10, and $5\text{mb}$. The freeze-out criteria ${\mathrm{Kn}}_{\mathrm{fr}}=2.5$ (left) and 4 (right) are compared with each other. The corresponding $v_2$ results are shown in the bottom panels. Solid lines represent fluid-dynamics results, while the dashed lines show BAMPS solutions.

Figure 17

The $v_2$ values as function of $k_T$ for different cross sections. Solid lines show the results for particles which have not undergone any interaction, while the dashed lines shows the values for all other particles. The order of the curves from top to bottom is the same as in the legend.

Figure 18

$k_T$-integrated elliptic flow $v_2$ for nBC initial conditions as a function of the cross section varying from $\sigma =20$ to $1\text{mb}$. The fluid-dynamical results are calculated with different decoupling conditions: constant Knudsen number (left), constant mean free path (middle), and constant temperature (right). The corresponding constants are shown in the legend. Solid lines represent fluid-dynamical results, while the dashed lines show BAMPS solutions.

Figure 19

Average inverse Reynolds number on the freeze-out surface, average initial Knudsen number, relative magnitude of $\delta f_{\mathbf{k}}$ in $v_2$, and relative difference of $v_2$ from fluid-dynamical and BAMPS calculations.