Study of early time attractor with longitudinal forces with Bjorken symmetry

We study the effect of external proper time-dependent longitudinal forces on the evolution of the distribution function using the Boltzmann equation with a relaxation time collision kernel under Bjorken flow. We derive an exact solution and study the early time attractor behavior of different components of energy-momentum tensor under Bjorken symmetry. We show that the arbitrary initial data approaches the attractor solution but is influenced by the external force with slightly broken Bjorken symmetry.

Figure 1

Normalized force ($\max (F)=1$) for different timescales. ${\tau }_F=1\mathrm{fm}/\mathrm{c}$ (dot), ${\tau }_F=2.5\mathrm{fm}/\mathrm{c}$ (dashed), ${\tau }_F=5\mathrm{fm}/\mathrm{c}$ (dot-dashed).

Figure 2

Evolution of $T$, ${\overline{P}}_L$, ${\overline{P}}_T$, and ${\overline{T}}^{03}$ for $\alpha =[200,100,50,0]$, ${\tau }_m=1$, and for ${\overline{\mathrm{\Pi }}}_0=0$, ${\overline{\pi }}_0=-1$.

Figure 3

Evolution of $T$, ${\overline{P}}_L$, ${\overline{P}}_T$, and ${\overline{T}}^{03}$ for $\alpha =[200,100,50,0]$, ${\tau }_m=1$, and for ${\overline{\mathrm{\Pi }}}_0=0$, ${\overline{\pi }}_0=0.99$.

Figure 4

Evolution of $T$, ${\overline{P}}_L$, ${\overline{P}}_T$, and ${\overline{T}}^{03}$ for $\alpha =[200,100,50,0]$, ${\tau }_m=1$ and for ${\overline{\mathrm{\Pi }}}_0=0.85,{\overline{\pi }}_0=0$.

Figure 5

Evolution of $T$, ${\overline{P}}_L$, ${\overline{P}}_T$, and ${\overline{T}}^{03}$ for $\alpha =[200,100,50,0]$, ${\tau }_m=1$ and for ${\overline{\mathrm{\Pi }}}_0=0.25,{\overline{\pi }}_0=0$.

Figure 6

Evolution of $T$,${\overline{P}}_L$,${\overline{P}}_T$ and ${\overline{T}}^{03}$ for $\alpha =[10,5,2,0]$,${\tau }_m=1$ and for ${\overline{\mathrm{\Pi }}}_0=-0.37,{\overline{\pi }}_0=-1$.

Figure 7

Evolution of ${\overline{P}}_L$ for $\alpha =[0,50,100]$,${\tau }_m=1$ and for various initial conditions.

Figure 8

Scaled transverse and longitudinal pressure plotted for varying initial conditions (given in Table 1) in the presence of an external force $\alpha =100$ with $m=0$.