Anisotropic hydrodynamics with number-conserving kernels

We compare anisotropic hydrodynamics (aHydro) results obtained using the relaxation-time approximation (RTA) and leading-order (LO) scalar $\lambda {\varphi }^4$ collisional kernels. We extend previous work by explicitly enforcing number conservation through the incorporation of a dynamical chemical potential (fugacity) in the underlying aHydro distribution function. We focus on the case of a transversally homogenous and boost-invariant system obeying classical statistics and compare the relevant moments of the two collisional kernels. We then compare the time evolution of the aHydro microscopic parameters and components of the energy-momentum tensor. We also determine the nonequilibrium attractor using both the RTA and LO massless $\lambda {\varphi }^4$ number-conserving kernels. We find that the aHydro dynamics receives quantitatively important corrections when enforcing number conservation; however, the aHydro attractor itself is not modified substantially.

Figure 1

Comparison of $\mathcal{W}$ from the LO scalar and RTA kernels. Panel (a) shows the result for small values of $\xi$ and panel (b) shows the result for large values of $\xi$. The RTA kernel results at $\mu \ne 0$ and $\mu =0$ are indicated by solid red and dashed red lines, respectively. The scalar kernel results at $\mu \ne 0$ and $\mu =0$ are indicated by solid black and dashed black lines, respectively.

Figure 2

The evolution of $\xi$ (a)–(b), the transverse temperature scale $\mathrm{\Lambda }$ in GeV (c)–(d), and the fugacity $\gamma$ (e)–(f). The left column panels (a), (c), and (e) show the case when $\overline{\eta }=0.2$ and the right column panels (b), (d), and (f) show $\overline{\eta }=1$. For this figure we assumed isotropic initial conditions with ${\xi }_0={10}^{-8}$, ${\tau }_0=0.25\mathrm{fm}/c$, ${\mathrm{\Lambda }}_0=0.5\mathrm{GeV}$, and ${\gamma }_0=1$.

Figure 3

Same as Fig. 2 except for this figure we assumed anisotropic initial conditions ${\xi }_0=100$.

Figure 4

Evolution of the effective temperature (a)–(b) and pressure ansiotropy (c)–(d) using both the RTA and scalar collisional kernels with and without number conservation enforced. In the top row, we plot the scaled temperature multiplied by ${(\tau /{\tau }_0)}^{1/3}$ in order to better see the small deviations between the different approaches. The left column panels (a) and (c) show the case $\overline{\eta }=0.2$ and the right column panels (b) and (d) show the case $\overline{\eta }=1$. The initial conditions were taken to be isotropic with the same parameters as in Fig. 2.

Figure 5

Same as Fig. 4 except with anisotropic initial conditions. The initial conditions and parameters are the same as in Fig. 3.

Figure 6

The left panel (a) shows the attractor solution for the amplitude $\phi$ and the right panel (b) shows the associated pressure anisotropy. The four lines show results obtained using the RTA and scalar collisional kernels for both $\mu \ne 0$ and $\mu =0$.

Figure 7

Pressure anisotropy evolution for a variety of different initial conditions (dashed lines) together with the corresponding attractor (solid line). The left panel (a) shows the results obtained using the scalar collisional kernel and the right panel (b) shows the results obtained using the RTA collision kernel. For both panels we show the case $\mu \ne 0$.

Figure 8

Comparison of the pressure anisotropy evolution for a variety of different initial conditions (dashed lines) together with the corresponding attractor (solid lines) for both the RTA and scalar collisional kernels.